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Geometry: Concepts and Applications

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Study Guide

ParallelogramsA special kind of quadrilateral in which both pairs of oppositesides are parallel is called a parallelogram.

The following theorems all concern parallelograms.

Opposite sides of a parallelogram are congruent.Opposite angles of a parallelogram are congruent.Consecutive angles of a parallelogram are supplementary.The diagonals of a parallelogram bisect each other.

Example: If the quadrilateral in the figure is a parallelogram,find the values of x, y, and z.

Since opposite angles of a parallelogram arecongruent, x = 72.

Since consecutive angles of a parallelogram aresupplementary, y + 72 = 180. Therefore, y = 108.

Since opposite sides of a parallelogram arecongruent, z = 8.

If each quadrilateral is a parallelogram, find the values of x, y,andz.l.N·78·

z·

29·z.2.

29,73,102 31,44,105 73,73,1074. In parallelogram ABCD, mL A = 3x

and mL B = 4x + 40. Find the measureof angles A, B, C, and D.mL A = 60, mL B = 120,mL C = 60,mL D = 120

5. In parallelogram RSTV, diagonalsRT and VS intersect at Q. IfRQ = 5x + 1 and QT = 3x + 15,findQT. QT = 36

Explain why it is impossible for each figure to be aparallelogram.

'f5j;J11 The opposite angles

should be congruent.Each pair of opposite sidesshould be congruent.

© Glencoe/McGraw-Hili 312 Geometry: Concepts and Applications

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Study GuideNAME DATE PERIOD __

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Tests for ParallelogramsYou can show that a quadrilateral is a parallelogram if you canshow that one of the following is true.1. Both pairs of opposite sides are parallel.

2. Both pairs of opposite sides are congruent.

3. Diagonals bisect each other.

4. Both pairs of opposite angles are congruent.

5. A pair of opposite sides is both parallel and congruent.

Example: AP = 3x - 4, AC = 46, PE = 3y, and DP =5y - 12. Find the values of x and y.that wouldmake ABCD a parallelogram.

For the diagonals to bisect each other, 2(3x - 4) =46 and 3y = 5y - 12. Solve for each variable.

2(3x - 4) = 466x - 8 = 46 Dist. Prop.

6x = 54 Add 8.x = 9 Divide by 6.,

So, x = 9 and y = 6.

3y = 5y - 12-2y = -12

y'= 6

Determine whether each quadrilateral is a parallelogram.Write yes or no. Give a reason for your answer.

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Subtract 5y.Divide by -2.

""--="----~_'52____'0 / 3. ~

Yes; both pairsof opposite sidesare congruent.

No; the top andbottom sides areparallel, but theother pair may not be.

Find the values of x and y that ensure each quadrilateral is aparallelogram.

4. 5. ""'--(x + y)"2x

5yO

600 (3x + 60t

20,12 12,8© Glencoe/McGraw-Hili 317

No; top andbottom sides are

. not parallel.

20

Geometry: Concepts and Applications

NAME _

Skills Practice N \ W

Tests for Paralielograms ~ ~r€/Yl~ bu.+ you should.Determine whether each quadrilateral is a parallelogram. Write yes or no. If yes, ~give a reason for your answer. -1' ",e

3. L-! 0:0-----'1 ~.I;o 2.~1.

yes, Theorem 8-7 yes, Theorem 8-9 yes, definition

($\ I) (*3) of parallelogramI

4. 5'L] 6. »I \ / d ::Jtn:e(1)

no yes, definition no ~of parallelogram tn

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9. (Jl

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yes, Theorem 8-8 yes, Theorem 8-9 no(: D (

10. 11.

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12. t:::J1050

50

yes, definition no yes, T eorem 8-8of parallelogram, $5jt D

© Glencoe/McGraw-Hili 318 Geometry: Concepts and Applications