High School Geometry Past Paper PDF

7 Quadrilaterals and Other Polygons 7.1 Angles of Polygons 7.2 Properties of Parallelograms 7.3 Proving That a Quadrilateral Is a Parallelogram 7.4 Pr...

7 Quadrilaterals and Other Polygons 7.1 Angles of Polygons 7.2 Properties of Parallelograms 7.3 Proving That a Quadrilateral Is a Parallelogram 7.4 Properties of Special Parallelograms 7.5 Properties of Trapezoids and Kites SEE the Big Idea Di d ((p. Diamond p. 40 406) 6) Window Wi d ((p. 395) (p. 377) Amusement Park Ride (p Arrow (p. 373) Gazebo (p. 365) hs_geo_pe_07co.indd 356 1/19/15 11:41 AM  Maintaining Mathematical Proficiency Using Structure to Solve a Multi-Step Equation Example 1 Solve 3(2 + x) = −9 by interpreting the expression 2 + x as a single quantity. 3(2 + x) = −9 Write the equation. 3(2 + x) −9 —=— Divide each side by 3. 3 3 2 + x = −3 Simplify. −2 −2 Subtract 2 from each side. x = −5 Simplify. Solve the equation by interpreting the expression in parentheses as a single quantity. 1. 4(7 − x) = 16 2. 7(1 − x) + 2 = −19 3. 3(x − 5) + 8(x − 5) = 22 Identifying Parallel and Perpendicular Lines Example 2 Determine which of the lines are parallel and which are perpendicular. a y c b Find the slope of each line. (−4, 3) d 3 − (−3) (2, 2) Line a: m = — = −3 2 (3, 2) −4 − (−2) −1 − (−4) Line b: m = — = −3 −2 (1, −1) 4 x 1−2 (−4, 0) (4, −2) −2 2 − (−2) Line c: m = — = −4 3−4 (−2, −3) (2, −4) −4 2−0 1 Line d: m=—=— 2 − (−4) 3 Because lines a and b have the same slope, lines a and b are parallel. Because 1 —3 (−3) = −1, lines a and d are perpendicular and lines b and d are perpendicular. Determine which lines are parallel and which are perpendicular. 4. a y d 5. c y b 6. (−4, 4) b y c 4 4 4 (3, 4) (−2, 4) a (0, 3) (−4, 2) (−2, 2) d 2 2 (−2, 2) (4, −2) d (0, 1) (3, 1) c (−3, 0) (1, 0) (2, 1) (3, 1) −4 x 1 3 x −4 2 4 x (−3, −3) b −2 −2 (−3, −2) −2 (2, −3) (−3, −2) (0, −4) (0, −3) (−2, −2) a (3, −3) (4, −4) (4, −4) 7. ABSTRACT REASONING Explain why interpreting an expression as a single quantity does not contradict the order of operations. Dynamic Solutions available at BigIdeasMath.com 357 hs_geo_pe_07co.indd 357 1/19/15 11:41 AM  Mathematical Practices Mathematically proficient students use diagrams to show relationships. Mapping Relationships Core Concept Classifications of Quadrilaterals quadrilaterals parallelograms trapezoids rhombuses squares rectangles kites Writing Statements about Quadrilaterals Use the Venn diagram above to write three true statements about different types of quadrilaterals. SOLUTION Here are three true statements that can be made about the relationships shown in the Venn diagram. All rhombuses are parallelograms. Some rhombuses are rectangles. No trapezoids are parallelograms. Monitoring Progress Use the Venn diagram above to decide whether each statement is true or false. Explain your reasoning. 1. Some trapezoids are kites. 2. No kites are parallelograms. 3. All parallelograms are rectangles. 4. Some quadrilaterals are squares. 5. Example 1 lists three true statements based on the Venn diagram above. Write six more true statements based on the Venn diagram. 6. A cyclic quadrilateral is a quadrilateral that can be circumscribed by a circle so that the circle touches each vertex. Redraw the Venn diagram so that it includes cyclic quadrilaterals. 358 Chapter 7 Quadrilaterals and Other Polygons hs_geo_pe_07co.indd 358 1/19/15 11:41 AM  7.1 Angles of Polygons Essential Question What is the sum of the measures of the interior angles of a polygon? The Sum of the Angle Measures of a Polygon Work with a partner. Use dynamic geometry software. a. Draw a quadrilateral and a pentagon. Find the sum of the measures of the interior angles of each polygon. Sample B G F A C H E I D b. Draw other polygons and find the sums of the measures of their interior angles. Record your results in the table below. CONSTRUCTING Number of sides, n 3 4 5 6 7 8 9 VIABLE ARGUMENTS To be proficient in math, Sum of angle measures, S you need to reason c. Plot the data from your table in a coordinate plane. inductively about data. d. Write a function that fits the data. Explain what the function represents. Measure of One Angle in a Regular Polygon Work with a partner. a. Use the function you found in Exploration 1 to write a new function that gives the measure of one interior angle in a regular polygon with n sides. b. Use the function in part (a) to find the measure of one interior angle of a regular pentagon. Use dynamic geometry software to check your result by constructing a regular pentagon and finding the measure of one of its interior angles. c. Copy your table from Exploration 1 and add a row for the measure of one interior angle in a regular polygon with n sides. Complete the table. Use dynamic geometry software to check your results. Communicate Your Answer 3. What is the sum of the measures of the interior angles of a polygon? 4. Find the measure of one interior angle in a regular dodecagon (a polygon with 12 sides). Section 7.1 Angles of Polygons 359 hs_geo_pe_0701.indd 359 1/19/15 11:50 AM  7.1 Lesson What You Will Learn Use the interior angle measures of polygons. Use the exterior angle measures of polygons. Core Vocabul Vocabulary larry diagonal, p. 360 Using Interior Angle Measures of Polygons equilateral polygon, p. 361 In a polygon, two vertices that are endpoints of Polygon ABCDE equiangular polygon, p. 361 the same side are called consecutive vertices. regular polygon, p. 361 A diagonal of a polygon is a segment that C Previous joins two nonconsecutive vertices. B D polygon As you can see, the diagonals from one vertex convex diagonals divide a polygon into triangles. Dividing a interior angles polygon with n sides into (n − 2) triangles A E exterior angles shows that the sum of the measures of the A and B are consecutive vertices. interior angles of a polygon is a multiple Vertex B has two diagonals, — BD and — BE. of 180°. Theorem Theorem 7.1 Polygon Interior Angles Theorem The sum of the measures of the interior angles REMEMBER ⋅ of a convex n-gon is (n − 2) 180°. 1 2 3 A polygon is convex when m∠1 + m∠2 +... + m∠n = (n − 2) 180° ⋅ 6 5 4 no line that contains a side of the polygon contains Proof Ex. 42 (for pentagons), p. 365 n=6 a point in the interior of the polygon. Finding the Sum of Angle Measures in a Polygon Find the sum of the measures of the interior angles of the figure. SOLUTION The figure is a convex octagon. It has 8 sides. Use the Polygon Interior Angles Theorem. ⋅ (n − 2) 180° = (8 − 2) 180°⋅ Substitute 8 for n. ⋅ = 6 180° Subtract. = 1080° Multiply. The sum of the measures of the interior angles of the figure is 1080°. Monitoring Progress Help in English and Spanish at BigIdeasMath.com 1. The coin shown is in the shape of an 11-gon. Find the sum of the measures of the interior angles. 360 Chapter 7 Quadrilaterals and Other Polygons hs_geo_pe_0701.indd 360 1/19/15 11:50 AM  Finding the Number of Sides of a Polygon The sum of the measures of the interior angles of a convex polygon is 900°. Classify the polygon by the number of sides. SOLUTION Use the Polygon Interior Angles Theorem to write an equation involving the number of sides n. Then solve the equation to find the number of sides. ⋅ (n − 2) 180° = 900° Polygon Interior Angles Theorem n−2=5 Divide each side by 180°. n=7 Add 2 to each side. The polygon has 7 sides. It is a heptagon. Corollary Corollary 7.1 Corollary to the Polygon Interior Angles Theorem The sum of the measures of the interior angles of a quadrilateral is 360°. Proof Ex. 43, p. 366 Finding an Unknown Interior Angle Measure 108° 121° Find the value of x in the diagram. SOLUTION x° 59° The polygon is a quadrilateral. Use the Corollary to the Polygon Interior Angles Theorem to write an equation involving x. Then solve the equation. x° + 108° + 121° + 59° = 360° Corollary to the Polygon Interior Angles Theorem x + 288 = 360 Combine like terms. x = 72 Subtract 288 from each side. The value of x is 72. Monitoring Progress Help in English and Spanish at BigIdeasMath.com 2. The sum of the measures of the interior angles of a convex polygon is 1440°. Classify the polygon by the number of sides. 3. The measures of the interior angles of a quadrilateral are x°, 3x°, 5x°, and 7x°. Find the measures of all the interior angles. In an equilateral polygon, In an equiangular A regular polygon is all sides are congruent. polygon, all angles in the a convex polygon that interior of the polygon are is both equilateral and congruent. equiangular. Section 7.1 Angles of Polygons 361 hs_geo_pe_0701.indd 361 1/19/15 11:50 AM  Finding Angle Measures in Polygons A home plate for a baseball field is shown. A B a. Is the polygon regular? Explain your reasoning. b. Find the measures of ∠C and ∠E. E C SOLUTION D a. The polygon is not equilateral or equiangular. So, the polygon is not regular. b. Find the sum of the measures of the interior angles. ⋅ ⋅ (n − 2) 180° = (5 − 2) 180° = 540° Polygon Interior Angles Theorem Then write an equation involving x and solve the equation. x° + x° + 90° + 90° + 90° = 540° Write an equation. 2x + 270 = 540 Combine like terms. x = 135 Solve for x. So, m∠C = m∠E = 135°. Q Monitoring Progress Help in English and Spanish at BigIdeasMath.com P R 156° 93° 85° 4. Find m∠S and m∠ T in the diagram. 5. Sketch a pentagon that is equilateral but not equiangular. T S Using Exterior Angle Measures of Polygons Unlike the sum of the interior angle measures of a convex polygon, the sum of the exterior angle measures does not depend on the number of sides of the polygon. The diagrams suggest that the sum of the measures of the exterior angles, one angle at each vertex, of a pentagon is 360°. In general, this sum is 360° for any convex polygon. 2 1 360° 5 1 5 1 5 3 4 2 4 2 4 3 JUSTIFYING STEPS 3 To help justify this Step 1 Shade one Step 2 Cut out the Step 3 Arrange the conclusion, you can exterior angle exterior angles. exterior angles visualize a circle containing at each vertex. to form 360°. two straight angles. So, there are 180° + 180°, or 360°, in a circle. Theorem Theorem 7.2 Polygon Exterior Angles Theorem 180° 2 The sum of the measures of the exterior angles of a 3 convex polygon, one angle at each vertex, is 360°. 180° 4 m∠1 + m∠2 + · · · + m∠n = 360° 1 5 Proof Ex. 51, p. 366 n=5 362 Chapter 7 Quadrilaterals and Other Polygons hs_geo_pe_0701.indd 362 1/19/15 11:50 AM  Finding an Unknown Exterior Angle Measure Find the value of x in the diagram. 89° 2x° 67° x° SOLUTION Use the Polygon Exterior Angles Theorem to write and solve an equation. x° + 2x° + 89° + 67° = 360° Polygon Exterior Angles Theorem 3x + 156 = 360 Combine like terms. x = 68 Solve for x. The value of x is 68. REMEMBER A dodecagon is a Finding Angle Measures in Regular Polygons polygon with 12 sides and 12 vertices. The trampoline shown is shaped like a regular dodecagon. a. Find the measure of each interior angle. b. Find the measure of each exterior angle. SOLUTION a. Use the Polygon Interior Angles Theorem to find the sum of the measures of the interior angles. ⋅ (n − 2) 180° = (12 − 2) 180° ⋅ = 1800° Then find the measure of one interior angle. A regular dodecagon has 12 congruent interior angles. Divide 1800° by 12. 1800° — = 150° 12 The measure of each interior angle in the dodecagon is 150°. b. By the Polygon Exterior Angles Theorem, the sum of the measures of the exterior angles, one angle at each vertex, is 360°. Divide 360° by 12 to find the measure of one of the 12 congruent exterior angles. 360° — = 30° 12 The measure of each exterior angle in the dodecagon is 30°. Monitoring Progress Help in English and Spanish at BigIdeasMath.com 6. A convex hexagon has exterior angles with measures 34°, 49°, 58°, 67°, and 75°. What is the measure of an exterior angle at the sixth vertex? 7. An interior angle and an adjacent exterior angle of a polygon form a linear pair. How can you use this fact as another method to find the measure of each exterior angle in Example 6? Section 7.1 Angles of Polygons 363 hs_geo_pe_0701.indd 363 1/19/15 11:50 AM  7.1 Exercises Dynamic Solutions available at BigIdeasMath.com Vocabulary and Core Concept Check 1. VOCABULARY Why do vertices connected by a diagonal of a polygon have to be nonconsecutive? 2. WHICH ONE DOESN’T BELONG? Which sum does not belong with the other three? Explain your reasoning. the sum of the measures of the interior the sum of the measures of the exterior angles of a quadrilateral angles of a quadrilateral the sum of the measures of the interior the sum of the measures of the exterior angles of a pentagon angles of a pentagon Monitoring Progress and Modeling with Mathematics In Exercises 3–6, find the sum of the measures of the In Exercises 15–18, find the value of x. interior angles of the indicated convex polygon. 15. 16. (See Example 1.) 102° 146° 140° 3. nonagon 4. 14-gon 120° 86° 138° 5. 16-gon 6. 20-gon 124° x° 158° x° 59° In Exercises 7–10, the sum of the measures of the interior angles of a convex polygon is given. Classify the 17. 96° 18. 121° 2x° 152° polygon by the number of sides. (See Example 2.) 143° 116° x° x° 125° 7. 720° 8. 1080° 101° 139° 140° 162° 9. 2520° 10. 3240° In Exercises 19 –22, find the measures of ∠X and ∠Y. In Exercises 11–14, find the value of x. (See Example 3.) (See Example 4.) 11. X 12. H J 19. X Z 20. 100° Y 103° 133° 164° V W Y 130° 102° 47° x° 66° Z x° 58° W Z G K 119° Y X W V 13. M 14. B x° 92° 21. 22. W L Z Y 154° A V Y 110° x° 171° 99° 88° K 29° V 100° 149° U 101° 68° X N 159° D C 91° W U X Z 364 Chapter 7 Quadrilaterals and Other Polygons hs_geo_pe_0701.indd 364 1/19/15 11:51 AM  In Exercises 23–26, find the value of x. (See Example 5.) 34. MODELING WITH MATHEMATICS The floor of the gazebo shown is shaped like a regular decagon. 23. 24. x° Find the measure of each interior angle of the 48° 50° regular decagon. Then find the measure of each 65° 59° exterior angle. 39° 78° 106° 58° x° x° 25. 26. 85° 40° x° 71° 45° 44° 2x° 77° 3x° 2x° 35. WRITING A FORMULA Write a formula to find the number of sides n in a regular polygon given that the measure of one interior angle is x°. In Exercises 27–30, find the measure of each interior angle and each exterior angle of the indicated regular 36. WRITING A FORMULA Write a formula to find the polygon. (See Example 6.) number of sides n in a regular polygon given that the 27. pentagon 28. 18-gon measure of one exterior angle is x°. 29. 45-gon 30. 90-gon REASONING In Exercises 37– 40, find the number of sides for the regular polygon described. ERROR ANALYSIS In Exercises 31 and 32, describe and 37. Each interior angle has a measure of 156°. correct the error in finding the measure of one exterior angle of a regular pentagon. 38. Each interior angle has a measure of 165°. ✗ 31. ⋅ (n − 2) 180° = (5 − 2) 180° ⋅ 39. Each exterior angle has a measure of 9°. ⋅ = 3 180° 40. Each exterior angle has a measure of 6°. = 540° 41. DRAWING CONCLUSIONS Which of the following The sum of the measures of the angle measures are possible interior angle measures angles is 540°. There are five of a regular polygon? Explain your reasoning. Select angles, so the measure of one 540° all that apply. exterior angle is —5 = 108°. A 162° ○ B 171° ○ C 75° ○ D 40° ○ ✗ 32. There are a total of 10 exterior 42. PROVING A THEOREM The Polygon Interior Angles angles, two at each vertex, so the Theorem (Theorem 7.1) states that the sum of the measure of one exterior angle is measures of the interior angles of a convex n-gon is 360° —10 = 36°. ⋅ (n − 2) 180°. Write a paragraph proof of this theorem for the case when n = 5. A 33. MODELING WITH MATHEMATICS The base of a jewelry box is shaped like a regular hexagon. What E B is the measure of each interior angle of the jewelry box base? D C Section 7.1 Angles of Polygons 365 hs_geo_pe_0701.indd 365 1/19/15 11:51 AM  43. PROVING A COROLLARY Write a paragraph proof of 49. MULTIPLE REPRESENTATIONS The formula for the the Corollary to the Polygon Interior Angles Theorem measure of each interior angle in a regular polygon (Corollary 7.1). can be written in function notation. a. Write a function h(n), where n is the number of 44. MAKING AN ARGUMENT Your friend claims that to sides in a regular polygon and h(n) is the measure find the interior angle measures of a regular polygon, of any interior angle in the regular polygon. you do not have to use the Polygon Interior Angles Theorem (Theorem 7.1). You instead can use the b. Use the function to find h(9). Polygon Exterior Angles Theorem (Theorem 7.2) and c. Use the function to find n when h(n) = 150°. then the Linear Pair Postulate (Postulate 2.8). Is your friend correct? Explain your reasoning. d. Plot the points for n = 3, 4, 5, 6, 7, and 8. What happens to the value of h(n) as n gets larger? 45. MATHEMATICAL CONNECTIONS In an equilateral hexagon, four of the exterior angles each have a measure of x°. The other two exterior angles each 50. HOW DO YOU SEE IT? Is the hexagon a regular have a measure of twice the sum of x and 48. Find the hexagon? Explain your reasoning. measure of each exterior angle. 59° 60° 46. THOUGHT PROVOKING For a concave polygon, is it true that at least one of the interior angle measures 60° must be greater than 180°? If not, give an example. 59° If so, explain your reasoning. 61° 61° 47. WRITING EXPRESSIONS Write an expression to find the sum of the measures of the interior angles for a concave polygon. Explain your reasoning. 51. PROVING A THEOREM Write a paragraph proof of the Polygon Exterior Angles Theorem (Theorem 7.2). (Hint: In a convex n-gon, the sum of the measures of an interior angle and an adjacent exterior angle at any vertex is 180°.) 52. ABSTRACT REASONING You are given a convex polygon. You are asked to draw a new polygon by increasing the sum of the interior angle measures by 540°. How many more sides does your new polygon 48. ANALYZING RELATIONSHIPS Polygon ABCDEFGH have? Explain your reasoning. — and CD is a regular octagon. Suppose sides AB — are extended to meet at a point P. Find m∠BPC. Explain your reasoning. Include a diagram with your answer. Maintaining Mathematical Proficiency Reviewing what you learned in previous grades and lessons Find the value of x. (Section 3.2) 53. 54. 55. 56. 113° (3x + 10)° x° (8x − 16)° (6x − 19)° x° (3x + 20)° 79° 366 Chapter 7 Quadrilaterals and Other Polygons hs_geo_pe_0701.indd 366 1/19/15 11:51 AM  7.2 Properties of Parallelograms Essential Question What are the properties of parallelograms? Discovering Properties of Parallelograms Work with a partner. Use dynamic geometry software. a. Construct any parallelogram and label it ABCD. Explain your process. Sample B C A D b. Find the angle measures of the parallelogram. What do you observe? c. Find the side lengths of the parallelogram. What do you observe? d. Repeat parts (a)–(c) for several other parallelograms. Use your results to write conjectures about the angle measures and side lengths of a parallelogram. Discovering a Property of Parallelograms Work with a partner. Use dynamic geometry software. a. Construct any parallelogram and label it ABCD. b. Draw the two diagonals of the parallelogram. Label the point of intersection E. Sample B C E A MAKING SENSE OF PROBLEMS To be proficient in math, D you need to analyze givens, constraints, c. Find the segment lengths AE, BE, CE, and DE. What do you observe? relationships, and goals. d. Repeat parts (a)–(c) for several other parallelograms. Use your results to write a conjecture about the diagonals of a parallelogram. Communicate Your Answer 3. What are the properties of parallelograms? Section 7.2 Properties of Parallelograms 367 hs_geo_pe_0702.indd 367 1/19/15 11:51 AM  7.2 Lesson What You Will Learn Use properties to find side lengths and angles of parallelograms. Use parallelograms in the coordinate plane. Core Vocabul Vocabulary larry parallelogram, p. 368 Using Properties of Parallelograms Previous A parallelogram is a quadrilateral with both pairs Q R quadrilateral —  RS of opposite sides parallel. In ▱PQRS, PQ — and diagonal —  PS QR — by definition. The theorems below describe interior angles other properties of parallelograms. segment bisector P S Theorems Theorem 7.3 Parallelogram Opposite Sides Theorem If a quadrilateral is a parallelogram, then Q R its opposite sides are congruent. — ≅ RS If PQRS is a parallelogram, then PQ — — — and QR ≅ SP. P S Proof p. 368 Theorem 7.4 Parallelogram Opposite Angles Theorem If a quadrilateral is a parallelogram, then Q R its opposite angles are congruent. If PQRS is a parallelogram, then ∠P ≅ ∠R and ∠Q ≅ ∠S. P S Proof Ex. 37, p. 373 Parallelogram Opposite Sides Theorem Q R Given PQRS is a parallelogram. Prove — ≅ RS PQ —, QR — ≅ SP — — to form △PQS and △RSQ. a. Draw diagonal QS Plan P S for Proof b. Use the ASA Congruence Theorem (Thm. 5.10) to show that △PQS ≅ △RSQ. — ≅ RS c. Use congruent triangles to show that PQ — and QR — ≅ SP —. Plan STATEMENTS REASONS in Action 1. PQRS is a parallelogram. 1. Given —. a. 2. Draw QS 2. Through any two points, there exists exactly one line. — —— — 3. PQ  RS , QR  PS 3. Definition of parallelogram b. 4. ∠PQS ≅ ∠RSQ, 4. Alternate Interior Angles Theorem ∠PSQ ≅ ∠RQS (Thm. 3.2) — — 5. QS ≅ SQ 5. Reflexive Property of Congruence (Thm. 2.1) 6. △PQS ≅ △RSQ 6. ASA Congruence Theorem (Thm. 5.10) c. 7. — ≅ RS PQ —, QR — ≅ SP — 7. Corresponding parts of congruent triangles are congruent. 368 Chapter 7 Quadrilaterals and Other Polygons hs_geo_pe_0702.indd 368 1/19/15 11:51 AM  Using Properties of Parallelograms A x+4 B Find the values of x and y. y° SOLUTION 65° ABCD is a parallelogram by the definition of a parallelogram. D 12 C Use the Parallelogram Opposite Sides Theorem to find the value of x. AB = CD Opposite sides of a parallelogram are congruent. x + 4 = 12 Substitute x + 4 for AB and 12 for CD. x=8 Subtract 4 from each side. By the Parallelogram Opposite Angles Theorem, ∠A ≅ ∠C, or m∠A = m∠C. So, y° = 65°. In ▱ABCD, x = 8 and y = 65. Monitoring Progress Help in English and Spanish at BigIdeasMath.com 1. Find FG and m∠G. 2. Find the values of x and y. G H K L 50° 8 18 y+3 60° 2x° F E J M The Consecutive Interior Angles Theorem (Theorem 3.4) states that if two parallel lines are cut by a transversal, then the pairs of consecutive interior angles formed are supplementary. x° A pair of consecutive angles in a parallelogram is like a pair of consecutive interior angles between parallel lines. This y° similarity suggests the Parallelogram Consecutive Angles Theorem. Theorems Theorem 7.5 Parallelogram Consecutive Angles Theorem If a quadrilateral is a parallelogram, then its Q R consecutive angles are supplementary. x° y° If PQRS is a parallelogram, then x° + y° = 180°. y° x° Proof Ex. 38, p. 373 P S Theorem 7.6 Parallelogram Diagonals Theorem If a quadrilateral is a parallelogram, then its Q R diagonals bisect each other. — ≅ SM If PQRS is a parallelogram, then QM — M — ≅ RM and PM —. P S Proof p. 370 Section 7.2 Properties of Parallelograms 369 hs_geo_pe_0702.indd 369 1/19/15 11:51 AM  Parallelogram Diagonals Theorem — and QS Given PQRS is a parallelogram. Diagonals PR — Q R intersect at point M. — and PR Prove M bisects QS —. M P S STATEMENTS REASONS 1. PQRS is a parallelogram. 1. Given — — 2. PQ  RS 2. Definition of a parallelogram 3. ∠QPR ≅ ∠SRP, ∠PQS ≅ ∠RSQ 3. Alternate Interior Angles Theorem (Thm. 3.2) — — 4. PQ ≅ RS 4. Parallelogram Opposite Sides Theorem 5. △PMQ ≅ △RMS 5. ASA Congruence Theorem (Thm. 5.10) — — — — 6. QM ≅ SM , PM ≅ RM 6. Corresponding parts of congruent triangles are congruent. 7. — and PR M bisects QS —. 7. Definition of segment bisector Using Properties of a Parallelogram As shown, part of the extending arm of a desk lamp is a parallelogram. The angles B of the parallelogram change as the lamp is raised and lowered. Find m∠BCD when m∠ADC = 110°. C A SOLUTION D By the Parallelogram Consecutive Angles Theorem, the consecutive angle pairs in ▱ABCD are supplementary. So, m∠ADC + m∠BCD = 180°. Because m∠ADC = 110°, m∠BCD = 180° − 110° = 70°. Writing a Two-Column Proof C Write a two-column proof. E B Given ABCD and GDEF are parallelograms. D F Prove ∠B ≅ ∠F A G STATEMENTS REASONS 1. ABCD and GDEF are 1. Given parallelograms. 2. ∠CDA ≅ ∠B, ∠EDG ≅ ∠F 2. If a quadrilateral is a parallelogram, then its opposite angles are congruent. 3. ∠CDA ≅ ∠EDG 3. Vertical Angles Congruence Theorem (Thm. 2.6) 4. ∠B ≅ ∠F 4. Transitive Property of Congruence (Thm. 2.2) Monitoring Progress Help in English and Spanish at BigIdeasMath.com 3. WHAT IF? In Example 2, find m∠BCD when m∠ADC is twice the measure of ∠BCD. 4. Using the figure and the given statement in Example 3, prove that ∠C and ∠F are supplementary angles. 370 Chapter 7 Quadrilaterals and Other Polygons hs_geo_pe_0702.indd 370 1/19/15 11:51 AM  Using Parallelograms in the Coordinate Plane Using Parallelograms in the Coordinate Plane JUSTIFYING STEPS Find the coordinates of the intersection of the diagonals of ▱LMNO with vertices In Example 4, you can use L(1, 4), M(7, 4), N(6, 0), and O(0, 0). either diagonal to find the coordinates of the SOLUTION intersection. Using diagonal — helps simplify the OM By the Parallelogram Diagonals Theorem, the diagonals of a parallelogram bisect each calculation because — and OM other. So, the coordinates of the intersection are the midpoints of diagonals LN —. one endpoint is (0, 0). —= — 7+0 4+0 coordinates of midpoint of OM 2 ( 7 , — = —, 2 2 2 ) ( ) Midpoint Formula The coordinates of the intersection y ( ) of the diagonals are —72 , 2. You can 4 L M check your answer by graphing ▱LMNO and drawing the diagonals. The point of 2 intersection appears to be correct. O 2 4 N 8 x Using Parallelograms in the Coordinate Plane REMEMBER Three vertices of ▱WXYZ are W(−1, −3), X(−3, 2), and Z(4, −4). Find the When graphing a polygon coordinates of vertex Y. in the coordinate plane, the name of the polygon gives SOLUTION the order of the vertices. Step 1 Graph the vertices W, X, and Z. —. Step 2 Find the slope of WX y X −2 —=— slope of WX 2 – (–3) 5 5 = — = −— 2 −2 –3 – (–1) –2 2 Y(2, 1) Step 3 Start at Z(4, −4). Use the rise and run −4 5 2 x from Step 2 to find vertex Y. 5 A rise of 5 represents a change of 5 units W up. A run of −2 represents a change of −4 Z 2 units left. So, plot the point that is 5 units up and 2 units left from Z(4, −4). The point is (2, 1). Label it as vertex Y. — and WZ Step 4 Find the slopes of XY — to verify that they are parallel. —=— slope of XY 1−2 −1 1 = — = −— —=— slope of WZ −4 − (−3) −1 1 = — = −— 2 − (−3) 5 5 4 − (−1) 5 5 So, the coordinates of vertex Y are (2, 1). Monitoring Progress Help in English and Spanish at BigIdeasMath.com 5. Find the coordinates of the intersection of the diagonals of ▱STUV with vertices S(−2, 3), T(1, 5), U(6, 3), and V(3, 1). 6. Three vertices of ▱ABCD are A(2, 4), B(5, 2), and C(3, −1). Find the coordinates of vertex D. Section 7.2 Properties of Parallelograms 371 hs_geo_pe_0702.indd 371 1/19/15 11:51 AM  7.2 Exercises Dynamic Solutions available at BigIdeasMath.com Vocabulary and Core Concept Check 1. VOCABULARY Why is a parallelogram always a quadrilateral, but a quadrilateral is only sometimes a parallelogram? 2. WRITING You are given one angle measure of a parallelogram. Explain how you can find the other angle measures of the parallelogram. Monitoring Progress and Modeling with Mathematics In Exercises 3–6, find the value of each variable in the In Exercises 17–20, find the value of each variable in the parallelogram. (See Example 1.) parallelogram. 3. y 4. m+1 17. 18. 70° d° c° x n 9 12 15 2m° n° (b − 10)° (b + 10)° 6 5. (d − 21)° 6. (g + 4)° 19. k+4 8 z−8 16 − h 20 7 105° 65° m 11 In Exercises 7 and 8, find the measure of the indicated 20. 5u − 10 angle in the parallelogram. (See Example 2.) 6 2u + 2 7. Find m∠B. 8. Find m∠N. v B 3 C M N 95° ERROR ANALYSIS In Exercises 21 and 22, describe and 51° correct the error in using properties of parallelograms. A D L P ✗ 21. T U In Exercises 9–16, find the indicated measure in ▱LMNQ. Explain your reasoning. 50° 9. LM S V L M 10. LP 8.2 Because quadrilateral STUV is a 100° P 7 8 parallelogram, ∠S ≅ ∠V. So, m∠V = 50°. 11. LQ 29° Q N ✗ 13 22. 12. MQ G H 13. m∠LMN F 14. m∠NQL K J 15. m∠MNQ Because quadrilateral GHJK is — ≅ FH a parallelogram, GF —. 16. m∠LMQ 372 Chapter 7 Quadrilaterals and Other Polygons hs_geo_pe_0702.indd 372 1/19/15 11:51 AM  PROOF In Exercises 23 and 24, write a two-column 34. ATTENDING TO PRECISION ∠J and ∠K are proof. (See Example 3.) consecutive angles in a parallelogram, m∠J = (3x + 7)°, and m∠K = (5x −11)°. 23. Given ABCD and CEFD B C Find the measure of each angle. are parallelograms. — ≅ FE Prove AB — A 35. CONSTRUCTION Construct any parallelogram and D E label it ABCD. Draw diagonals AC— and BD—. Explain how to use paper folding to verify the Parallelogram F Diagonals Theorem (Theorem 7.6) for ▱ABCD. 24. Given ABCD, EBGF, and HJKD are parallelograms. 36. MODELING WITH MATHEMATICS The feathers on Prove ∠2 ≅ ∠3 an arrow form two congruent parallelograms. The A E B parallelograms are reflections of each other over the 1 line that contains their shared side. Show that 2 m∠2 = 2m∠1. F G H J 3 4 1 2 D K C 37. PROVING A THEOREM Use the diagram to write a In Exercises 25 and 26, find the coordinates of the two-column proof of the Parallelogram Opposite intersection of the diagonals of the parallelogram with Angles Theorem (Theorem 7.4). the given vertices. (See Example 4.) B C 25. W(−2, 5), X(2, 5), Y(4, 0), Z(0, 0) 26. Q(−1, 3), R(5, 2), S(1, −2), T(−5, −1) A D In Exercises 27–30, three vertices of ▱DEFG are Given ABCD is a parallelogram. given. Find the coordinates of the remaining vertex. (See Example 5.) Prove ∠A ≅ ∠C, ∠B ≅ ∠D 27. D(0, 2), E(−1, 5), G(4, 0) 38. PROVING A THEOREM Use the diagram to write a two-column proof of the Parallelogram Consecutive 28. D(−2, −4), F(0, 7), G(1, 0) Angles Theorem (Theorem 7.5). 29. D(−4, −2), E(−3, 1), F(3, 3) Q R x° y° 30. E(−1, 4), F(5, 6), G(8, 0) y° x° MATHEMATICAL CONNECTIONS In Exercises 31 and 32, P S find the measure of each angle. Given PQRS is a parallelogram. 31. The measure of one interior angle of a parallelogram Prove x° + y° = 180° is 0.25 times the measure of another angle. 39. PROBLEM SOLVING The sides of ▱MNPQ are 32. The measure of one interior angle of a parallelogram represented by the expressions below. Sketch is 50 degrees more than 4 times the measure of ▱MNPQ and find its perimeter. another angle. MQ = −2x + 37 QP = y + 14 33. MAKING AN ARGUMENT In quadrilateral ABCD, NP = x − 5 MN = 4y + 5 m∠B = 124°, m∠A = 56°, and m∠C = 124°. Your friend claims quadrilateral ABCD could be 40. PROBLEM SOLVING In ▱LMNP, the ratio of a parallelogram. Is your friend correct? Explain LM to MN is 4 : 3. Find LM when the perimeter your reasoning. of ▱LMNP is 28. Section 7.2 Properties of Parallelograms 373 hs_geo_pe_0702.indd 373 1/19/15 11:52 AM  41. ABSTRACT REASONING Can you prove that two 44. THOUGHT PROVOKING Is it possible that any triangle parallelograms are congruent by proving that all can be partitioned into four congruent triangles that their corresponding sides are congruent? Explain can be rearranged to form a parallelogram? Explain your reasoning. your reasoning. 42. HOW DO YOU SEE IT? The mirror shown is attached 45. CRITICAL THINKING Points W(1, 2), X(3, 6), and to the wall by an arm that can extend away from Y(6, 4) are three vertices of a parallelogram. How the wall. In the figure, points P, Q, R, and S are the many parallelograms can be created using these three vertices of a parallelogram. This parallelogram is vertices? Find the coordinates of each point that could one of several that change shape as the mirror be the fourth vertex. is extended. — 46. PROOF In the diagram, EK bisects ∠FEH, and FJ — — — bisects ∠EFG. Prove that EK ⊥ FJ. (Hint: Write equations using the angle measures of the triangles and quadrilaterals formed.) E F Q P R P S H J K G a. What happens to m∠P as m∠Q increases? Explain. 47. PROOF Prove the Congruent Parts of Parallel Lines Corollary: If three or more parallel lines cut off b. What happens to QS as m∠Q decreases? Explain. congruent segments on one transversal, then they cut c. What happens to the overall distance between off congruent segments on every transversal. the mirror and the wall when m∠Q decreases? Explain. G P H J Q K 43. MATHEMATICAL CONNECTIONS In ▱STUV, m∠TSU = 32°, m∠USV = (x 2)°, m∠TUV = 12x°, L M and ∠TUV is an acute angle. Find m∠USV. T U ⃖ ⃗  ⃖JK⃗  LM Given GH — ≅ JL ⃖ ⃗, GJ — — ≅ KM Prove HK — S V — and MQ (Hint: Draw KP — such that quadrilateral GPKJ and quadrilateral JQML are parallelograms.) Maintaining Mathematical Proficiency Reviewing what you learned in previous grades and lessons Determine whether lines ℓ and m are parallel. Explain your reasoning. (Section 3.3) 48. m 49. 50. 132° m 81° 81° 58° m 374 Chapter 7 Quadrilaterals and Other Polygons hs_geo_pe_0702.indd 374 1/19/15 11:52 AM  7.3 Proving That a Quadrilateral Is a Parallelogram Essential Question How can you prove that a quadrilateral is a parallelogram? Proving That a Quadrilateral Is a Parallelogram Work with a partner. C 4 Sample Use dynamic geometry software. Points