Geometry: A Complete Course (with Trigonometry)

Save this PDF as:
Size: px
Start display at page:

Download "Geometry: A Complete Course (with Trigonometry)"

Transcription

1 Geometry: omplete ourse with Trigonometry) Module - Student WorkTet Written by: Thomas. lark Larry. ollins

2 Geometry: omplete ourse with Trigonometry) Module Student Worktet opyright 014 by VideotetInteractive Send all inquiries to: VideotetInteractive P.O. o Indianapolis, IN 4619 ll rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior permission of the publisher. Printed in the United States of merica. ISN RPInc

3 Table of ontents Unit V - Other Polygons Part - Properties of Polygons LSSON 1 - asic Terms LSSON - Parallelograms LSSON 3 - Special Parallelograms Rectangle, Rhombus, Square) LSSON 4 - Trapezoids LSSON 5 - Kites LSSON 6 - Midsegments LSSON 7 - General Polygons Part - reas of Polygons LSSON 1 - Postulate 14 - rea LSSON - Triangles LSSON 3 - Parallelograms LSSON 4 - Trapezoids LSSON 5 - Regular Polygons Part - pplications LSSON 1 - Using reas in Proofs LSSON - Schedules Unit VI - ircles Part - Fundamental Terms LSSON 1 - Lines and Segments LSSON - rcs and ngles LSSON 3 - ircle Relationships Part - ngle and rc Relationships LSSON 1 - Theorem 65 - If, in the same circle, or in congruent circles, two central angles are congruent, then their intercepted minor arcs are congruent. Theorem 66 - If, in the same circle, or in congruent circles,two minor arcs are congruent, then the central angles which intercept those minor arcs are congruent. LSSON - Theorem 67 - If you have an inscribed angle of a circle, then the measure of that angle, is one-half the measure of its intercepted arc. LSSON 3 - Theorem 68 - If, in a circle, you have an angle formed by a secant ray, and a tangent ray, both drawn from a point on the circle,then the measure of that angle, is one-half the measure of the intercepted arc. Module - Table of ontents i

4 LSSON 4 - Theorem 69 - If, for a circle, two secant lines intersect inside the circle, then the measure of an angle formed by the two secant lines,or its vertical angle), is equal to one-half the sum of the measures of the arcs intercepted by the angle, and its vertical angle. Theorem 70 - If, for a circle, two secant lines intersect outside the circle, then the measure of an angle formed by the two secant lines, or its vertical angle), is equal to one-half the difference of the measures of the arcs intercepted by the angle. LSSON 5 - Theorem 71 - If, for a circle, a secant line and a tangent line intersect...77 outside a circle, then the measure of the angle formed, is equal to one-half the difference of the measures of the arcs intercepted by the angle. Theorem 7 - If, for a circle, two tangent lines intersect outside the circle, then the measure of the angle formed, is equal to one-half the difference of the measures of the arcs intercepted by the angle. Part - Line and Segment Relationships LSSON 1 - Theorem 73 - If a diameter of a circle is perpendicular to a chord of that circle, then that diameter bisects that chord. LSSON - Theorem 74 - If a diameter of a circle bisects a chord of the circle which is not a diameter of the circle, then that diameter is perpendicular to that chord. Theorem 75 - If a chord of a circle is a perpendicular bisector of another chord of that circle, then the original chord must be a diameter of the circle. LSSON 3 - Theorem 76 - If two chords intersect within a circle, then the product of the lengths of the segments of one chord, is equal to the product of the lengths of the segments of the other chord. LSSON 4 - Theorem 77 - If two secant segments are drawn to a circle from a single point outside the circle, the product of the lengths of one secant segment and its eternal segment, is equal to the product of the lengths of the other secant segment and its eternal segment. Theorem 78 - If a secant segment and a tangent segment are drawn to a circle, from a single point outside the circle, then the length of that tangent segment is the mean proportional between the length of the secant segment, and the length of its eternal segment. LSSON 5 - Theorem 79 - If a line is perpendicular to a diameter of a circle at one....9 of its endpoints, then the line must be tangent to the circle,at that endpoint, LSSON 6 - Theorem 80 - If two tangent segments are drawn to a circle from the same point outside the circle, then those tangent segments are congruent. LSSON 7 - Theorem 81 - If two chords of a circle are congruent, then their intercepted minor arcs are congruent. Theorem 8 - If two minor arcs of a circle are congruent, then the chords which intercept them are congruent. Part - ircles and oncurrency LSSON 1 - Theorem 83 - If you have a triangle, then that triangle is cyclic LSSON - Theorem 84 - If the opposite angles of a quadrilateral are supplementary, then the quadrilateral is cyclic. ii Module - Table of ontents

5 For ercises 11 and 1, use the diagram to the right. 11. State the theorem that justifies each of the following conclusions. a) b) c) d) e) and are supplementary f) omplete each of the following statements. a) If = 18, = b) If = 105, then m = c) If = 4, then = d) If = 15, then = e) If m = 65, then m = f) If m = 30, then m 3 = g) If = 8 and = 6, then = and = h) If m = m ), then m = i) If m = 130, and m 1 = 35, then m = j) If = 3 5 and = + 9, then = k) If m = and m = 6 1, then m = 1 In ercises 13 18, determine if each statement is true or false. Give a reason for your answer, drawing a sketch if necessary. Refer to the figure to the right for ercise If RSTU is a parallelogram and UT, then TU is also a parallelogram. R U S T Refer to the figure to the right for ercises 14 and If F is a parallelogram, and is a parallelogram, then F is a parallelogram. 15. If is not in the same plane as and F, and if F and are parallelograms, then F is a parallelogram. F 16. If a diagonal of a quadrilateral is drawn so that two congruent triangles are formed, then the quadrilateral is a parallelogram. 17. If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. 18. If two sides of a quadrilateral are congruent and the other two sides are parallel, then the quadrilateral is a parallelogram. 464 Unit V Other Polygons

6 Unit V Other Polygons Part Properties of Polygons Lesson 3 Special Parallelograms Rectangle, Rhombus, Square) Objective: To understand, and be able to prove, the relationships between special parallelograms, and to see the relationships ehibited by their parts. Important Terms: uadrilateral polygon made with four line segments. Further, the segments are called the sides of the quadrilateral, and the endpoints of the segments are called the vertices of the quadrilateral. Rectangle parallelogram in which there are four right angles. the standard symbol for a rectangle is ) Rhombus quadrilateral in which all four sides are congruent. Square quadrilateral in which all four sides are congruent. iagonal of a Polygon line segment joining any two non-consecutive vertices of a polygon. uadrilateral Hierarchy Theorem If a polygon is one of the seven types of quadrilaterals, then it is related to all other quadrilaterals, using the diagram below. Theorem 48 If a quadrilateral is a rectangle, then it is a parallelogram. Theorem 49 If a quadrilateral is a rectangle, then its diagonals are congruent. Theorem 50 If a quadrilateral is a rhombus, then it is a parallelogram Theorem 51 If a quadrilateral is a rhombus, then its diagonals are perpendicular. Theorem 5 If a quadrilateral is a rhombus, then the diagonals bisect the interior angles of the rhombus. 466 Unit V Other Polygons

7 For ercises 0, use the diagram below. G U R 0. ssume GUR is a rectangle. a) If G is 6, find U. b) If is 9, find RU. c) If m RG = 40, find m GR. d) If m UG = 65, find m UG. 1. ssume GUR is a rhombus. a) If m GR = 53, find m GR. b) If m GU = 86, find m GUR.. ssume GUR is a square. a) If G = and U = 4 10, find G. b) Find m UR. 3. If quadrilateral is a rectangle, with coordinates,5),,1), 7,1), and 7,5), then find the lengths of the diagonals and verify that they are congruent. 4. In quadrilateral, determine the coordinates of point, so that is a square, given that is 3, 1), is 1,3), is 5, 1), and is,y). 5. uadrilateral is a parallelogram, with = + 4, = 3 11, and = Show that is a rhombus. 6. Given: is a parallelogram; m 3 = m 4 Prove: is a rhombus Given: is a rectangle, is a parallelogram Prove: is isosceles 470 Unit V Other Polygons

8 ample 3: F. The area of F is 17 square units. Find the area of and state the assumptions) of Postulate 14 which apply. F Solution: Since the two polygons are congruent and the area of F is 17 square units, the area of is 17 square units. Postulate 14 - Second ssumption - ongruent polygons have the same area. ample 4: raw as many rectangles as you can that are separated into 1 one-centimeter squares. Name the dimensions for each rectangle. Solution: Unit V Other Polygons

9 4. Find the area of each given rectangle. a) b) c) o 4 60 o 5. In the figure at the right, quadrilateral GHIJ is divided into triangles by diagonals GI and HJ which intersect at point M. If you know that the area of GHI = 7, the area of GHJ = 9, and the area of JMI = 8, then will the area of quadrilateral GHIJ = ? Why or why not? G J M H I 6. The area of PST is 14, the area of RU is 15, and the area of PU is 6. TUP and PU are congruent. What is the area of polygon PRS? State the assumptions) of Postulate 14 which justify your answer. P S T U R For ercises 7 1, find the area of each figure. ll adjacent sides are perpendicular Part reas of Polygons 499

10 7. How is the area of related to the areas of two right riangles in the figure below? If = 7 inches, = 15 inches, and = 5 inches, what is the area of? What parts) of Postulate 14 are involved in the process of finding the area of? M 8. Find the area of MN 9. Find the area of RST M 3 U R.1 10 N 10 P 4 T S 10. Using the figure shown at the right, find the area of: a) MRN b) MN c) MT d) MR M y N R T 11. Find the area of 1. Find the area of RP O 15 R 60 O 10 P 13. Find the area of F 14. Find the area of MNT 45 M 9 O F O 1 N 14 T 504 Unit V Other Polygons

11 Lesson 3 ercises: 1. State Theorem 59 and sketch two figures, one using one side as the base and another using an adjacent side as the base, to illustrate the conditions of the theorem.. Find the area of the parallelogram in each part shown below a) List all the measures in the figure you did not need, to find the area of parallelogram b) List all the measures in the figure you did not need, to find the area of parallelogram RSTU. V 30 S 4 T F 0 6 U R 3. The base of a parallelogram is 18 units long and the area is 54 square units. a) Find the measure of the height. b) How many of the other three sides of the parallelogram can you determine? 4. In a parallelogram, the ratio of the altitude to its base is 4 to 5. If its area is 16 square units, what are the dimensions of the parallelogram? In ercises 5 8, find the area of the given parallelogram o o o 1 Part reas of Polygons 507

12 Lesson 5 ercises: In eercises 1 6, find the area of each regular polygon. Note: the perpendicular segment shown is drawn to the center of the polygon In eercises 7 9, find the measure of 1 and in each regular polygon. Note: the perpendicular segment is drawn to the center of the polygon Refer to the equilateral triangle at the right for parts a c. The perpendicular segment is drawn to the center of the polygon. a) If r = 1, find s, P, a, and. b) If s =, find r, P, a, and. c) If P =, find s, r, a, and. r S a 11. Refer to the square at the right for parts a c. The perpendicular segment is drawn to the center of the polygon. a) If s = 16, find r, P, a, and. b) If r =, find s, P, a, and. c) If a =, find s, r, P, and. r a S Part reas of Polygons 513

13 1. Refer to the heagon at the right for parts a c. The perpendicular segment is drawn to the center of the polygon. a) If r = 10, find s, P, a, and. b) If a =, find r, P, s, and. c) If s = 6, find r, P, a, and. r S a In ercises 13 15, find the area of the shaded region. ll polygons shown are regular polygons Unit V Other Polygons Part pplications Lesson 1 Using reas in Proofs Objective: To understand how the concept of the area of a polygon can be used to demonstrate theorems. Important Terms: Theorem 6 If you have a median of a triangle, then that median separates the points inside the triangle into two polygonal regions with the same area. Theorem 63 If you have a rhombus, then the area enclosed by that rhombus, is equal to one-half the product of the measures of the diagonals of the rhombus. Theorem 64 If you have two similar polygons, then the ratio of the areas of the two polygons is equal to the square of the ratio of any pair of corresponding sides. 514 Unit V Other Polygons

14 Lesson 1 ercises: 1. State Theorem 6. Then sketch and label a triangle to show the conditions of the theorem.. State Theorem 63. Then sketch and label a rhombus to show the conditions of the theorem. 3. State Theorem 64. Then sketch and label two triangles which demonstrate the conditions of the theorem. 4. The lengths of the sides of two squares are 4 and 8. Find the ratio of the areas of the two squares. 5. Two regular pentagons have side lengths in the ratio of 13 to 0. Find the ratio of the areas of the two pentagons. 6. If = 16 and h = 5, find the area of and the area of M. Note: M is a median of. h M W X 7. Find the area of rhombus WXYZ Z Y R 6 S 8. Find the area of rhombus RSTU T 6 U 9. FInd the area of a rhombus with a 10 O interior angle and sides of length Find the area of a rhombus with a perimeter of 68 and one diagonal of length Unit V Other Polygons

15 13. Use the given O to find each measure, if mw = , mwz =, mzy = + 10, and my = 4. a) m WOZ b) myz c) m WO d) m YOZ W Z O Y 14. Use the given O to name a major arc for each minor arc named below. a) b) c) d) O W 15. Given: WZ is a diameter of mwx = mxy = n 3 1 X Prove: m Z = n Z 4 5 Y Unit VI ircles Part Fundamental Terms Lesson 3 ircle Relationships Objective: To recognize and understand the relationships between circles according to their relative positions, and to see how they may be related to other polygons. Important Terms: oncentric ircles From two Latin words meaning, with the same center, concentric circles share the same center point. Formally, in our Geometry, two circles are concentric, if and only if, they lie in the same plane, and share the same center. See the illustration below. Part Fundamental Terms 57

16 ample : M N R S 1 U T Theorem 66 states that, if M N and, then. It further states that in, if RS UT, then. Solution: M N; 1. ample 3: Solution: In the given pair of concentric circles, find mxy and muv. oes mxy = muv? re the arcs congruent? plain. mxy = muv = 37 O XY UV U X O 37 Y V If you walked from point U to point V, you would walk a longer distance than if you walked from point X to point Y. The arcs UV and XY have different lengths. rc length is not the same as arc measure. rc length indicates a distance. rc measure indicates an amount of turn. The length of UV or XY) is a distance that is measured in units such as inches or feet, but muv or mxy) is measured in degrees. The units of measure are different. Lesson 1 ercises: 1. Prove Theorem 65 - In a circle or in congruent circles, if two central angles are congruent, then their corresponding intercepted arcs are congruent. Note: This is the same theorem we proved in the lesson. We are using it as an eercise to make sure you understand its proof. Use your ourse Notes to check.) a) State the theorem. b) raw and label a diagram to accurately show the conditions of the theorem. c) List the given information. d) Write the statement we wish to prove. e) emonstrate the direct proof using the two column format.. Using the outline given in ercise 1, prove Theorem Unit VI ircles

17 Inscribed ngle of a ircle n angle formed by any two chords with a common endpoint. Formally, in our Geometry, an angle is an inscribed angle of a circle, if and only if, the verte of the angle is on the circle, and the sides are chords of the circle. Tangent Line of a ircle From the Latin word, tangere, meaning, to touch, a tangent line of a circle is a line which intersects the circle in eactly one point, called the point of tangency, or the point of contact. Note: We sometimes refer to a line segment as a tangent segment, if that segment is contained in a tangent line and intersects the circle in such a way that the point of tangency is one of its endpoints. Intercepted rc of a ircle n angle of a cirlce intercepts an arc of a circle, if and only if, each of the following conditions hold: 1) The endpoints of the arc lie on the sides of the angles. ) ach side of the angle contains one endpoint of the arc. 3) ll points on the arc, ecept the endpoints, lie in the interior of the angle. Measure of an rc of a ircle ased on its relationship with the central angles of a circle, this is defined as the measure of the central angle which intercepts the arc. Measure of a Semicircle ecause we can consider a semicircle to be the intercepted arc of a central angle of 180, its rays are radii of the same diameter), we say that the measure of a semicircle is 180. orollary 68a If, in a circle, a diameter is drawn to a tangent line, at the point of tangency, then that diameter is perpendicular to the tangent line, at that point. ample 1: In the figure at the right, is a tangent line to circle at point. If is a secant line intersecting at point, and m is 84 O, find m. Solution: 1 m = m Theorem 68) 1 84 = m 84= 1 m 168 = m 54 Unit VI ircles

18 ample : In the figure at the right, is a tangent line to at point. is a diameter of circle. Find m, m, and m if the radius is 6 units and = 1. Solution: is a right triangle since = 1 /. Therefore, m = 60, m = 30, and m = 90. ample 3: In the figure at the right, is a tangent line to at point. If is a secant line intersecting at point and mx = 6, find m. X Solution: m 1 = mx 1 = = = 113 ample 4: JK is tangent to P and. Find JK. M J K P 8 3 Solution: Using Postulate For any two points, there is eactly one line containing them.), we can draw PJ and K. Using Postulate 9 In a plane, through a point not on a given line, there is eactly one line parallel to the given line.), we can draw M parallel to JK. y orollary 68a, JP JK and K JK. So JKM is a rectangle and MP is a right triangle. Use the pythagorean theorem to find M. M JK. MP) + M) = P) JP JM) + M) = P) 8 3 ) + M) = ) 5) + M) = 13) 5 + M) = 169 M) = 144 M = 1 Therefore, JK = 1 Part ngle and rc Relationships 543

19 Lesson 3 ercises: 1. In shown at the right, is a tangent line to at point. Name the angles that are formed by a tangent line and a secant, or by a tangent line and a chord, intersecting on the circle.. Using in eercise 1, name the arcs intercepted by each angle named in eercise In shown at the right, UV is a tangent line to at point Y. WY and XY are chords of such that mwy = 8 and mxw = 96. Find m VYW, m VYX, m WYX, and m UYX. 4. In the figure to the right, chord intersects tangent line at point, m = 10, and m = 00. Use this information to find the following: a) m b) m c) m d) m F e) m f) m U H G 00 Z F Y F X W V In the figure to the right, F is a diameter of, is a tangent to at point, m = 80, and m = 70. Use this information to find the following: a) m b) m F c) m F d) m F e) mf f) m F g) m F 6. Use the figure to the right for the following: Given: RT is a common internal tangent to P and P R U F Prove: T 544 Unit VI ircles

20 7. In the figure to the right, SR is tangent to P and at points S and T respectively. lso, T = 6, TR = 8, and PR = 30. Find P, PS, and ST. Hint: Use the Pythagorean Theorem) P S T R 8. T S P In the figure above, PT is tangent to at point T and TS omplete the following statements. P. a) TS is the geometric mean between and. b) T is the geometric mean between and. c) If S = 6 and SP = 4, TS = and TP =. R 9. In the figure to the right, RS is a tangent segment to, and R is a radius of. If RS = 17 and ST = 7, find the length of the radius of. T S Use the figure to the right for eercises 10 and Given: at center. Prove: m = 90 P F 11. Given: is tangent to P and at point Prove: m = 1 / m 1. In shown to the right, PT is a tangent line at point S. m RST = 45 and m USP = 85. Find msu, msr, and the measure of each angle of URS. U P S R T Part ngle and rc Relationships 545

21 Lesson 5 ercises: 1. Prove Theorem 71. Prove Theorem 7 In eercises 3 through 1, find the value of for each indicated angle or arc of y Given: m = 180 m = 90 + a y Note: for this eercse, find the values of and y a 80 Part ngle and rc Relationships 553

22 Use the figure to the right, and the given information for eercises 13 through 5. is tangent to at point. is a secant of intersecting the circle at points and. m = 18, m = 9 Note: m > m) 13. Find m 14. Find m F G 15. Find m F 16. Find m 17. Find m 18. Find m 19. Find m 0. G and are. 1. Find m. Find m 3. Find m 4. Find m 5. Find m 554 Unit VI ircles

23 14. Find the value of in. 15. Find the value of on Find the measure of XY. 17. Find the measure of M. X Y Given: X = 7 = 4 XY M Given: = 4 = Find the measure of Y. 19. Find the measure of VW. X Y Given: = 9 XY = 18 XY V W X Given: muxw = 88 XV UW U 0. Find the measure of X, Y, and F. 1. Find the measure of and. X Y F Given: F = 16 X = 7 m = 58 F H I G J Given: = 18 I = 1 J = 10 mg = Unit VI ircles

24 8. Prove: If a diameter of a circle bisects one of two parallel chords which are not diameters), it bisects the other chord. 9. Given: is a diameter of.. Prove: 10. How many chords may be drawn from a point on a circle? How many diameters? 11. What is the greatest chord in a circle? n arc has how many chords? chord has how many arcs? 1. If a chord were etended at either or both ends, what would it become? 13. How long is the chord which is perpendicular to a tangent to a circle having a radius with a measure of 9 inches? Justify your answer. 14. Through what point does the perpendicular bisector of a chord pass? 15. Prove: If a diameter of a circle bisects each of two chords which are not diameters, the chords are parallel to each other. 16. Write eercises 8 and 15 as a biconditional. 17. Prove Theorem 75 using the two column format. Given: hord bisects chord at point. at point Prove: is a diameter of. 18. In eercise 1, eercise 8, and eercise 15, all contain the phrase which are not diameters. plain why it is necessary to include this phrase in the statement of these three conditionals. raw a diagram. 56 Unit VI ircles

25 Unit VI ircles Part Line and Segment Relationships Lesson 3 Theorem 76 - If two chords intersect within a circle, then the product of the lengths of the segments of one chord, is equal to the product of the lengths of the segments of the other chord. Objective: To investigate the relationship between ordinary chords as they intersect within a circle, and to prove this theorem directly, using previously accepted definitions, postulates and theorems. Important Terms: hord of a ircle line segment whose endpoints are two points on a circle. Means-tremes Property of a Proportion property of a valid, standard proportion, which states that the product of the means is equal to the product of the etremes. ample 1: Solution: The segments of one of two intersecting chords of a circle are 4 inches and 8 inches respectively. One segment of the other chord is 6 inches. Find the measure of the second chord. X X = X X 8 4= 6 X 3 = 6 X 3 = X = 3 3 = X The measure of the second chord is given by the epression X + X. = X + X 8 6 X 4 = = = or 11 1 inches 3 3 Part Line and Segment Relationships 563

26 Lesson 4 ercises: 1. Prove Theorem 77 - If two secant segments are drawn to a circle from a single point outside the circle, the product of the lengths of one secant segment and its eternal segment, is equal to the product of the lengths of the other secant segment and its eternal segment. Note: This is the same theorem we proved in the lesson. We are using it as an eercise to make sure you understand its proof. Use your ourse Notes to check.) a) State the theorem. b) raw and label a diagram to accurately show the conditions of the theorem. c) List the given information. d) Write the statement we wish to prove. e) emonstrate the direct proof using the two column format.. Prove Theorem 78 - If a secant segment and a tangent segment are drawn to a circle, from a single point outside the circle, then the length of that tangent segment is the mean proportional between the length of the secant segment, and the length of its eternal segment. Use the outline given in ercise 1. In eercises 3 through 14, find the value of Unit VI ircles

27 Lesson 5 ercises: 1. Prove Theorem 79 using the two column format.. Prove Theorem 79 using the indirect proof method. Use the figure to answer questions 3 through 15. P ; P ; P G; ; ; G G = 9; P = 6; FI = ; FH = 3 J P 6 I F 3 H G 9 O 48 K 3. m =. 4. FG is a segment. State the Theorem which justifies this statement. 5. P =. 6. m GF =. 7. FG =. 8. JK =. 9. m FP =. 10. F is a segment. State the Theorem which justifies this statement. 11. F =. 1. =. 13. m PF =. 14. PF =. 15. is a tangent segment to circle and circle. State the Theorem which justifies this statement. 16. Theorem 79 is the converse of orollary 68a. State the relationship as a biconditional. Part Line and Segment Relationships 571

28 1. m = 150 Find m. 13. X UV WX VW Find mvw. 150 U X V W 14. V = 9 F = 9 F = 7 V UW F G Find UW 7 9 F U 9 G V W Use the following information in eercises 15 through 17. X 15. Given: P GH G F P Y H Given: is a diameter of F is a diameter of P. F GH Prove: GH 16. Given: P X PY Prove: GH 17. Given: X PY P Prove: GH 578 Unit VI ircles

29 Unit VI ircles Part ircles and oncurrency Lesson 1 Theorem 83 - If you have a triangle, then that triangle is cyclic. Objective: To investigate a relationship between triangles and circles, and to prove this theorem directly, using previously accepted definitions, postulates and theorems. Important Terms: oncurrent Lines From two Latin words meaning to run together, this term deals primarily with intersecting lines, or line segments. Formally, if two or more lines contain the same point, they are said to be concurrent. Note: In Geometry, because it is more significant when three or more lines contain the same point, the more frequently used geometric definition of concurrent lines relates to three or more lines.) oncyclic Points From two Latin words meaning together on a circle, this term deals primarily with points on a circle. Formally, if two or more points lie on the same circle, they are said to be concyclic. Note: In Geometry, because it is more significant when three of more points lie on the same circle, the more frequently used geometric definition of concyclic points relates to three or more points.) yclic Polygons specific term which refers to a relationship between polygons and circles. Formally, when there is a circle which contains all of the vertices of a polygon, then that polygon is cyclic. orollary 83a If you have a triangle, then the perpendicular bisectors of the sides are concurrent. orollary 83b If you have a triangle, then the bisectors of the angles are concurrent. 580 Unit VI ircles

30 ample 1: The angle bisectors of meet at point from orollary 83b. Given that T = 15 and = 17. a) Which segments are congruent? b) Find T and S. T R S Solution: a) T, S and R are congruent since the angle bisectors meet at a point equidistant from the sides of the triangle. Using Theorem 80, we can conclude that: R T, R S, and S T. b) T T by corollary 68a. T) + T) = ) 15) + T) = 17) 5 + T) = 89 T) = 64 T =± 64 T cannot be negative) T = 8 T = S S = 8 Lesson 1 ercises: 1. Prove Theorem 83 - If you have a triangle, then that triangle is cyclic.. Find the values of and y. 3. Find the values of and y. O 55 6y) O 6y) O O y O 4. Write a two column proof for orollary 83a - If you have a triangle, then the perpendicular bisectors of the sides are concurrent. 5. Write a two column proof for orollary 83b - If you have a triangle, then the bisectors of the angles are concurrent. ) O 4) O Part ircles and oncurrency 581

31 S 11. The perpendicular bisectors of the sides of RST meet at point. S = 11 and M = 4. Find R. Give a reason for your answer. P M R N T 1. is given. Perpendicular bisectors of the sides,, F and G are shown. an you conclude that = G? If not, eplain, and state a correct conclusion that can be deduced from the diagram. F G 13. Traingle PMN is given. ngle bisectors PX, MY, and NZ are shown. an you conclude that Z = X? If not, eplain, and state a correct conclusion that can be deduces from the diagram. P Y Z X M 14. The three perpendicular bisectors of the sides of a triangle are concurrent in a point which can be inside the triangle, on the triangle, or outside the triangle. Sketch an obtuse triangle, an acute triangle, and a right triangle showing the perpendicular bisectors of the sides in each to verify each relationship. 15. Triangle is an obtuse triangle. Perpendicular bisectors of the sides meet at point M. MP = 1 and M = 13. Find. P N M R In eercises 16 through 0, complete the statement using always, sometimes, or never. 16. perpendicular bisector of a side of a triangle passes through the midpoint of a side of the triangle. 17. The angle bisectors of the angle of a triangle intersect at a single point. 18. The angle bisectors of the angle of a triangle meet at a point outside the triangle. 19. The perpendicular bisectors of the sides of a triangle meet at a point which lies outside the triangle.. 0. The midpoint of the hypotenuse of a right triangle is equidistant from all vertices of the triangle. Part ircles and oncurrency 583

32 Y 1. XP and ZP are angle bisectors of X and Z in XYZ. m XYZ = 11. Find m XPZ. X P Z Use the graph of and eercises through 4 to illustrate orollary 83a, about the concurrency of perpendicular bisectors of the sides of a triangle. 1,8) 0,0) 16,0). Find the midpoint of each side of. Use the midpoints to find the equations of the perpendicular bisectors of the sides of. 3. Using your equations from eercise, find the intersection of two of the lines. 4. Show that the point in eercise 3 is equidistant from the vertices of. Unit VI ircles Part ircles and oncurrency Lesson Theorem 84 - If the opposite angles of a quadrilateral are supplementary, then the quadrilateral is cyclic. Objective: To investigate a relationship between quadrilaterals and circles, and to prove this theorem directly, using previously accepted definitions, postulates and theorems. Important Terms: oncurrent Lines From two Latin words meaning to run together, this term deals primarily with intersecting lines, or line segments. Formally, if two or more lines contain the same point, they are said to be concurrent. Note: In Geometry, because it is more significant when three or more lines contain the same point, the more frequently used geometric definition of concurrent lines relates to three or more lines.) oncyclic Points From two Latin words meaning together on a circle, this term deals primarily with points on a circle. Formally, if two or more points lie on the same circle, they are said to be concyclic. Note: In Geometry, because it is more significant when three of more points lie on the same circle, the more frequently used geometric definition of concyclic points relates to three or more points.) yclic Polygons specific term which refers to a relationship between polygons and circles. Formally, when there is a circle which contains all of the vertices of a polygon, then that polygon is cyclic. 584 Unit VI ircles

33 Lesson ercises: 1. Prove Theorem 84 - If the opposite angles of a quadrilateral are supplementary, then the quadrilateral is cyclic. onverse of orollary 67b). Find the values of, y, and z. 3. Find the values of and y. m = 136 O m = z 10 y y z Find the values of and y. 5. Find the values of and y. m = z y y 9y y 4 6. Find the values of and y. 7. Find the measure of the angles of quadrilateral. 6y 1y Prove that trapezoid inscribed in a circle, is an isosceles trapezoid. 9. Suppose that is a quadrilateral inscribed in a circle, and that is a diameter of the circle. If m is three times m, what are the measure of all four angles? Part ircles and oncurrency 585

34 10. Show that the ratio of the radius of the inscribed circle to the radius of the circle circumscribed about a square is to. 11. ll regular simple polygons are cyclic. circle contains 360 degrees. How many degrees are in each arc of a circle circumscribed about: a) n equilateral triangle? b) square? c) regular heagon? d) regular octagon? 1. Given: uadrilaterl XYWZ is cyclic. ZY is a diameter of circle. XY ZW X Y Prove: XY ZW Z W 13. Square is cyclic a) Find m b) Find c) Find the distance from point to. 10 In eercises 14 through 19, tell if the given quadrilateral can always be inscribed in a circle. plain each answer. 14. Square 15. Rectangle 16. Parallelogram 17. Kite 18. Rhombus 19. Isosceles Trapezoid 0. The line segment in a quadrilateral drawn from a midpoint of a side perpendicular to the opposite side is called a maltitude. In a cyclic quadrilateral, maltitudes are concurrent. Using a straightedge, sketch a cyclic quadrilateral and its maltitudes and verify this relationship. 586 Unit VI ircles

Geometry: A Complete Course

Geometry: A Complete Course Geometry: omplete ourse with Trigonometry) Module Progress Tests Written by: Larry. ollins Geometry: omplete ourse with Trigonometry) Module - Progress Tests opyright 2014 by VideotextInteractive Send

More information

Geometry: A Complete Course

Geometry: A Complete Course Geometry: omplete ourse with Trigonometry) Module Instructor's Guide with etailed Solutions for Progress Tests Written by: Larry. ollins RRT /010 Unit V, Part, Lessons 1, uiz Form ontinued. Match each

More information

Geometry: A Complete Course

Geometry: A Complete Course Geometry: omplete ourse with Trigonometry) Module E Solutions Manual Written by: Larry E. ollins Geometry: omplete ourse with Trigonometry) Module E Solutions Manual opyright 2014 by VideotextInteractive

More information

Geometry: A Complete Course

Geometry: A Complete Course Geometry: Complete Course with Trigonometry) Module E - Course Notes Written by: Thomas E. Clark Geometry: Complete Course with Trigonometry) Module E - Course Notes Copyright 2014 by VideotextInteractive

More information

Geometry Definitions and Theorems. Chapter 9. Definitions and Important Terms & Facts

Geometry Definitions and Theorems. Chapter 9. Definitions and Important Terms & Facts Geometry Definitions and Theorems Chapter 9 Definitions and Important Terms & Facts A circle is the set of points in a plane at a given distance from a given point in that plane. The given point is the

More information

Videos, Constructions, Definitions, Postulates, Theorems, and Properties

Videos, Constructions, Definitions, Postulates, Theorems, and Properties Videos, Constructions, Definitions, Postulates, Theorems, and Properties Videos Proof Overview: http://tinyurl.com/riehlproof Modules 9 and 10: http://tinyurl.com/riehlproof2 Module 9 Review: http://tinyurl.com/module9livelesson-recording

More information

SOME IMPORTANT PROPERTIES/CONCEPTS OF GEOMETRY (Compiled by Ronnie Bansal)

SOME IMPORTANT PROPERTIES/CONCEPTS OF GEOMETRY (Compiled by Ronnie Bansal) 1 SOME IMPORTANT PROPERTIES/CONCEPTS OF GEOMETRY (Compiled by Ronnie Bansal) 1. Basic Terms and Definitions: a) Line-segment: A part of a line with two end points is called a line-segment. b) Ray: A part

More information

Geometry Third Quarter Study Guide

Geometry Third Quarter Study Guide Geometry Third Quarter Study Guide 1. Write the if-then form, the converse, the inverse and the contrapositive for the given statement: All right angles are congruent. 2. Find the measures of angles A,

More information

Examples: The name of the circle is: The radii of the circle are: The chords of the circle are: The diameter of the circle is:

Examples: The name of the circle is: The radii of the circle are: The chords of the circle are: The diameter of the circle is: Geometry P Lesson 10-1: ircles and ircumference Page 1 of 1 Objectives: To identify and use parts of circles To solve problems involving the circumference of a circle Geometry Standard: 8 Examples: The

More information

Postulates, Theorems, and Corollaries. Chapter 1

Postulates, Theorems, and Corollaries. Chapter 1 Chapter 1 Post. 1-1-1 Through any two points there is exactly one line. Post. 1-1-2 Through any three noncollinear points there is exactly one plane containing them. Post. 1-1-3 If two points lie in a

More information

Unit 10 Circles 10-1 Properties of Circles Circle - the set of all points equidistant from the center of a circle. Chord - A line segment with

Unit 10 Circles 10-1 Properties of Circles Circle - the set of all points equidistant from the center of a circle. Chord - A line segment with Unit 10 Circles 10-1 Properties of Circles Circle - the set of all points equidistant from the center of a circle. Chord - A line segment with endpoints on the circle. Diameter - A chord which passes through

More information

Instructional Unit CPM Geometry Unit Content Objective Performance Indicator Performance Task State Standards Code:

Instructional Unit CPM Geometry Unit Content Objective Performance Indicator Performance Task State Standards Code: 306 Instructional Unit Area 1. Areas of Squares and The students will be -Find the amount of carpet 2.4.11 E Rectangles able to determine the needed to cover various plane 2. Areas of Parallelograms and

More information

Thomas Jefferson High School for Science and Technology Program of Studies TJ Math 1

Thomas Jefferson High School for Science and Technology Program of Studies TJ Math 1 Course Description: This course is designed for students who have successfully completed the standards for Honors Algebra I. Students will study geometric topics in depth, with a focus on building critical

More information

MATH 30 GEOMETRY UNIT OUTLINE AND DEFINITIONS Prepared by: Mr. F.

MATH 30 GEOMETRY UNIT OUTLINE AND DEFINITIONS Prepared by: Mr. F. 1 MTH 30 GEMETRY UNIT UTLINE ND DEFINITINS Prepared by: Mr. F. Some f The Typical Geometric Properties We Will Investigate: The converse holds in many cases too! The Measure f The entral ngle Tangent To

More information

Geometry (H) Worksheet: 1st Semester Review:True/False, Always/Sometimes/Never

Geometry (H) Worksheet: 1st Semester Review:True/False, Always/Sometimes/Never 1stSemesterReviewTrueFalse.nb 1 Geometry (H) Worksheet: 1st Semester Review:True/False, Always/Sometimes/Never Classify each statement as TRUE or FALSE. 1. Three given points are always coplanar. 2. A

More information

Geometry Spring Semester Review

Geometry Spring Semester Review hapter 5 Geometry Spring Semester Review 1. In PM,. m P > m. m P > m M. m > m P. m M > m P 7 M 2. Find the shortest side of the figure QU. Q Q 80 4. QU. U. 50 82 U 3. In EFG, m E = 5 + 2, m F = -, and

More information

Pre-AP Geometry Spring Semester Exam Review 2015

Pre-AP Geometry Spring Semester Exam Review 2015 hapter 8 1. Find.. 25.4. 11.57. 3 D. 28 3. Find.. 3.73. 4. 2 D. 8.77 5. Find, y, k, and m. = k= Pre-P Geometry Spring Semester Eam Review 2015 40 18 25 y= m= 2. Find.. 5 2.. 5 D. 2 4. Find.. 3 2. 2. D.

More information

MANHATTAN HUNTER SCIENCE HIGH SCHOOL GEOMETRY CURRICULUM

MANHATTAN HUNTER SCIENCE HIGH SCHOOL GEOMETRY CURRICULUM COORDINATE Geometry Plotting points on the coordinate plane. Using the Distance Formula: Investigate, and apply the Pythagorean Theorem as it relates to the distance formula. (G.GPE.7, 8.G.B.7, 8.G.B.8)

More information

High School Mathematics Geometry Vocabulary Word Wall Cards

High School Mathematics Geometry Vocabulary Word Wall Cards High School Mathematics Geometry Vocabulary Word Wall Cards Table of Contents Reasoning, Lines, and Transformations Basics of Geometry 1 Basics of Geometry 2 Geometry Notation Logic Notation Set Notation

More information

6.1 Circles and Related Segments and Angles

6.1 Circles and Related Segments and Angles Chapter 6 Circles 6.1 Circles and Related Segments and Angles Definitions 32. A circle is the set of all points in a plane that are a fixed distance from a given point known as the center of the circle.

More information

Modeling with Geometry

Modeling with Geometry Modeling with Geometry 6.3 Parallelograms https://mathbitsnotebook.com/geometry/quadrilaterals/qdparallelograms.html Properties of Parallelograms Sides A parallelogram is a quadrilateral with both pairs

More information

STANDARDS OF LEARNING CONTENT REVIEW NOTES HONORS GEOMETRY. 3 rd Nine Weeks,

STANDARDS OF LEARNING CONTENT REVIEW NOTES HONORS GEOMETRY. 3 rd Nine Weeks, STANDARDS OF LEARNING CONTENT REVIEW NOTES HONORS GEOMETRY 3 rd Nine Weeks, 2016-2017 1 OVERVIEW Geometry Content Review Notes are designed by the High School Mathematics Steering Committee as a resource

More information

Geometry Final Exam - Study Guide

Geometry Final Exam - Study Guide Geometry Final Exam - Study Guide 1. Solve for x. True or False? (questions 2-5) 2. All rectangles are rhombuses. 3. If a quadrilateral is a kite, then it is a parallelogram. 4. If two parallel lines are

More information

Name Honors Geometry Final Exam Review

Name Honors Geometry Final Exam Review 2014-2015 Name Honors Geometry Final Eam Review Chapter 5 Use the picture at the right to answer the following questions. 1. AC= 2. m BFD = 3. m CAE = A 29 C B 71⁰ 19 D 16 F 65⁰ E 4. Find the equation

More information

West Windsor-Plainsboro Regional School District Basic Geometry Grades 9-12

West Windsor-Plainsboro Regional School District Basic Geometry Grades 9-12 West Windsor-Plainsboro Regional School District Basic Geometry Grades 9-12 Unit 1: Basics of Geometry Content Area: Mathematics Course & Grade Level: Basic Geometry, 9 12 Summary and Rationale This unit

More information

Unit 2: Triangles and Quadrilaterals Lesson 2.1 Apply Triangle Sum Properties Lesson 4.1 from textbook

Unit 2: Triangles and Quadrilaterals Lesson 2.1 Apply Triangle Sum Properties Lesson 4.1 from textbook Unit 2: Triangles and Quadrilaterals Lesson 2.1 pply Triangle Sum Properties Lesson 4.1 from textbook Objectives Classify angles by their sides as equilateral, isosceles, or scalene. Classify triangles

More information

Geometry Rules. Triangles:

Geometry Rules. Triangles: Triangles: Geometry Rules 1. Types of Triangles: By Sides: Scalene - no congruent sides Isosceles - 2 congruent sides Equilateral - 3 congruent sides By Angles: Acute - all acute angles Right - one right

More information

NOTES: Tangents to Circles

NOTES: Tangents to Circles Unit# ssign # TS: Tangents to ircles GL Identify segments and lines related to circles and use properties of a tangent to a circle VULRY circle is the set of all points in a plane that are equidistant

More information

STANDARDS OF LEARNING CONTENT REVIEW NOTES GEOMETRY. 3 rd Nine Weeks,

STANDARDS OF LEARNING CONTENT REVIEW NOTES GEOMETRY. 3 rd Nine Weeks, STANDARDS OF LEARNING CONTENT REVIEW NOTES GEOMETRY 3 rd Nine Weeks, 2016-2017 1 OVERVIEW Geometry Content Review Notes are designed by the High School Mathematics Steering Committee as a resource for

More information

Lesson 13.1 The Premises of Geometry

Lesson 13.1 The Premises of Geometry Lesson 13.1 The remises of Geometry 1. rovide the missing property of equality or arithmetic as a reason for each step to solve the equation. Solve for x: 5(x 4) 2x 17 Solution: 5(x 4) 2x 17 a. 5x 20 2x

More information

Notes Circle Basics Standard:

Notes Circle Basics Standard: Notes Circle Basics M RECALL EXAMPLES Give an example of each of the following: 1. Name the circle 2. Radius 3. Chord 4. Diameter 5. Secant 6. Tangent (line) 7. Point of tangency 8. Tangent (segment) DEFINTION

More information

Properties of Triangles

Properties of Triangles Properties of Triangles Perpendiculars and isectors segment, ray, line, or plane that is perpendicular to a segment at its midpoint is called a perpendicular bisector. point is equidistant from two points

More information

Angles. Classification Acute Right Obtuse. Complementary s 2 s whose sum is 90 Supplementary s 2 s whose sum is 180. Angle Addition Postulate

Angles. Classification Acute Right Obtuse. Complementary s 2 s whose sum is 90 Supplementary s 2 s whose sum is 180. Angle Addition Postulate ngles Classification cute Right Obtuse Complementary s 2 s whose sum is 90 Supplementary s 2 s whose sum is 180 ngle ddition Postulate If the exterior sides of two adj s lie in a line, they are supplementary

More information

Geometry Vocabulary Math Fundamentals Reference Sheet Page 1

Geometry Vocabulary Math Fundamentals Reference Sheet Page 1 Math Fundamentals Reference Sheet Page 1 Acute Angle An angle whose measure is between 0 and 90 Acute Triangle A that has all acute Adjacent Alternate Interior Angle Two coplanar with a common vertex and

More information

Chapter 10 Similarity

Chapter 10 Similarity Chapter 10 Similarity Def: The ratio of the number a to the number b is the number. A proportion is an equality between ratios. a, b, c, and d are called the first, second, third, and fourth terms. The

More information

Unit Number of Days Dates. 1 Angles, Lines and Shapes 14 8/2 8/ Reasoning and Proof with Lines and Angles 14 8/22 9/9

Unit Number of Days Dates. 1 Angles, Lines and Shapes 14 8/2 8/ Reasoning and Proof with Lines and Angles 14 8/22 9/9 8 th Grade Geometry Curriculum Map Overview 2016-2017 Unit Number of Days Dates 1 Angles, Lines and Shapes 14 8/2 8/19 2 - Reasoning and Proof with Lines and Angles 14 8/22 9/9 3 - Congruence Transformations

More information

SOL Chapter Due Date

SOL Chapter Due Date Name: Block: Date: Geometry SOL Review SOL Chapter Due Date G.1 2.2-2.4 G.2 3.1-3.5 G.3 1.3, 4.8, 6.7, 9 G.4 N/A G.5 5.5 G.6 4.1-4.7 G.7 6.1-6.6 G.8 7.1-7.7 G.9 8.2-8.6 G.10 1.6, 8.1 G.11 10.1-10.6, 11.5,

More information

algebraic representation algorithm alternate interior angles altitude analytical geometry x x x analytical proof x x angle

algebraic representation algorithm alternate interior angles altitude analytical geometry x x x analytical proof x x angle Words PS R Comm CR Geo R Proof Trans Coor Catogoriers Key AA triangle similarity Constructed Response AAA triangle similarity Problem Solving AAS triangle congruence Resoning abscissa Communication absolute

More information

Reteaching Inequalities in Two Triangles

Reteaching Inequalities in Two Triangles Name ate lass Inequalities in Two Triangles INV You have worked with segments and angles in triangles. Now ou will eplore inequalities with triangles. Hinge Theorem If two sides of one triangle are congruent

More information

Course: Geometry PAP Prosper ISD Course Map Grade Level: Estimated Time Frame 6-7 Block Days. Unit Title

Course: Geometry PAP Prosper ISD Course Map Grade Level: Estimated Time Frame 6-7 Block Days. Unit Title Unit Title Unit 1: Geometric Structure Estimated Time Frame 6-7 Block 1 st 9 weeks Description of What Students will Focus on on the terms and statements that are the basis for geometry. able to use terms

More information

a B. 3a D. 0 E. NOTA m RVS a. DB is parallel to EC and AB=3, DB=5, and

a B. 3a D. 0 E. NOTA m RVS a. DB is parallel to EC and AB=3, DB=5, and Geometry Individual Test NO LULTOR!!! Middleton Invitational February 18, 006 The abbreviation "NOT" denotes "None of These nswers." iagrams are not necessarily drawn to scale. 1. Which set does not always

More information

Geometry Curriculum Map

Geometry Curriculum Map Geometry Curriculum Map Unit 1 st Quarter Content/Vocabulary Assessment AZ Standards Addressed Essentials of Geometry 1. What are points, lines, and planes? 1. Identify Points, Lines, and Planes 1. Observation

More information

Geometry. Instructional Activities:

Geometry. Instructional Activities: GEOMETRY Instructional Activities: Geometry Assessment: A. Direct Instruction A. Quizzes B. Cooperative Learning B. Skill Reviews C. Technology Integration C. Test Prep Questions D. Study Guides D. Chapter

More information

Carnegie Learning High School Math Series: Geometry Indiana Standards Worktext Correlations

Carnegie Learning High School Math Series: Geometry Indiana Standards Worktext Correlations Carnegie Learning High School Math Series: Logic and Proofs G.LP.1 Understand and describe the structure of and relationships within an axiomatic system (undefined terms, definitions, axioms and postulates,

More information

3. Radius of incenter, C. 4. The centroid is the point that corresponds to the center of gravity in a triangle. B

3. Radius of incenter, C. 4. The centroid is the point that corresponds to the center of gravity in a triangle. B 1. triangle that contains one side that has the same length as the diameter of its circumscribing circle must be a right triangle, which cannot be acute, obtuse, or equilateral. 2. 3. Radius of incenter,

More information

The Research- Driven Solution to Raise the Quality of High School Core Courses. Geometry. Course Outline

The Research- Driven Solution to Raise the Quality of High School Core Courses. Geometry. Course Outline The Research- Driven Solution to Raise the Quality of High School Core Courses Course Outline Course Outline Page 2 of 5 0 1 2 3 4 5 ACT Course Standards A. Prerequisites 1. Skills Acquired by Students

More information

Index COPYRIGHTED MATERIAL. Symbols & Numerics

Index COPYRIGHTED MATERIAL. Symbols & Numerics Symbols & Numerics. (dot) character, point representation, 37 symbol, perpendicular lines, 54 // (double forward slash) symbol, parallel lines, 54, 60 : (colon) character, ratio of quantity representation

More information

Pearson Mathematics Geometry

Pearson Mathematics Geometry A Correlation of Pearson Mathematics Geometry Indiana 2017 To the INDIANA ACADEMIC STANDARDS Mathematics (2014) Geometry The following shows where all of the standards that are part of the Indiana Mathematics

More information

Lines Plane A flat surface that has no thickness and extends forever.

Lines Plane A flat surface that has no thickness and extends forever. Lines Plane A flat surface that has no thickness and extends forever. Point an exact location Line a straight path that has no thickness and extends forever in opposite directions Ray Part of a line that

More information

A Solution: The area of a trapezoid is height (base 1 + base 2) = ( 6) (8 + 18) = ( 6) ( 26) = 78

A Solution: The area of a trapezoid is height (base 1 + base 2) = ( 6) (8 + 18) = ( 6) ( 26) = 78 10.0 ompute areas of polygons, including rectangles, scalene triangles, equilateral triangles, rhombi, parallelograms, and trapezoids. (cont) Eample 2 Find the area of Trapezoid 8 Solution: The area of

More information

CST Geometry Practice Problems

CST Geometry Practice Problems ST Geometry Practice Problems. Which of the following best describes deductive reasoning? using logic to draw conclusions based on accepted statements accepting the meaning of a term without definition

More information

Honors Geometry CHAPTER 7. Study Guide Final Exam: Ch Name: Hour: Try to fill in as many as possible without looking at your book or notes.

Honors Geometry CHAPTER 7. Study Guide Final Exam: Ch Name: Hour: Try to fill in as many as possible without looking at your book or notes. Honors Geometry Study Guide Final Exam: h 7 12 Name: Hour: Try to fill in as many as possible without looking at your book or notes HPTER 7 1 Pythagorean Theorem: Pythagorean Triple: 2 n cute Triangle

More information

ALLEGHANY COUNTY SCHOOLS CURRICULUM GUIDE

ALLEGHANY COUNTY SCHOOLS CURRICULUM GUIDE GRADE/COURSE: Geometry GRADING PERIOD: 1 Year Course Time SEMESTER 1: 1 ST SIX WEEKS Pre-Test, Class Meetings, Homeroom Chapter 1 12 days Lines and Angles Point Line AB Ray AB Segment AB Plane ABC Opposite

More information

U4 Polygon Notes January 11, 2017 Unit 4: Polygons

U4 Polygon Notes January 11, 2017 Unit 4: Polygons Unit 4: Polygons 180 Complimentary Opposite exterior Practice Makes Perfect! Example: Example: Practice Makes Perfect! Def: Midsegment of a triangle - a segment that connects the midpoints of two sides

More information

WAYNESBORO AREA SCHOOL DISTRICT CURRICULUM ACCELERATED GEOMETRY (June 2014)

WAYNESBORO AREA SCHOOL DISTRICT CURRICULUM ACCELERATED GEOMETRY (June 2014) UNIT: Chapter 1 Essentials of Geometry UNIT : How do we describe and measure geometric figures? Identify Points, Lines, and Planes (1.1) How do you name geometric figures? Undefined Terms Point Line Plane

More information

Geometry. (F) analyze mathematical relationships to connect and communicate mathematical ideas; and

Geometry. (F) analyze mathematical relationships to connect and communicate mathematical ideas; and (1) Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is (A) apply mathematics to problems arising in everyday life,

More information

added to equal quantities, their sum is equal. Same holds for congruence.

added to equal quantities, their sum is equal. Same holds for congruence. Mr. Cheung s Geometry Cheat Sheet Theorem List Version 6.0 Updated 3/14/14 (The following is to be used as a guideline. The rest you need to look up on your own, but hopefully this will help. The original

More information

South Carolina College- and Career-Ready (SCCCR) Geometry Overview

South Carolina College- and Career-Ready (SCCCR) Geometry Overview South Carolina College- and Career-Ready (SCCCR) Geometry Overview In South Carolina College- and Career-Ready (SCCCR) Geometry, students build on the conceptual knowledge and skills they mastered in previous

More information

Maintaining Mathematical Proficiency

Maintaining Mathematical Proficiency Name ate hapter 7 Maintaining Mathematical Proficiency Solve the equation by interpreting the expression in parentheses as a single quantity. 1. 5( 10 x) = 100 2. 6( x + 8) 12 = 48 3. ( x) ( x) 32 + 42

More information

Suggested List of Mathematical Language. Geometry

Suggested List of Mathematical Language. Geometry Suggested List of Mathematical Language Geometry Problem Solving A additive property of equality algorithm apply constraints construct discover explore generalization inductive reasoning parameters reason

More information

Geometry First Semester Practice Final (cont)

Geometry First Semester Practice Final (cont) 49. Determine the width of the river, AE, if A. 6.6 yards. 10 yards C. 12.8 yards D. 15 yards Geometry First Semester Practice Final (cont) 50. In the similar triangles shown below, what is the value of

More information

Lesson 13.1 The Premises of Geometry

Lesson 13.1 The Premises of Geometry Lesson 13.1 he remises of Geometry Name eriod ate 1. rovide the missing property of equality or arithmetic as a reason for each step to solve the equation. olve for x: 5(x 4) 15 2x 17 olution: 5(x 4) 15

More information

Geometry Foundations Pen Argyl Area High School 2018

Geometry Foundations Pen Argyl Area High School 2018 Geometry emphasizes the development of logical thinking as it relates to geometric problems. Topics include using the correct language and notations of geometry, developing inductive and deductive reasoning,

More information

The Research- Driven Solution to Raise the Quality of High School Core Courses. Geometry. Instructional Units Plan

The Research- Driven Solution to Raise the Quality of High School Core Courses. Geometry. Instructional Units Plan The Research- Driven Solution to Raise the Quality of High School Core Courses Instructional Units Plan Instructional Units Plan This set of plans presents the topics and selected for ACT s rigorous course.

More information

Aldine ISD Benchmark Targets /Geometry SUMMER 2004

Aldine ISD Benchmark Targets /Geometry SUMMER 2004 ASSURANCES: By the end of Geometry, the student will be able to: 1. Use properties of triangles and quadrilaterals to solve problems. 2. Identify, classify, and draw two and three-dimensional objects (prisms,

More information

0613ge. Geometry Regents Exam 0613

0613ge. Geometry Regents Exam 0613 wwwjmaporg 0613ge 1 In trapezoid RSTV with bases and, diagonals and intersect at Q If trapezoid RSTV is not isosceles, which triangle is equal in area to? 2 In the diagram below, 3 In a park, two straight

More information

Geometry CP Pen Argyl Area High School 2018

Geometry CP Pen Argyl Area High School 2018 Geometry emphasizes the development of logical thinking as it relates to geometric problems. Topics include using the correct language and notations of geometry, developing inductive and deductive reasoning,

More information

Chapter 6. Sir Migo Mendoza

Chapter 6. Sir Migo Mendoza Circles Chapter 6 Sir Migo Mendoza Central Angles Lesson 6.1 Sir Migo Mendoza Central Angles Definition 5.1 Arc An arc is a part of a circle. Types of Arc Minor Arc Major Arc Semicircle Definition 5.2

More information

Reteaching Exploring Angles of Polygons

Reteaching Exploring Angles of Polygons Name Date lass Eploring Angles of Polygons INV X 3 You have learned to identify interior and eterior angles in polygons. Now you will determine angle measures in regular polygons. Interior Angles Sum of

More information

Name Honors Geometry Final Exam Review. 1. The following figure is a parallelogram. Find the values of x and y.

Name Honors Geometry Final Exam Review. 1. The following figure is a parallelogram. Find the values of x and y. 2013-2014 Name Honors Geometr Final Eam Review Chapter 5 Questions 1. The following figure is a parallelogram. Find the values of and. (+)⁰ 130⁰ (-)⁰ 85⁰ 2. Find the value of in the figure below. D is

More information

Plane Geometry. Paul Yiu. Department of Mathematics Florida Atlantic University. Summer 2011

Plane Geometry. Paul Yiu. Department of Mathematics Florida Atlantic University. Summer 2011 lane Geometry aul Yiu epartment of Mathematics Florida tlantic University Summer 2011 NTENTS 101 Theorem 1 If a straight line stands on another straight line, the sum of the adjacent angles so formed is

More information

Perimeter. Area. Surface Area. Volume. Circle (circumference) C = 2πr. Square. Rectangle. Triangle. Rectangle/Parallelogram A = bh

Perimeter. Area. Surface Area. Volume. Circle (circumference) C = 2πr. Square. Rectangle. Triangle. Rectangle/Parallelogram A = bh Perimeter Circle (circumference) C = 2πr Square P = 4s Rectangle P = 2b + 2h Area Circle A = πr Triangle A = bh Rectangle/Parallelogram A = bh Rhombus/Kite A = d d Trapezoid A = b + b h A area a apothem

More information

Indicate whether the statement is true or false.

Indicate whether the statement is true or false. Math 121 Fall 2017 - Practice Exam - Chapters 5 & 6 Indicate whether the statement is true or false. 1. The simplified form of the ratio 6 inches to 1 foot is 6:1. 2. The triple (20,21,29) is a Pythagorean

More information

GEOMETRY is the study of points in space

GEOMETRY is the study of points in space CHAPTER 5 Logic and Geometry SECTION 5-1 Elements of Geometry GEOMETRY is the study of points in space POINT indicates a specific location and is represented by a dot and a letter R S T LINE is a set of

More information

ACTM Geometry Exam State 2010

ACTM Geometry Exam State 2010 TM Geometry xam State 2010 In each of the following select the answer and record the selection on the answer sheet provided. Note: Pictures are not necessarily drawn to scale. 1. The measure of in the

More information

NEW YORK GEOMETRY TABLE OF CONTENTS

NEW YORK GEOMETRY TABLE OF CONTENTS NEW YORK GEOMETRY TABLE OF CONTENTS CHAPTER 1 POINTS, LINES, & PLANES {G.G.21, G.G.27} TOPIC A: Concepts Relating to Points, Lines, and Planes PART 1: Basic Concepts and Definitions...1 PART 2: Concepts

More information

CURRICULUM GUIDE. Honors Geometry

CURRICULUM GUIDE. Honors Geometry CURRICULUM GUIDE Honors Geometry This level of Geometry is approached at an accelerated pace. Topics of postulates, theorems and proofs are discussed both traditionally and with a discovery approach. The

More information

Name Class Date. Lines that appear to be tangent are tangent. O is the center of each circle. What is the value of x?

Name Class Date. Lines that appear to be tangent are tangent. O is the center of each circle. What is the value of x? 12-1 Practice Tangent Lines Lines that appear to be tangent are tangent. O is the center of each circle. What is the value of x? 1. To start, identify the type of geometric figure formed by the tangent

More information

Use throughout the course: for example, Parallel and Perpendicular Lines Proving Lines Parallel. Polygons and Parallelograms Parallelograms

Use throughout the course: for example, Parallel and Perpendicular Lines Proving Lines Parallel. Polygons and Parallelograms Parallelograms Geometry Correlated to the Texas Essential Knowledge and Skills TEKS Units Lessons G.1 Mathematical Process Standards The student uses mathematical processes to acquire and demonstrate mathematical understanding.

More information

Math 3315: Geometry Vocabulary Review Human Dictionary: WORD BANK

Math 3315: Geometry Vocabulary Review Human Dictionary: WORD BANK Math 3315: Geometry Vocabulary Review Human Dictionary: WORD BANK [acute angle] [acute triangle] [adjacent interior angle] [alternate exterior angles] [alternate interior angles] [altitude] [angle] [angle_addition_postulate]

More information

Theorems & Postulates Math Fundamentals Reference Sheet Page 1

Theorems & Postulates Math Fundamentals Reference Sheet Page 1 Math Fundamentals Reference Sheet Page 1 30-60 -90 Triangle In a 30-60 -90 triangle, the length of the hypotenuse is two times the length of the shorter leg, and the length of the longer leg is the length

More information

Unit 9 Syllabus: Circles

Unit 9 Syllabus: Circles ate Period Unit 9 Syllabus: ircles ay Topic 1 Tangent Lines 2 hords and rcs and Inscribed ngles 3 Review/Graded lasswork 4 Review from before reak 5 Finding ngle Measures 6 Finding Segment Lengths 7 Review

More information

Geometry Mathematics. Grade(s) 10th - 12th, Duration 1 Year, 1 Credit Required Course

Geometry Mathematics. Grade(s) 10th - 12th, Duration 1 Year, 1 Credit Required Course Scope And Sequence Timeframe Unit Instructional Topics 9 Week(s) 9 Week(s) 9 Week(s) Geometric Structure Measurement Similarity Course Overview GENERAL DESCRIPTION: In this course the student will become

More information

Lincoln Public Schools GEOMETRY REVIEW - Semester One CALCULATOR Revised 12/2007

Lincoln Public Schools GEOMETRY REVIEW - Semester One CALCULATOR Revised 12/2007 Lincoln Public chools GOMY VIW - emester One LULO evised /007. escribe the lines in the sketch.. coplanar and intersecting. coplanar and nonintersecting. noncoplanar and intersecting. noncoplanar and nonintersecting.

More information

Slide 1 / 343 Slide 2 / 343

Slide 1 / 343 Slide 2 / 343 Slide 1 / 343 Slide 2 / 343 Geometry Quadrilaterals 2015-10-27 www.njctl.org Slide 3 / 343 Table of ontents Polygons Properties of Parallelograms Proving Quadrilaterals are Parallelograms Rhombi, Rectangles

More information

Geometry. Quadrilaterals. Slide 1 / 189. Slide 2 / 189. Slide 3 / 189. Table of Contents. New Jersey Center for Teaching and Learning

Geometry. Quadrilaterals. Slide 1 / 189. Slide 2 / 189. Slide 3 / 189. Table of Contents. New Jersey Center for Teaching and Learning New Jersey enter for Teaching and Learning Slide 1 / 189 Progressive Mathematics Initiative This material is made freely available at www.njctl.org and is intended for the non-commercial use of students

More information

Section Congruence Through Constructions

Section Congruence Through Constructions Section 10.1 - Congruence Through Constructions Definitions: Similar ( ) objects have the same shape but not necessarily the same size. Congruent ( =) objects have the same size as well as the same shape.

More information

Geometry Quarter 4 Test Study Guide

Geometry Quarter 4 Test Study Guide Geometry Quarter 4 Test Study Guide 1. Write the if-then form, the converse, the inverse and the contrapositive for the given statement: All right angles are congruent. 2. Find the measures of angles A,

More information

104, 107, 108, 109, 114, 119, , 129, 139, 141, , , , , 180, , , 128 Ch Ch1-36

104, 107, 108, 109, 114, 119, , 129, 139, 141, , , , , 180, , , 128 Ch Ch1-36 111.41. Geometry, Adopted 2012 (One Credit). (c) Knowledge and skills. Student Text Practice Book Teacher Resource: Activities and Projects (1) Mathematical process standards. The student uses mathematical

More information

Preliminary: First you must understand the relationship between inscribed and circumscribed, for example:

Preliminary: First you must understand the relationship between inscribed and circumscribed, for example: 10.7 Inscribed and Circumscribed Polygons Lesson Objective: After studying this section, you will be able to: Recognize inscribed and circumscribed polygons Apply the relationship between opposite angles

More information

11-1 Study Guide and Intervention

11-1 Study Guide and Intervention 11-1 Study Guide and Intervention reas of Parallelograms reas of Parallelograms parallelogram is a quadrilateral with both pairs of opposite sides parallel. ny side of a parallelogram can be called a base.

More information

Performance Objectives Develop dictionary terms and symbols

Performance Objectives Develop dictionary terms and symbols Basic Geometry Course Name: Geometry Unit: 1 Terminology & Fundamental Definitions Time Line: 4 to 6 weeks Building Blocks of Geometry Define & identify point, line, plane angle, segment, ray, midpoint,

More information

What is a(n); 2. acute angle 2. An angle less than 90 but greater than 0

What is a(n); 2. acute angle 2. An angle less than 90 but greater than 0 Geometry Review Packet Semester Final Name Section.. Name all the ways you can name the following ray:., Section.2 What is a(n); 2. acute angle 2. n angle less than 90 but greater than 0 3. right angle

More information

PA Core Standards For Mathematics Curriculum Framework Geometry

PA Core Standards For Mathematics Curriculum Framework Geometry Patterns exhibit relationships How can patterns be used to describe Congruence G.1.3.1.1 and Similarity G.1.3.1.2 described, and generalized. situations? G.1.3.2.1 Use properties of congruence, correspondence,

More information

2) Prove that any point P on the perpendicular bisector of AB is equidistant from both points A and B.

2) Prove that any point P on the perpendicular bisector of AB is equidistant from both points A and B. Seattle Public Schools Review Questions for the Washington State Geometry End of ourse Exam 1) Which term best defines the type of reasoning used below? bdul broke out in hives the last four times that

More information

Lesson 4.3 Ways of Proving that Quadrilaterals are Parallelograms

Lesson 4.3 Ways of Proving that Quadrilaterals are Parallelograms Lesson 4.3 Ways of Proving that Quadrilaterals are Parallelograms Getting Ready: How will you know whether or not a figure is a parallelogram? By definition, a quadrilateral is a parallelogram if it has

More information

Geometry. Cluster: Experiment with transformations in the plane. G.CO.1 G.CO.2. Common Core Institute

Geometry. Cluster: Experiment with transformations in the plane. G.CO.1 G.CO.2. Common Core Institute Geometry Cluster: Experiment with transformations in the plane. G.CO.1: Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of

More information

, Geometry, Quarter 1

, Geometry, Quarter 1 2017.18, Geometry, Quarter 1 The following Practice Standards and Literacy Skills will be used throughout the course: Standards for Mathematical Practice Literacy Skills for Mathematical Proficiency 1.

More information

Madison County Schools Suggested Geometry Pacing Guide,

Madison County Schools Suggested Geometry Pacing Guide, Madison County Schools Suggested Geometry Pacing Guide, 2016 2017 Domain Abbreviation Congruence G-CO Similarity, Right Triangles, and Trigonometry G-SRT Modeling with Geometry *G-MG Geometric Measurement

More information

LT 1.2 Linear Measure (*) LT 1.3 Distance and Midpoints (*) LT 1.4 Angle Measure (*) LT 1.5 Angle Relationships (*) LT 1.6 Two-Dimensional Figures (*)

LT 1.2 Linear Measure (*) LT 1.3 Distance and Midpoints (*) LT 1.4 Angle Measure (*) LT 1.5 Angle Relationships (*) LT 1.6 Two-Dimensional Figures (*) PS1 Tools of Geometry: Students will be able to find distances between points and midpoints of line segments; identify angle relationships; and find perimeters, areas, surface areas and volumes. LT 1.1

More information