HPTER 6 Vocabulary The table contains important vocabulary terms from hapter 6. s you work through the chapter, fill in the page number, definition, and a clarifying example. Term Page efinition larifying Example base angle of a trapezoid base of a trapezoid concave (polygon) convex (polygon) diagonal (of a polygon) isosceles trapezoid kite leg of a trapezoid 124 Geometry
HPTER 6 Vocabulary The table contains important vocabulary terms from hapter 6. s you work through the chapter, fill in the page number, definition, and a clarifying example. Term Page efinition larifying Example base angle of a trapezoid 429 One of a pair of consecutive angles whose common side is a base of the trapezoid. ase angles ase angles base of a trapezoid 429 One of the two parallel sides of the trapezoid. ase ase concave (polygon) 383 polygon in which a diagonal can be drawn such that part of the diagonal contains points in the exterior of the polygon. diagonal convex (polygon) 383 polygon in which no diagonal contains points in the exterior of the polygon. diagonal diagonal (of a polygon) isosceles trapezoid 382 429 segment connecting two nonconsecutive vertices of a polygon. trapezoid in which the legs are congruent. E iagonal kite 427 quadrilateral with exactly two pairs of congruent consecutive sides. leg of a trapezoid 429 One of the two nonparallel sides of the trapezoid. leg leg 124 Geometry
HPTER 6 VOULRY ONTINUE Term Page efinition larifying Example midsegment of a trapezoid parallelogram rectangle regular polygon rhombus side of a polygon square trapezoid vertex of a polygon 125 Geometry
HPTER 6 VOULRY ONTINUE Term Page efinition larifying Example midsegment of a trapezoid 431 The segment whose endpoints are the midpoints of the legs of the trapezoid. midsegment parallelogram 391 quadrilateral with two pairs of parallel sides. rectangle 408 quadrilateral with four right angles. regular polygon 382 polygon that is both equilateral and equiangular. rhombus 409 quadrilateral with four congruent sides. side of a polygon 382 One of the segments that form a polygon. Side E square 410 quadrilateral with four congruent sides and four right angles. trapezoid 429 quadrilateral with exactly one pair of parallel sides. vertex of a polygon 382 The intersection of two sides of the polygon. Vertex E 125 Geometry
HPTER 6 hapter Review 6-1 Properties and ttributes of Polygons Tell whether each figure is a polygon. If it is a polygon, name it by the number of its sides. 1. 2. 3. 4. 5. Find the sum of the interior angle measures of a convex 24-gon. 6. The surface of a trampoline is in the shape of a regular octagon. Find the measure of each interior angle of the trampoline. 7. flower garden is surrounded by paths as shown. Find the measure of each exterior angle of the flower garden. (7z 5) (5z + 10) 4z (4z + 15) 8. Find the measure of each exterior angle of a regular hexagon. 140 Geometry
HPTER 6 hapter Review 6-1 Properties and ttributes of Polygons Tell whether each figure is a polygon. If it is a polygon, name it by the number of its sides. 1. 2. Yes; hexagon No 3. 4. Yes; quadrilateral No 5. Find the sum of the interior angle measures of a convex 24-gon. 3960 6. The surface of a trampoline is in the shape of a regular octagon. Find the measure of each interior angle of the trampoline. 135 7. flower garden is surrounded by paths as shown. Find the measure of each exterior angle of the flower garden. 95, 68, 114, 83 (7z 5) (5z + 10) 4z (4z + 15) 8. Find the measure of each exterior angle of a regular hexagon. 60 140 Geometry
HPTER 6 REVIEW ONTINUE 6-2 Properties of Parallelograms In parallelogram, 20, E 9, and m 50. Find each measure. 9. 10. 11. E E 12. m 13. m 14. m 15. Three vertices of EFG are E( 1, 3), F(4, 1), and G( 2, 1). Find the coordinates of vertex. PQRS is a parallelogram. Find each measure. 16. PS 17. QR 18. m P 19. m S 6-3 onditions for Parallelograms 20. Show that is a parallelogram for x 3 and y 9. S R (8x 7) 7y + 3 (3x + 22) P 4y + 15 Q 6y 30 10x + 11 8x + 17 y + 15 141 Geometry
HPTER 6 REVIEW ONTINUE 6-2 Properties of Parallelograms In parallelogram, 20, E 9, and m 50. Find each measure. 9. 10. 11. E E 18 20 9 12. m 13. m 14. m 130 50 130 15. Three vertices of EFG are E( 1, 3), F(4, 1), and G( 2, 1). Find the coordinates of vertex. ( 7, 1) PQRS is a parallelogram. Find each measure. 16. PS 31 17. QR 18. m P 113 19. m S 6-3 onditions for Parallelograms 20. Show that is a parallelogram for x 3 and y 9. 10x + 11 6y 30 8x + 17 31 67 S 7y + 3 (8x 7) R (3x + 22) P 10x 11 8x 17 10(3) 11 8(3) 17 41 41 4y + 15 Q y + 15 6y 30 y 15 6(9) 30 9 15 24 24 141 Geometry
HPTER 6 REVIEW ONTINUE 21. Show that JKLM is a parallelogram for a 5 and b 11. M (8a + 13) L (14b 27) J (12a 7) K etermine if each quadrilateral must be a parallelogram. Justify your answer. 22. W X 23. X 24. X Y W Y Z Y Z W Z 25. Show that a quadrilateral with vertices ( 3, 6), ( 5, 0), (4, 3) and (2, 3) is a parallelogram. 142 Geometry
HPTER 6 REVIEW ONTINUE 21. Show that JKLM is a parallelogram for a 5 and b 11. M (8a + 13) L 8a 13 12a 7 8(5) 13 12(3) 7 53 53 m M m K (14b 27) J (12a 7) K m J 14b 27 14(11) 27 127 m J 127 m L 360 (53 53 127) m L 360 233 127 m J m L etermine if each quadrilateral must be a parallelogram. Justify your answer. 22. W X 23. X 24. X Y W Y Z Y Z W Z No Theorem 6-3-3 (If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.) Theorem 6-3-1 (If one pair of opposite sides of a quadrilateral are parallel and congruent, then the quadrilateral is a parallelogram.) 25. Show that a quadrilateral with vertices ( 3, 6), ( 5, 0), (4, 3) and (2, 3) is a parallelogram. The distance from ( 3, 6) to ( 5, 0) and (4, 3) to (2, 3) are equal. d ( 3 ( 5) 2 (6 0) 2 4 6 3 40 d (4 ) 2 3 2 ( 3)) ( 2 4 6 3 40 The distance from ( 3, 6) to (4, 3) and ( 5, 0) to (2, 3) are equal. d ( 3 4) 2 3) (6 2 49 9 58 d ( 5 2) 2 ( 3)) (0 2 49 9 58 142 Geometry
HPTER 6 REVIEW ONTINUE 6-4 Properties of Special Parallelograms In rectangle QRST, RT 55, and QR 45. Find each length. Q R 26. PR 27. RS P 28. QS 29. QP WXYZ is a rhombus. Find each measure. 30. YZ T S X 5a + 6 Y 31. m WXZ and m XYZ if m YVZ (7b 27) and m WYZ (3b + 7). 9a 10 V W Z 143 Geometry
HPTER 6 REVIEW ONTINUE 6-4 Properties of Special Parallelograms In rectangle QRST, RT 55, and QR 45. Find each length. Q R 26. PR 27.5 27. RS 31.6 P 28. QS 55 29. QP 27.5 T S WXYZ is a rhombus. Find each measure. 30. YZ 26 X 5a + 6 Y 31. m WXZ and m XYZ if m YVZ (7b 27) and m WYZ (3b + 7). 9a 10 V m WXZ 56 and m XYZ 68 W Z 143 Geometry
HPTER 6 REVIEW ONTINUE 32. Given: and WXYZ are congruent rhombi. M is the midpoint of and WX and N is the midpoint of and WZ. M X Y Prove: WMN is a rhombus. W N Z 144 Geometry
HPTER 6 REVIEW ONTINUE 32. Given: and WXYZ are congruent rhombi. M is the midpoint of and WX and N is the midpoint of and WZ. M X Y Prove: WMN is a rhombus. W N Z Statements 1. and WXYZ are congruent rhombi. M is the midpoint of and WX and N is the midpoint of and WZ. 2. WX WZ 3. M M WM MX WX N N WN NZ WZ 4. M M, WM MX N N, WN NZ 5. M M N N WM MX WN NZ 6. M M N N WM WM WN WN 7. M N WM WN 8. WMN is a rhombus Reasons 1. Given 2. ef. of congruent rhombi 3. Segment addition postulate 4. ef. of midpoint 5. Substitution 6. Substitution 7. ivision Prop of Equality 8. efinition of rhombus 144 Geometry
HPTER 6 REVIEW ONTINUE 6-5 onditions for Special Parallelograms etermine if the conclusion is valid. If not, tell what additional information is needed to make it valid. 33. Given: ; onclusion: is a parallelogram. 34. Given: bisects. onclusion: is a rectangle. Use the diagonals to determine whether a parallelogram with the given vertices is a rectangle, rhombus, or square. Give all the names that apply. 35. W( 4, 1), X( 1, 3), Y(3, 0), Z(0, 4) 36. M(0, 1), N(2, 6), P(4, 1), Q(2, 4) 145 Geometry
HPTER 6 REVIEW ONTINUE 6-5 onditions for Special Parallelograms etermine if the conclusion is valid. If not, tell what additional information is needed to make it valid. 33. Given: ; onclusion: is a parallelogram. No; or 34. Given: bisects. onclusion: is a rectangle. No; bisects, Use the diagonals to determine whether a parallelogram with the given vertices is a rectangle, rhombus, or square. Give all the names that apply. 35. W( 4, 1), X( 1, 3), Y(3, 0), Z(0, 4) rectangle, rhombus, square 36. M(0, 1), N(2, 6), P(4, 1), Q(2, 4) rhombus 145 Geometry
HPTER 6 REVIEW ONTINUE 37. Given: WXYZ is a rhombus; E, F, G and H are midpoints; m FEH 90 Prove: EFGH is a rectangle. X F E Y G H Z W 6-6 Properties of Kites and Trazezoids In kite MNPQ, m NQP 70 and m NMQ 60. Find each measure. 38. m NMR 39. m MQR M N Q R P 40. m NPR 41. m MQP S 10 T 42. Find MN M N R 18 U 146 Geometry
HPTER 6 REVIEW ONTINUE 37. Given: WXYZ is a rhombus; E, F, G and H are midpoints; m FEH 90 Prove: EFGH is a rectangle. X F E Y G H Z W Statements 1. WXYZ is a rhombus, E, F, G, and H are midpoints, m FEH 90 2. m Y m W m X m Z 3. XF FY; YG GZ ZH HW; WE EX 4. YGF WEH XFE ZHG 5. FG HE GH EF 6. EFGH is a rectangle Reasons 1. Given 2. ef. of rhombus 3. ef. of midpoint 4. SS 5. PT 6. ef. of rectangle 6-6 Properties of Kites and Trazezoids In kite MNPQ, m NQP 70 and m NMQ 60. Find each measure. 38. m NMR 30 39. m MQR 60 M N Q R P 40. m NPR 20 41. m MQP S 10 T 42. Find MN M 14 R 18 N 130 U 146 Geometry
HPTER 6 Postulates and Theorems Theorem 6-1-1 Theorem 6-1-2 Theorem 6-2-1 Theorem 6-2-2 Theorem 6-2-3 Theorem 6-2-4 Theorem 6-3-1 Theorem 6-3-2 Theorem 6-3-3 Theorem 6-3-4 Theorem 6-3-5 Theorem 6-4-1 Theorem 6-4-2 Theorem 6-4-3 Theorem 6-4-4 Theorem 6-4-5 Theorem 6-5-1 (Polygon ngle Sum Theorem) The sum of the interior angle measures of a convex polygon with n sides is (n 2)180. (Polygon Exterior ngle Sum Theorem) The sum of the exterior angle measures, one angle at each vertex, of a convex polygon is 360. If a quadrilateral is a parallelogram, then its opposite sides are congruent. If a quadrilateral is a parallelogram, then its opposite angles are congruent. If a quadrilateral is a parallelogram, then its consecutive angles are supplementary. If a quadrilateral is a parallelogram, then its diagonals bisect each other. If one pair of opposite sides of a quadrilateral are parallel and congruent, then the quadrilateral is a parallelogram. If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram. If an angle of a quadrilateral is supplementary to both of its consecutive angles, then the quadrilateral is a parallelogram. If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. If a quadrilateral is a rectangle, then it is a parallelogram. If a parallelogram is a rectangle, then its diagonals are congruent. If a quadrilateral is a rhombus, then it is a parallelogram. If a parallelogram is a rhombus, then its diagonals are perpendicular. If a parallelogram is a rhombus, then each diagonal bisects a pair of opposite angles. If one angle of a parallelogram is a right angle, then the parallelogram is a rectangle. 147 Geometry
HPTER 6 Postulates and Theorems Theorem 6-1-1 Theorem 6-1-2 Theorem 6-2-1 Theorem 6-2-2 Theorem 6-2-3 Theorem 6-2-4 Theorem 6-3-1 Theorem 6-3-2 Theorem 6-3-3 Theorem 6-3-4 Theorem 6-3-5 Theorem 6-4-1 Theorem 6-4-2 Theorem 6-4-3 Theorem 6-4-4 Theorem 6-4-5 Theorem 6-5-1 (Polygon ngle Sum Theorem) The sum of the interior angle measures of a convex polygon with n sides is (n 2)180. (Polygon Exterior ngle Sum Theorem) The sum of the exterior angle measures, one angle at each vertex, of a convex polygon is 360. If a quadrilateral is a parallelogram, then its opposite sides are congruent. If a quadrilateral is a parallelogram, then its opposite angles are congruent. If a quadrilateral is a parallelogram, then its consecutive angles are supplementary. If a quadrilateral is a parallelogram, then its diagonals bisect each other. If one pair of opposite sides of a quadrilateral are parallel and congruent, then the quadrilateral is a parallelogram. If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram. If an angle of a quadrilateral is supplementary to both of its consecutive angles, then the quadrilateral is a parallelogram. If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. If a quadrilateral is a rectangle, then it is a parallelogram. If a parallelogram is a rectangle, then its diagonals are congruent. If a quadrilateral is a rhombus, then it is a parallelogram. If a parallelogram is a rhombus, then its diagonals are perpendicular. If a parallelogram is a rhombus, then each diagonal bisects a pair of opposite angles. If one angle of a parallelogram is a right angle, then the parallelogram is a rectangle. 147 Geometry
HPTER 6 POSTULTES N THEOREMS ONTINUE Theorem 6-5-2 Theorem 6-5-3 Theorem 6-5-4 Theorem 6-5-5 Theorem 6-6-1 Theorem 6-6-2 Theorem 6-6 3 Theorem 6-6-4 Theorem 6-6-5 Theorem 6-6-6 If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle. If one pair of consecutive sides of a parallelogram are congruent, then the parallelogram is a rhombus. If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus. If one diagonal of a parallelogram bisects a pair of opposite angles, then the parallelogram is a rhombus. If a quadrilateral is a kite, then its diagonals are perpendicular. If a quadrilateral is a kite, then exactly one pair of opposite angles are congruent. If a quadrilateral is an isosceles trapezoid, then each pair of base angles are congruent. If a trapezoid has one pair of congruent base angles, then the trapezoid is isosceles. trapezoid is isosceles if and only if its diagonals are congruent. The midsegment of a trapezoid is parallel to each base, and its length is one half the sum of the lengths of the bases. 148 Geometry
HPTER 6 POSTULTES N THEOREMS ONTINUE Theorem 6-5-2 Theorem 6-5-3 Theorem 6-5-4 Theorem 6-5-5 Theorem 6-6-1 Theorem 6-6-2 Theorem 6-6 3 Theorem 6-6-4 Theorem 6-6-5 Theorem 6-6-6 If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle. If one pair of consecutive sides of a parallelogram are congruent, then the parallelogram is a rhombus. If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus. If one diagonal of a parallelogram bisects a pair of opposite angles, then the parallelogram is a rhombus. If a quadrilateral is a kite, then its diagonals are perpendicular. If a quadrilateral is a kite, then exactly one pair of opposite angles are congruent. If a quadrilateral is an isosceles trapezoid, then each pair of base angles are congruent. If a trapezoid has one pair of congruent base angles, then the trapezoid is isosceles. trapezoid is isosceles if and only if its diagonals are congruent. The midsegment of a trapezoid is parallel to each base, and its length is one half the sum of the lengths of the bases. 148 Geometry
HPTER 6 ig Ideas nswer these questions to summarize the important concepts from hapter 6 in your own words. 1. Name five ways to determine if a quadrilateral is a parallelogram. 2. Explain why a square is a rectangle but a rectangle is not necessarily a square. 3. Explain the difference between a rhombus and a kite. 4. Explain the difference between a regular polygon and a polygon that is not regular. For more review of hapter 6: omplete the hapter 6 Study Guide and Review on pages 438 441 of your textbook. omplete the Ready to Go On quizzes on pages 407 and 437 of your textbook. 149 Geometry
HPTER 6 ig Ideas nswer these questions to summarize the important concepts from hapter 6 in your own words. 1. Name five ways to determine if a quadrilateral is a parallelogram. nswers will vary. Possible answer: Five ways to determine if a quadrilateral is a parallelogram are: both pairs of opposite sides are parallel, one pair of opposite sides are parallel and congruent, both pairs of opposite sides are congruent, both pairs of opposite angles are congruent, one angle is supplementary to both of its consecutive angles and the diagonals bisect each other. 2. Explain why a square is a rectangle but a rectangle is not necessarily a square. nswers will vary. Possible answer: square is a parallelogram with 4 right angles, both are conditions for a rectangle. rectangle does not meet the condition for a square of 4 congruent sides. 3. Explain the difference between a rhombus and a kite. nswers will vary. Possible answer: rhombus has 4 congruent sides. kite has two pairs of congruent sides. 4. Explain the difference between a regular polygon and a polygon that is not regular. nswers will vary. Possible answer: regular polygon has all sides congruent and all angles congruent. Polygons without these conditions are not considered regular. For more review of hapter 6: omplete the hapter 6 Study Guide and Review on pages 438 441 of your textbook. omplete the Ready to Go On quizzes on pages 407 and 437 of your textbook. 149 Geometry