Showing quadrilaterals are parallelograms

Bell Ringer

Use Opposite SidesExample 1

Tell whether the quadrilateral is a parallelogram. Explain your reasoning.

SOLUTION

The quadrilateral is not a parallelogram. It has two pairs of congruent sides, but opposite sides are not congruent.

Use Opposite AnglesExample 2

SOLUTION

The quadrilateral is a parallelogram because both pairs of opposite angles are congruent.

Tell whether the quadrilateral is a parallelogram. Explain your reasoning.

Now You Try

Tell whether the quadrilateral is a parallelogram. Explain your reasoning.

1.

In quadrilateral WXYZ, WX = 15, YZ = 20, XY = 15, and ZW = 20. Is WXYZ a parallelogram? Explain your reasoning.

3.

2.

ANSWERYes; both pairs of oppositesides are congruent.

No; opposite angles arenot congruent.ANSWER

ANSWER No; opposite sides are not congruent.

Example 3 Tell whether the quadrilateral is a parallelogram. Explain your reasoning.

a. b. c.

SOLUTION

U is supplementary to T and V (85° + 95° = 180°). So, by Theorem 6.8, TUVW is a parallelogram.

a.

G is supplementary to F (55° + 125° = 180°), but G is not supplementary to H (55° + 120° ≠ 180°). So, EFGH is not a parallelogram.

b.

Use Consecutive AnglesExample 3

D is supplementary to C (90° + 90° = 180°), but you are not given any information about A or B. Therefore, you cannot conclude that ABCD is a parallelogram.

c.

Example 4Tell whether the quadrilateral is a parallelogram. Explain your reasoning.

b.

SOLUTION

a. The diagonals of JKLM bisect each other. So, by Theorem 6.9, JKLM is a parallelogram.

a.

The diagonals of PQRS do not bisect each other. So, PQRS is not a parallelogram.

b.

Now You Try

4.

5.

ANSWER No; opposite angles arenot congruent (orconsecutive angles arenot supplementary).

Yes; one angle issupplementary to both ofits consecutive angles.

ANSWER

Tell whether the quadrilateral is a parallelogram. Explain your reasoning.

Now You Try

ANSWER Yes; the diagonals bisect each other.

No; the diagonals do notbisect each other.

ANSWER

7.

6.

Tell whether the quadrilateral is a parallelogram. Explain your reasoning.

Now You Try

Now You Try

Now You Try

Now You Try

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