Geometry: A Complete Course (with Trigonometry)


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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 onehalf 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 onehalf 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 onehalf 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 onehalf 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 onehalf 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 onehalf 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 nonconsecutive 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 onecentimeter 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 onehalf 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. Meanstremes 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
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