Study Guide and Intervention - prosseracademy.org

1 Chapter 66 Glencoe GeometryStudy Guide and InterventionAngles of PolygonsNAME _____ DATE _____ PERIOD _____6-1 Copyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, of Measures of Interior AnglesThe segments that connect thenonconsecutive sides of a polygon are called diagonals. Drawing all of the diagonals from one vertex of an n-gonseparates 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 AngleIf a convex polygon has nsides, and Sis the sum of the measures of its interior angles, Sum Theoremthen S 180(n 2).A convex polygon has 13 sides. Find the sum of the measuresof the interior 180(n 2) 180(13 2) 180(11) 1980 The measure of aninterior angle of a regular polygon is120.

2 Find the number of number of sides is n, so the sum of themeasures of the interior angles is 180(n 2)120n 180(n 2)120n 180n 360 60n 360n 6 Find the sum of the measures of the interior angles of each convex measure of an interior angle of a regular polygon is given. Find the numberof sides in each (4x 10) (4x 5) (6x 10) (5x 5) ABCDEE xample 1 Example 2 Exercises05-56 Geo-06-873963 4/3/06 2:05 PM Page 6 Chapter 67 Glencoe GeometryCopyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Guide and Intervention (continued)Angles of PolygonsNAME _____ DATE _____ PERIOD _____6-1 Sum of Measures of Exterior AnglesThere is a simple relationship among the exterior angles of a convex AngleIf a polygon is convex, then the sum of the measures of the exterior angles, Sum Theoremone at each vertex, is the sum of the measures of the exterior angles, one at each vertex, of a convex anyconvex polygon, the sum of the measures of its exterior angles, one at each vertex, is the measure of each exterior angle of regular hexagon sum of the measures of the exterior angles is 360 and a hexagon has 6 angles.

3 If nis the measure of each exterior angle, then6n 360n 60 Find the sum of the measures of the exterior angles of each convex the measure of an exterior angle for each convex regular the measure of an exterior angle given the number of sides of a 6-1 Example 1 Example 2 Exercises05-56 Geo-06-873963 4/3/06 2:05 PM Page 7 Chapter 613 Glencoe GeometryCopyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Guide and InterventionParallelogramsNAME _____ DATE _____ PERIOD _____6-2 Sides and Angles of ParallelogramsA quadrilateral with both pairs of opposite sides parallel is a parallelogram. Here are fourimportant properties of PQRSis a parallelogram, thenThe opposite sides of a P Q S R and P S Q R parallelogram are opposite angles of a P Rand S Qparallelogram are consecutive angles of a Pand Sare supplementary; Sand Rare supplementary; parallelogram are supplementary.

4 Rand Qare supplementary; Qand Pare a parallelogram has one right If m P 90, then m Q 90, m R 90, and m S , then it has four right ABCDis a parallelogram, find aand B and C D are opposite sides, so A B C D .2a 34a 17 Aand Care opposite angles, so A 112b 14 Find xand yin each 55 5x 2y 6x 12x 3y 3y126x8y886x 4y 3x BADC342a8b 112 PQRSL esson 6-2 ExampleExercises05-56 Geo-06-873963 4/3/06 2:05 PM Page 13 Chapter 614 Glencoe GeometryStudy Guide and Intervention (continued)ParallelogramsNAME _____ DATE _____ PERIOD _____6-2 Copyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, of ParallelogramsTwo important properties of parallelograms deal with their ABCDis a parallelogram, then:The diagonals of a parallelogram bisect each PCand DP PBEach diagonal separates a parallelogram ACD CABand ADB CBDinto two congruent xand yin parallelogram diagonals bisect each other, so AE CEand DE 244y 18x 4y xand yin each each statement about your BAC E 9.

5 ADC D ||ABCDEx174y3x 2y123x30 10y2x 60 4y 4x2y283x4y812 ABECD6x18244yABCDPE xampleExercises05-56 Geo-06-873963 4/3/06 2:05 PM Page 14 Chapter 620 Glencoe GeometryStudy Guide and InterventionTests for ParallelogramsNAME _____ DATE _____ PERIOD _____6-3 Copyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, for a ParallelogramThere are many ways to establish that a quadrilateral is a :If:both pairs of opposite sides are parallel,A B ||D C and A D ||B C ,both pairs of opposite sides are congruent,A B D C and A D B C ,both pairs of opposite angles are congruent, ABC ADCand DAB BCD,the diagonals bisect each other,A E C E and D E B E ,one pair of opposite sides is congruent and parallel,A B ||C D and A B C D , or A D ||B C and A D B C ,then:the figure is a :ABCDis a xand yso that FGHJis a a parallelogram if the lengths of the opposite sides are 3 154x 2y 26x 124(2) 2y 2x 28 2y 2 2y 6y 3 Find xand yso that each quadrilateral is a 3x (x y) 2x 30 24 9x 6y45 185x 5y 25 11x 5y 55 2x 22y812 JHGF6x 3154x 2y2 ABCDEE xampleExercises05-56 Geo-06-873963 4/3/06 2:06 PM Page 20 Chapter 621 Glencoe GeometryCopyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Guide and Intervention (continued)Tests for ParallelogramsNAME _____ DATE _____ PERIOD _____6-3 Parallelograms on the Coordinate PlaneOn the coordinate plane, the DistanceFormula and the Slope Formula can be used to test if a quadrilateral is a whether ABCDis a vertices are A( 2, 3),B(3, 2),C(2, 1), and D( 3, 0).

6 Method 1:Use the Slope Formula,m yx22 yx11 .slope of A D 23 (0 3) 31 3slope of B C 23 ( 21) 31 3slope of A B 32 ( 32) 15 slope of C D 2 1( 03) 15 Opposite sides have the same slope, so A B ||C D and A D ||B C . Both pairs of opposite sides are parallel, so ABCDis a 2:Use the Distance Formula,d (x2 x1)2 (y2 y1)2 .AB ( 2 3)2 (3 2)2 25 1 or 26 CD (2 ( 3))2 ( 1 0)2 25 1 or 26 AD ( 2 ( 3)) 2 (3 0)2 1 9 or 10 BC (3 2 )2 ( 2 ( 1))2 1 9 or 10 Both pairs of opposite sides have the same length, so ABCDis a whether a figure with the given vertices is a the method (0, 0),B(1, 3),C(5, 3),D(4, 0); ( 1, 1),E(2, 4),F(6, 4),G(3, 1);Slope FormulaSlope ( 1, 0),S(3, 0),T(2, 3),U( 3, 2); ( 3, 2),B( 1, 4),C(2, 1),D(0, 1);Distance FormulaDistance and Slope ( 2, 4),T( 1, 1),U(3, 4),V(2, 1); (3, 3),G(1, 2),H( 3, 1),I( 1, 4);Distance and Slope FormulasMidpoint parallelogram has vertices R( 2, 1),S(2, 1), and T(0, 3).

7 Find all possiblecoordinates for the fourth 6-3 ExampleExercises05-56 Geo-06-873963 4/3/06 2:06 PM Page 21 Chapter 627 Glencoe GeometryCopyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Guide and InterventionRectangles NAME _____ DATE _____ PERIOD _____6-4 Properties of RectanglesA rectangleis a quadrilateral with four right angles. Here are the properties of rectangle has all the properties of a parallelogram. Opposite sides are parallel. Opposite angles are congruent. Opposite sides are congruent. Consecutive angles are supplementary. The diagonals bisect each : All four angles are right angles. UTS, TSR, SRU, and RUTare right angles. The diagonals are R U S TSRUQIn rectangle RSTU above,US 6x 3 and RT 7x diagonals of a rectangle bisect eachother, so US 3 7x 23 x 25 xIn rectangle RSTU above,m STR 8x 3 and m UTR 16x 9.

8 Find m STR. UTSis a right angle, so m STR m UTR 3 16x 9 9024x 6 9024x 96x 4m STR 8x 3 8(4) 3 or 35 ABCDis a AE 36 and CE 2x 4, find BE 6y 2 and CE 4y 6, find BC 24 and AD 5y 1, find m BEA 62, find m m AED 12xand m BEC 10x 20, find m BD 8y 4 and AC 7y 3, find m DBC 10xand m ACB 4x2 6, find m AB 6yand BC 8y, find BDin terms of rectangle MNOP,m 1 40. Find the measure of each numbered 6-4 Example 1 Example 2 Exercises05-56 Geo-06-873963 4/3/06 2:06 PM Page 27 Chapter 628 Glencoe GeometryStudy Guide and Intervention (continued)Rectangles NAME _____ DATE _____ PERIOD _____6-4 Copyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, that Parallelograms Are RectanglesThe diagonals of a rectangle arecongruent, and the converse is also the diagonals of a parallelogram are congruent, then the parallelogram is a the coordinate plane you can use the Distance Formula, the Slope Formula, andproperties of diagonals to show that a figure is a whether A( 3, 0),B( 2, 3),C(4, 1),and D(3, 2) are the vertices of a 1.

9 Use the Slope of A B 23 (0 3) 31 or 3slope of A D 3 2( 03) 62 or 13 slope of C D 32 41 31 or 3slope of B C 41 ( 32) 62 or 13 Opposite sides are parallel, so the figure is a parallelogram. Consecutive sides areperpendicular, so ABCDis a 2:Use the Midpoint and Distance midpoint of A C is 32 4 , 0 21 12 , 12 and the midpoint of B D is 22 3 , 3 22 12 , 12 . The diagonals have the same midpoint so they bisect each ,ABCDis a ( 3 4)2 (0 1)2 49 1 50 or 5 2 BD ( 2 3)2 (3 ( 2))2 25 25 50 or 5 2 The diagonals are a parallelogram with diagonals that bisect eachother, so it is a whether ABCDis a rectangle given each set of vertices. Justify ( 3, 1),B( 3, 3),C(3, 3),D(3, 1) ( 3, 0),B( 2, 3),C(4, 5),D(3, 2) ( 3, 0),B( 2, 2),C(3, 0),D(2, 2) ( 1, 0),B(0, 2),C(4, 0),D(3, 2) ( 1, 5),B( 3, 0),C(2, 2),D(4, 3) ( 1, 1),B(0, 2),C(4, 3),D(3, 0) parallelogram has vertices R( 3, 1),S( 1, 2), and T(5, 2).

10 Find the coordinates ofUso that RSTUis a Geo-06-873963 4/3/06 2:06 PM Page 28 Chapter 636 Glencoe GeometryStudy Guide and InterventionRhombi and SquaresNAME _____ DATE _____ PERIOD _____6-5 Copyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, of RhombiA rhombusis a quadrilateral with four congruent sides. Opposite sides are congruent, so a rhombus is also aparallelogram and has all of the properties of a parallelogram. Rhombi also have the following diagonals are H R O Each diagonal bisects a pair of opposite H bisects RMOand O bisects MRHand the diagonals of a parallelogram are If RHOMis a parallelogram and R O M H ,perpendicular, then the figure is a RHOM is a rhombus ABCD,m BAC 32. Find the measure of each numbered a rhombus, so the diagonals are perpendicular and ABEis a right triangle.