Study Guide and Intervention
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1 8-1 NM T PRIO Study Guide and Intervention ngles of Polygons Sum of Measures of Interior ngles The segments that connect the nonconsecutive sides of a polygon are called diagonals. rawing all of the diagonals from one vertex of an n-gon separates the polygon into n 2 triangles. The sum of the measures of the interior angles of the polygon can be found by adding the measures of the interior angles of those n 2 triangles. Interior ngle If a convex polygon has n sides, and S is the sum of the measures of its interior angles, Sum Theorem then S 180(n 2). xample 1 xample 2 convex polygon has 13 sides. Find the sum of the measures of the interior angles. S 180(n 2) 180(13 2) 180(11) 1980 The measure of an interior angle of a regular polygon is 120. Find the number of sides. The number of sides is n, so the sum of the measures of the interior angles is 120n. S 180(n 2) 120n 180(n 2) 120n 180n n 360 n 6 Lesson 8-1 xercises Find the sum of the measures of the interior angles of each convex polygon gon gon gon 4. 8-gon gon 6. 3x-gon The measure of an interior angle of a regular polygon is given. Find the number of sides in each polygon Find x. (4x 5) 7x (5x 5) (4x 10) (6x 10) Glencoe/McGraw-Hill 417 Glencoe Geometry
2 8-1 NM T PRIO Study Guide and Intervention (continued) ngles of Polygons Sum of Measures of xterior ngles There is a simple relationship among the exterior angles of a convex polygon. xterior ngle If a polygon is convex, then the sum of the measures of the exterior angles, Sum Theorem one at each vertex, is 360. xample 1 Find the sum of the measures of the exterior angles, one at each vertex, of a convex 27-gon. For any convex polygon, the sum of the measures of its exterior angles, one at each vertex, is 360. xample 2 Find the measure of each exterior angle of regular hexagon F. The sum of the measures of the exterior angles is 360 and a hexagon has 6 angles. If n is the measure of each exterior angle, then 6n 360 n 60 F xercises Find the sum of the measures of the exterior angles of each convex polygon gon gon gon Find the measure of an exterior angle for each convex regular polygon gon gon 6. 2x-gon Find the measure of an exterior angle given the number of sides of a regular polygon Glencoe/McGraw-Hill 418 Glencoe Geometry
3 8-1 NM T PRIO Skills Practice ngles of Polygons Find the sum of the measures of the interior angles of each convex polygon. 1. nonagon 2. heptagon 3. decagon The measure of an interior angle of a regular polygon is given. Find the number of sides in each polygon Find the measure of each interior angle using the given information. Lesson (2x 15) x L M (2x 20) (3x 10) x (2x 15) 2x P (2x 10) N 9. quadrilateral STUW with S T, 10. hexagon FGHI with U W, m S 2x 16, G H, F I, m U x 14 m 7x, m F 4x S T I F W U H G Find the measures of an interior angle and an exterior angle for each regular polygon. 11. quadrilateral 12. pentagon 13. dodecagon Find the measures of an interior angle and an exterior angle given the number of sides of each regular polygon. Round to the nearest tenth if necessary Glencoe/McGraw-Hill 419 Glencoe Geometry
4 8-2 NM T PRIO Study Guide and Intervention Parallelograms Sides and ngles of Parallelograms quadrilateral with both pairs of opposite sides parallel is a parallelogram. Here are four important properties of parallelograms. The opposite sides of a parallelogram are congruent. The opposite angles of a parallelogram are congruent. The consecutive angles of a parallelogram are supplementary. If PQRS is a parallelogram, then P Q S R and P S Q R P R and S Q P and S are supplementary; S and R are supplementary; R and Q are supplementary; Q and P are supplementary. If a parallelogram has one right If m P 90, then m Q 90, m R 90, and m S 90. angle, then it has four right angles. S P R Q xample If is a parallelogram, find a and b. and are opposite sides, so. 2a 34 a 17 and are opposite angles, so. 8b 112 b 14 2a 8b Lesson 8-2 xercises Find x and y in each parallelogram x 4y 8y 88 6x 3. 6x 4. 3y 3y 12 6x 12x y 5x 2y 30x x Glencoe/McGraw-Hill 423 Glencoe Geometry
5 8-2 NM T PRIO Study Guide and Intervention (continued) Parallelograms iagonals of Parallelograms Two important properties of parallelograms deal with their diagonals. If is a parallelogram, then: P The diagonals of a parallelogram bisect each other. ach diagonal separates a parallelogram into two congruent triangles. P P and P P and xample Find x and y in parallelogram. The diagonals bisect each other, so and. 6x 24 4y 18 x 4 y 4.5 xercises Find x and y in each parallelogram. 18 6x 4y x 4y y 28 4x 2x 60 4y y 3x 12 3x 2y x 4 y 17 omplete each statement about. Justify your answer Glencoe/McGraw-Hill 424 Glencoe Geometry
6 8-2 NM T PRIO Skills Practice Parallelograms omplete each statement about FG. Justify your answer. H 1. G G F G H 4. F 5. FG is supplementary to. 6. G LGR Use WXYZ to find each measure or value. 7. m XYZ 8. m WZY 9. m WXY 10. a Z W a 70 Y X Lesson 8-2 OORINT GOMTRY Find the coordinates of the intersection of the diagonals of parallelogram HJKL given each set of vertices. 11. H(1, 1), J(2, 3), K(6, 3), L(5, 1) 12. H( 1, 4), J(3, 3), K(3, 2), L( 1, 1) 13. PROOF Write a paragraph proof of the theorem onsecutive angles in a parallelogram are supplementary. Glencoe/McGraw-Hill 425 Glencoe Geometry
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