Geometry: A Complete Course

Transcription

1 Geometry: omplete ourse with Trigonometry) Module Progress Tests Written by: Larry. ollins

2 Geometry: omplete ourse with Trigonometry) Module - Progress Tests opyright 2014 by VideotextInteractive Send all inquiries to: VideotextInteractive P.O. ox Indianapolis, IN 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 Instructional ids Program Overview iv Scope and Sequence Rationale vi Progress Tests Unit V - Other Polygons Part - Properties of Polygons LSSON 1 - asic Terms uiz uiz LSSON 2 - Parallelograms uiz uiz LSSON - Special Parallelograms Rectangle, Rhombus, Square) uiz uiz LSSON 4 - Trapezoids combined LSSON 5 - Kites uiz uiz LSSON 6 - Midsegments uiz uiz LSSON 7 - General Polygons uiz uiz Part - reas of Polygons LSSON 1 - Postulate 14 - rea combined LSSON 2 - Triangles uiz uiz LSSON - Parallelograms combined LSSON 4 - Trapezoids uiz uiz LSSON 5 - Regular Polygons uiz uiz Module - Table of ontents i

4 Part - pplications LSSON 1 - Using reas in Proofs combined LSSON 2 - Schedules uiz uiz Unit V Test - Form Unit V Test - Form Unit VI - ircles Part - Fundamental Terms LSSON 1 - Lines and Segments LSSON 2 - rcs and ngles combined LSSON - ircle Relationships uiz uiz 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 2 - 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 - 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. uiz uiz 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 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 72 - 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. uiz uiz combined combined ii Module - Table of ontents

5 Part - Line and Segment Relationships LSSON 1 - Theorem 7 - If a diameter of a circle is perpendicular to a chord of that circle, then that diameter bisects that chord. LSSON 2 - 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 - 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. uiz uiz LSSON 4 - LSSON 5 - LSSON 6 - LSSON 7 - 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 external segment, is equal to the product of the lengths of the other secant segment and its external 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 external segment. Theorem 79 - If a line is perpendicular to a diameter of a circle at one of its endpoints, then the line must be tangent to the circle, at that endpoint. Theorem 80 - If two tangent segments are drawn to a circle from the same point outside the circle, then those tangent segments are congruent. Theorem 81 - If two chords of a circle are congruent, then their intercepted minor arcs are congruent. Theorem 82 - If two minor arcs of a circle are congruent, then the chords which intercept them are congruent. uiz uiz Part - ircles and oncurrency LSSON 1 - Theorem 8 - If you have a triangle, then that triangle is cyclic. LSSON 2 - Theorem 84 - If the opposite angles of a quadrilateral are supplementary, then the quadrilateral is cyclic. uiz uiz Unit VI Test - Form Unit VI Test - Form Unit I-VI umulative Review - Form Unit I-VI umulative Review - Form combined combined combined Module - Table of ontents iii

6 Unit V, Part, Lessons 1, uiz Form ontinued 2. Match each statement in column I with a phrase in column II. olumn I olumn II Rectangle a) n equilateral parallelogram iagonal of a polygon b) parallelogram that has one right angle Polygon onvex Polygon Square c) closed path of four segments that does not cross itself d) quadrilateral that has exactly one pair of parallel sides e) n end point of a side of a polygon Parallelogram Trapezoid f) polygon in which any diagonal lies inside the polygon g) quadrilateral with opposite sides parallel Vertex of a polygon uadrilateral Rhombus h) simple closed curve made up entirely of line segments i) segment whose endpoints are two non-consecutive vertices of a polygon j) n equilangular equilateral quadrilateral. list of properties found in the group of seven special quadrilaterals is given below. Write the name of the special quadrilaterals) beside the given property for which that property is always present. a. oth pairs of opposite sides are parallel. b. xactly one pair of opposite sides are parallel. c. oth pairs of opposite sides are congruent. d. xactly one pair of oppsite sides are congruent. e. ll sides are congruent. f. ll angles are congruent VideoTextInteractive Geometry: omplete ourse

7 Unit V, Part, Lessons 1, uiz Form ontinued 4. Indicate whether each of the following is true or false. a) very square is a rhombus. b) very rhombus is a square. c) very square is a kite. d) very rhombus is a kite. e) If a quadrilateral has three sides of equal length, then it is a kite. f) very property of every square is a property of every rectangle. g) very property of every trapezoid is a property of every parallelogram. h) very property of a parallelogram is a property of every rhombus VideoTextInteractive Geometry: omplete ourse

8 uiz Form Unit V - Other Polygons Part - Properties of Polygons Lesson 2 - Parallelograms lass ate Score Use parallelogram to the right for problems two pairs of congruent sides. 2. two pairs of congruent angles.. pairs of congruent segments that are not sides of the parallelogram. 4. two pairs of supplementary angles. 5. If m = 40, find m. 6. If m = 95, find m and m. Use parallelogram shown to the right to complete each statement in problems If = x and = x + 10, then = 8. If = x + 15 and = 21, 10.If m = 100 O, then = then m = x 9. If = and = 2x 12, 11. If m = 85 O and m = 40 2 O then = then m = 2014 VideoTextInteractive Geometry: omplete ourse 9

9 uiz Form lass ate Score Unit V - Other Polygons Part - Properties of Polygons Lesson 4 - Trapezoids Lesson 5 - Kites 1. In trapezoid,, m = 8x 15 and m = 15x 12. Find m. m = S 2. PRS is a kite. Find SR and R SR = P R = S R 2014 VideoTextInteractive Geometry: omplete ourse 25

10 uiz Form lass ate Score Unit V - Other Polygons Part - Properties of Polygons Lesson 6 - Midsegments Use the figure to the right for problems 1 and Point H is the midpoint of GJ. GH = Point L is the midpoint of GK. HJ = If GJ = 12, and HL = 9, Find GH, HJ, and KJ. KJ = G L H K Problems 1 and 2 J 2. Point H is the midpoint of GJ. Point L is the midpoint of GK. m K = If m GLH is 21 degrees and KJ = 14 1 /2, find m K and HL. HL =. Using the figure to the right, find,, m, m, and m. = = m = m = m = 4 4 F O G VideoTextInteractive Geometry: omplete ourse

11 Unit V, Part, Lessons 6, uiz Form ontinued F 4. In the figure to the right, W, T, and S are midpoints of the sides of triangle F. If WT = 5, ST = 8, and W T SW = 7, What is the perimeter of F? S 5. Which of the following named quadrilaterals are parallelograms? a) b) W c) Y 5 Z 2 Permimeter of F = X J G I 4 2 H 6. In the figure to the right, is a trapezoid with median MN as shown. a) If = 10t and MN = 15t, find. = M N b) If = 5x and MN = 28x, find. = c) If = 9 and = 5 6, find MN. MN = VideoTextInteractive Geometry: omplete ourse

12 Unit V, Part, Lessons 6, uiz Form ontinued 4. In the figure to the right, point is the midpoint of, and point is the midpoint of. = x + 5, = 2y + 6, = 2x 5, and = y + 8. Find and. 5. Which of the following named quadrilaterals are parallelograms? M a) b) c) Z 7 W 7 5 Y 4 4 X 4 P N 4 F = = 6. In the figure to the right, is a trapezoid with median MN as shown. M N a) If = 2x + 5 and MN = 10x 1.2, find. b) If = 2 and = 7 2, find MN. = MN = c) If = 6.7 and = 14.4, find MN. MN = VideoTextInteractive Geometry: omplete ourse

13 uiz Form lass ate Score Unit V - Other Polygons Part - Properties of Polygons Lesson 7 - General Polygons In Problems 1 -, find the number of sides of a polygon if the sum of the measure of its angles is: O sides = O sides = O sides = In Problems 4-6, if the measure of each interior angle of a regular polygon is the given measure, how many sides does the polygon have? O sides = O sides = O sides = In Problems 7-9, find the sum of the measures of the interior angles of a polygon with the given number of sides sides sum = 8. 9 sides sum = sides sum = In Problems 10-12, find the measure of each exterior angle of a regular polygon with the given number of sides. 10. angle = angle = 12. x angle = 2014 VideoTextInteractive Geometry: omplete ourse 7

14 uiz Form Unit V - Other Polygons Part - reas of Polygons Lesson 1 - Postulate 14 - rea Lesson 2 - Triangles lass ate Score For problems 1 6, find the area of the given polygon using the appropriate Postulate, Theorem, or orollary from lesson 1 and = 2. 7 = O 0 O 5. = 4. = = 6. = O 2014 VideoTextInteractive Geometry: omplete ourse 4

15 uiz Form Unit V - Other Polygons Part - reas of Polygons Lesson - Parallelograms Lesson 4 - Trapezoids For problems 1 6, find the area of the given polygon using the appropriate Postulate, Theorem, or orollary from lessons and = 2. = Parallelogram) 11 Trapezoid) 5. = 4. = Trapezoid) 8 60 o Parallelogram) 9 5. = 6. = 45 o 8 Trapezoid) 15 0 o 6 Parallelogram) VideoTextInteractive Geometry: omplete ourse 45

16 Unit V, Part, Lessons &4, uiz Form ontinued For problems 7 and 8, find the area of each polygonal region = 8. = o 6 17 For Problems 9 and 10, find the area of the shaded region. 9. = = 60 o o VideoTextInteractive Geometry: omplete ourse

17 uiz Form Unit V - Other Polygons Part - reas of Polygons Lesson - Parallelograms Lesson 4 - Trapezoids For problems 1 6, find the area of the given polygon using the appropriate Postulate, Theorem, or orollary from lessons and = 2. = Trapezoid) 7 Parallelogram) 8. = 4. = Trapezoid) 45 o Parallelogram) 5 5. = 6. = Parallelogram) Trapezoid) 2014 VideoTextInteractive Geometry: omplete ourse 47

18 Unit V, Part, Lessons &4, uiz Form ontinued For Problems 7 and 8, find the area of each polygonal region. 7. = 8. = For Problems 9 and 10, find the area of the shaded region. 9. = o 0 o 10. = 6 0 o VideoTextInteractive Geometry: omplete ourse

19 uiz Form Unit V - Other Polygons Part - reas of Polygons Lesson 5 - Regular Polygons For problems 1-4, find the degree measure of each central angle of each regular polygon with the given number of sides. 1. degree measure = 2. 8 degree measure =. 12 degree measure = degree measure = For problems 5-7, complete the chart for each regular polygon described. n s P a units VideoTextInteractive Geometry: omplete ourse 49

20 Unit V, Part, Lesson 5, uiz Form ontinued 8. Find the area of an equilateral triangle inscribed in a circle, with a radius of 4 units. rea = 9. Find the area of a square with an apothem of 8 inches and a side of length 16 inches. rea = 10. Find the area of a regular hexagon with an apothem of 11 meters and a side of length 22 meters. rea = VideoTextInteractive Geometry: omplete ourse

21 Unit V, Part, Lessons 1&2, uiz Form ontinued X 9. is a right triangle with = 12 and = 5. is a median of the triangle. What is the area of? 10.Two similar triangles have areas of 81 square inches and 6 square inches. Find the length of a side of the larger triangle if a corresponding side of the smaller triangle is 6. Side = 11.Make a complete schedule for a tournament with 6 teams. Week 1 Week 2 Week Week 4 Week VideoTextInteractive Geometry: omplete ourse 55

22 Unit V, Unit Test Form lass ate Score Unit V - Other Polygons 1. the seven special quadrilaterals and sketch the network illustrating the hierarchy. 1_ 2 4_ 5_ 6_ 7_ 2. Tell whether each of the following statements is true or false. a) property of every rhombus is a property of every parallelogram. b) trapezoid can have three congruent sides. c) very quadrilateral is a convex polygon. d) If a quadrilateral has two consecutive sides of equal length, then it must be a kite. e) If a quadrilateral has three sides of equal length, then it must be a trapezoid. f) There exists a figure which is a rectangle and a parallelogram, but is not a square VideoTextInteractive Geometry: omplete ourse 61

23 Unit V, Unit Test Form ontinued. Find the length of the sides of parallelogram if the perimeter of the parallelogram is 110cm. and the measure of two consecutive sides is x 2 and 2x + 12 respectively. = = = = 4. RSTU is a parallelogram. RV = 8, and UV = 5. Find RT and US. Give a reason to justify your answers. R V S U T RT = US = 5. is a parallelogram. m = 7, find m, m, and m. Give a reason to justify your answers. m = m = m = VideoTextInteractive Geometry: omplete ourse

24 Unit V, Unit Test Form ontinued 6. MNOP is a rectangle as shown. Find MO and NP. Give a reason to justify your answers. M 11 N 5 P O MO = NP = 7. uadrilateral FGHI is a rhombus as shown. Find FG, GH, and HI. Give a reason to justify your answers. 6 F G I H FG = GH = HI = 8. uadrilateral STUV is a rhombus as shown. Find m 1, m 2, and m T if m V = 50. Give a reason to justify your answers. S 2 1 T V U m 1 = m 2 = m T = 2014 VideoTextInteractive Geometry: omplete ourse 6

25 Unit V, Unit Test Form ontinued 9. uadrilateral MNP is a rhombus. Find NR if P = 8 and MR = 4. Give a reasons) to justify your answers. M R N P NR = 10. is an isosceles trapezoid. If m = 60, find m m, and m. Give a reasons) to justify your answer. m = m = m = 11. WXYZ is an isosceles trapezoid. If WZ = 12 and WY = 16, find XY and XZ. Give a reasons) to justify your answer. W X Z XY = Y XZ = VideoTextInteractive Geometry: omplete ourse

26 Unit V, Unit Test Form ontinued 12. Is it possible for a trapezoid to have: a) Two right angles? b) Four congruent angles? c) Three congruent sides? d) Three acute angles? e) ases shorter than each leg? Use the diagram to the right for problems 1 and Find the area of kite RSTU, with diagonals of length 1 and 6. rea = R V S U T For problems 1 & Find the area of kite RSTU, if RT = 15 and VU =. rea = 15. The area of a kite is 180 square units. The length of one diagonal is 20. How long is the other diagonal? diagonal = 2014 VideoTextInteractive Geometry: omplete ourse 65

27 Unit V, Unit Test Form ontinued 16. a) Find the area of the kite shown to the right rea = b) If m FG = 8 O, Find m HFG H m HFG = 41 I 12 1 F c) Find m IF. m IF = G 17. Points P and are midpoints of the sides of F, shown to the right. omplete each of the following a) F = 18; P = b) F = 2x 2 7x + 10; P = x 2 9; F = ; P =. P c) P = x + ; F = 1 /x + 16; P = d) P = 18; F = F VideoTextInteractive Geometry: omplete ourse

28 Unit V, Unit Test Form ontinued 18. Find the sum of the measures of the interior angles of a 12-sided polygon. Sum = 19. The sum of the measures of the interior angles of a polygon is 1980 O. How many sides does the polygon have? 20. Find the measure of each angle of a regular 15-gon. 21. The measure of an exterior angle of a regular polygon is 18 O. How many sides does the polygon have? 2014 VideoTextInteractive Geometry: omplete ourse 67

29 Unit V, Unit Test Form ontinued 22. Find the area of each of the following labeled polygonal regions using the appropriate postulate, theorem, or corollary. Note: figures which appear to be regular are regular) a) rea = b) rea = Triangle) Trapezoid) c) rea = d) 11 rea = 60 o 4 Rhombus) 5 6 Paralleloram) 8 e) rea = f) rea = 4 Rectangle) 10 Regular Triangle) g) rea = h) rea = Regular Pentagon) 5 4 Square) VideoTextInteractive Geometry: omplete ourse

30 Unit V, Unit Test Form ontinued i) rea = j) rea = 6 6 Regular Hexagon) k) rea = l) rea = Trapezoid) 10 0 o Rhombus) m) rea = rea = 42; with median ) 2. In the figures below, ~ F; rea = 15 units. Find the area of F = 2 5 F 2014 VideoTextInteractive Geometry: omplete ourse 69

31 Unit V, Unit Test Form ontinued 24. omplete a schedule for a round robin tournament with 5 teams. Week 1 Week 2 Week Week 4 Week VideoTextInteractive Geometry: omplete ourse

32 Unit V, Unit Test Form ontinued 10y 4 F For problems 9 and 10, refer to parallelogram FGH shown to the right. 9y 9 H 7y + 5 G 9. HG =. a) 18 b) 26 c) 54 d) 10. FG =. a) 26 b) c) 54 d) The area of a trapezoid with bases 20 and 40 and height 18 is. a) 1080 b) 800 c) 540 d) The area of a regular octagon with side 2 and apothem 1+ 2is. a) 64 2 b) c) d) VideoTextInteractive Geometry: omplete ourse 7

33 UnitV, Unit Test Form ontinued 21. If is a parallelogram named in standard notation, which of the following must always be true? a) b) c) m + m = 180 d) e) d) ll of these 22. Find the area of the regular pentagon shown to the right. 8 rea = Note: P is the center of the pentagon) P 2. If four angles of a pentagon have measures of 105 O, 75 O, 145 O, and 10 O, then the measure of the fifth angle is? a) 95 O b) 80 O c) 100 O d) 85 O e) 145 O 24. The area of the parallelogram shown to the right is a) b) c) 45 d) 90 S P h 6 T 9 R 60 O VideoTextInteractive Geometry: omplete ourse

34 Unit VI, Part, Lessons 1,2&, uiz Form ontinued 2. Use the figure to the right to complete the following statements. In the figure, JT is tangent to at point T. T a) If T = 6 and J = 10, then JT = K J b) If T = 8 and JT = 15, then J = c) If m JT = 60 and T = 6, then J = d) If JK = 9 and K = 8, then JT = VideoTextInteractive Geometry: omplete ourse

35 Unit VI, Part, Lessons 4&5, uiz Form ontinued m 1 = 1 95 x m 1 = 7. m 1 = 8. y = 70 y m 2 = x = x 9. y x = 10. m 1 = x y = x = 88 1 y = F x 24 y F VideoTextInteractive Geometry: omplete ourse

36 Unit VI, Part, Lessons 1,2&, uiz Form ontinued 5. Find in, = 6. Find m in, m = if = 10 and = 9. if m = 96 O 7. Find in. = 8. Find and in. = = x 4 x Find in, given = 10. Find in. = that = 16, = 9 and = x VideoTextInteractive Geometry: omplete ourse

37 Unit VI, Part, Lessons 4,5,6&7, uiz Form ontinued. Find x in. x = 4. Find m in. m = 94 2 x -x Find in. = 6. Find and in. = 12 x+0 x = x+2 x+6 7. Find and in. = 8. and are m = 0.5x = tangents to, and m = 42 O. Find m. 0.4x VideoTextInteractive Geometry: omplete ourse

38 uiz Form Unit VI - ircles Part - ircle oncurrency Lesson 1 - Theorem 8 - If you have a triangle, then that triangle is cyclic. Lesson 2 - Theorem 84 - If the opposite angles of a quadrilateral are supplementary, then the quadrilateral is cyclic. 1. uadrilateral is cyclic. Find x and y. x = x y y = 2. uadrilateral Kite) is cyclic. Find m. m = VideoTextInteractive Geometry: omplete ourse 115

39 Unit VI, Part, Lessons 1&2, uiz Form ontinued. Given: uadrilateral XYWZ is cyclic. ZY is a diameter of. XY WZ Prove: XYZ WZY X Z Y W STTMNT RSON Z 4. The angle bisectors of the angles of XYZ meet at point. X = 75 and = 20. Find. xplain your answer. X W = omplete the following statements by choosing sometimes, always, or never. 5. Rectangles are cyclic quadrilaterals. 6. Irregular quadrilaterals are cyclic. 7. Regular polygons are cyclic. 8. kite is a cyclic quadrilateral. 9. Opposite angles of a cyclic quadrilateral add up to 180 degrees. 10. Isosceles trapezoids are cyclic quadrilaterals VideoTextInteractive Geometry: omplete ourse

40 UnitVI, Unit Test Form ontinued etermine whether each of the following is always, sometimes, or never true. 1. ongruent chords of different circles intercept congruent arcs. 14. n angle inscribed in a semicircle is a right angle. 15. Two circles are congruent if their radii are congruent. 16. Two externally tangent circles have only two common tangents. 17. radius is a segment that joins two points on a circle. 18. polygon inscribed in a circle is a regular polygon. 19. secant is a line that lies in the plane of a circle, and contains a chord of the circle. 20. The opposite angles of an inscribed quadrilateral are supplementary. 21. If point X is on, then mx + mx = mx. 22. The common tangent segments of two circles of unequal radii are congruent. 2. Tangent segments from an external point to two different circles are congruent. 24. yclic quadrilaterals are congruent. 25. If two circles are internally tangent, then the circles have three common tangents VideoTextInteractive Geometry: omplete ourse

41 UnitVI, Unit Test Form ontinued Use the given figure to answer problems 26 to 5. Note: is tangent to at point ) F G 26. If mf = 96, find m F. 27. If m = 62 and m GF = 110, find mf. m F = mf = 28. If mf = 96 and m = 40, find m F. 29. If mf = 170 and m = 110, find m. m F = m = 0. Find m. 1. If m = 26, find m. m = m = 2014 VideoTextInteractive Geometry: omplete ourse 121

42 UnitVI, Unit Test Form ontinued 2. If m = 26, find m F.. F. F G m F = 4. If m F = 18 and m = 80, find mf. 5. If m F = 90, find mf. mf = mf = For problems 6 to 41, find the value of x, or the indicated angle. 6. x = 7. x = x x VideoTextInteractive Geometry: omplete ourse

43 UnitVI, Unit Test Form ontinued H Use the figure to the right and the given information to answer problems 26 to 5. I G H is a diameter of and is tangent to at point. F mg = 24 m HG = 76 m = Find m H m H = 27. Find m F. m F = 28. Find m F m F = 29. Find m H m H = 0. Find m m = 1. Find m F m F = 2014 VideoTextInteractive Geometry: omplete ourse 127