Angles of Polygons Concept Summary

Transcription

1 Vocabulary and oncept heck diagonal (p. 404) isosceles trapezoid (p. 439) kite (p. 438) median (p. 440) parallelogram (p. 411) rectangle (p. 424) rhombus (p. 431) square (p. 432) trapezoid (p. 439) complete list of postulates and theorems can be found on pages R1 R8. Eercises State whether each sentence is true or false. If false, replace the underlined term to make a true sentence. 1. The diagonals of a rhombus are perpendicular. 2. ll squares are rectangles. 3. If a parallelogram is a rhombus, then the diagonals are congruent. 4. Every parallelogram is a quadrilateral. 5. (n) rhombus is a quadrilateral with eactly one pair of parallel sides.. Each diagonal of a rectangle bisects a pair of opposite angles. 7. If a quadrilateral is both a rhombus and a rectangle, then it is a square. 8. oth pairs of base angles in a(n) isosceles trapezoid are congruent ngles of Polygons oncept Summary If a conve polygon has n sides and the sum of the measures of its interior angles is S, then S 180(n 2). The sum of the measures of the eterior angles of a conve polygon is 30. Find the measure of an interior angle of a regular decagon. S 180(n 2) Interior ngle Sum Theorem 180(10 2) n (8) or 1440 Simplify. The measure of each interior angle is , or 144. Eercises Find the measure of each interior angle of a regular polygon given the number of sides. See 1 on page GER Find the measure of each interior angle. See 3 on page X 14. a (a 1 28) ( 2 a 8 W (a 2) Z ( ( 25) (1.5 3) (2 22) ( 27) E 452 hapter 8 Quadrilaterals

2 hapter Parallelograms oncept Summary In a parallelogram, opposite sides are parallel and congruent, opposite angles are congruent, and consecutive angles are supplementary. The diagonals of a parallelogram bisect each other. WXZ is a parallelogram. Find m ZW and m XWZ. m ZW m WX m ZW or 115 m XWZ m WX 180 m XWZ (82 33) 180 m XWZ m XWZ 5 Opp. of are. m WX m WXZ m XZ ons. in are suppl. m WX m WXZ m XZ Simplify. Subtract 115 from each side. W X Z Eercises Use to find each measure. See 2 on page m 1. F 17. m m F Tests for Parallelograms oncept Summary quadrilateral is a parallelogram if any one of the following is true. oth pairs of opposite sides are parallel and congruent. oth pairs of opposite angles are congruent. iagonals bisect each other. pair of opposite sides is both parallel and congruent. OORINTE GEOETR etermine whether the figure with vertices ( 5, 3), ( 1, 5), (, 1), and (2, 1) is a parallelogram. Use the istance and Slope Formulas. [ 5 ( 1)] (3 5) 2 y ( 4) 2 ) ( 2 or 20 ( ) 2 [1 )] ( or 20 Since,. 5 3 slope of or 1 1 ( 5) 2 slope of 1 1 or and have the same slope, so they are parallel. Since one pair of opposite sides is congruent and parallel, is a parallelogram. O hapter 8 453

3 hapter 8 Eercises etermine whether the figure with the given vertices is a parallelogram. Use the method indicated. See 5 on page ( 2, 5), (4, 4), (, 3), ( 1, 2); istance Formula 22. H(0, 4), J( 4, ), (5, ), (9, 4); idpoint Formula 23. S( 2, 1), T(2, 5), V( 10, 13), W( 14, 7); Slope Formula Rectangles oncept Summary rectangle is a quadrilateral with four right angles and congruent diagonals. If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle. Quadrilateral N is a rectangle. If P 2 1 and P 4 11, find. The diagonals of a rectangle are congruent and bisect each other, so P P. P P P P iag. are and bisect each other. ef. of angles ( 2)( ) 0 Factor The value of is 2 or. Subtract 4 from each side. Subtract 11 from each side P N Eercises is a rectangle. See s 1 and 2 on pages 425 and If 9 1 and F 2 7, find F. 25. If m and m , find m F 2. If F 4 1 and F 13, find. 27. If m and m , find m 5. OORINTE GEOETR etermine whether RSTV is a rectangle given each set of vertices. Justify your answer. See 4 on pages 42 and R( 3, 5), S(0, 5), T(3, 4), V(0, 4) 29. R(0, 0), S(, 3), T(4, 7), V( 2, 4) 454 hapter 8 Quadrilaterals

4 hapter Rhombi and Squares oncept Summary rhombus is a quadrilateral with each side congruent, diagonals that are perpendicular, and each diagonal bisecting a pair of opposite angles. quadrilateral that is both a rhombus and a rectangle is a square. Use rhombus J to find m J and m J. The opposite sides of a rhombus are parallel, so J. J because alternate interior angles are congruent. m J m 28 efinition of congruence The diagonals of a rhombus bisect the angles, so J. m J m J 180 m J (m J m ) 180 m J (28 28) 180 m J m J 124 ons. in are suppl. m J m J m dd. Subtract 5 from each side. 28 J Eercises Use rhombus with m , m 2 5 4, 15, and m 3 y 2 2. See 2 on page Find. 31. Find F. 32. Find y F 5 8- Trapezoids oncept Summary In an isosceles trapezoid, both pairs of base angles are congruent and the diagonals are congruent. The median of a trapezoid is parallel to the bases, and its measure is one-half the sum of the measures of the bases. RSTV is a trapezoid with bases R V and S T and median N. Find if N 0, ST 4 1, and RV 11. N 1 (ST RV) [(4 1) ( 11)] ultiply each side by Simplify Subtract 10 from each side. 11 ivide each side by 10. R S T N V hapter 8 455

5 hapter 8 Eercises Find the missing value for the given trapezoid. See 4 on page For isosceles trapezoid, 34. For trapezoid J, and are X and are midpoints of the legs. midpoints of the legs. If 57 Find m X if m 78. and 21, find J. 21 X J oordinate Proof with Quadrilaterals oncept Summary Position a quadrilateral so that a verte is at the origin and at least one side lies along an ais. Position and label rhombus RSTV on the coordinate plane. Then write a coordinate proof to prove that each pair of opposite sides is parallel. First, draw rhombus RSTV on the coordinate plane. abel the coordinates of the vertices. Given: Prove: Proof: RSTV is a rhombus. R V S T, R S V T y V(b, c) O R(0, 0) S(a, 0) T(a b, c) c 0 c slope of R V or b 0 b slope of R S 0 0 or 0 a 0 c 0 c slope of S T (a or b) a b c c slope of V T or 0 (a b) b R V and S T have the same slope. So R V S T. R S and V T have the same slope, and R S V T. Eercises Position and label each figure on the coordinate plane. Then write a coordinate proof for each of the following. See 3 on pages 448 and The diagonals of a square are perpendicular. 3. diagonal separates a parallelogram into two congruent triangles. Name the missing coordinates for each quadrilateral. See 2 on page y 38. y R(a, c) P(?,?) T(0, c) U(?,?) O (0, 0) N(4a, 0) W( a, 0) O V(b, 0) 45 hapter 8 Quadrilaterals