6-3 Proving That a Quadrilateral Is a Parallelogram

Quick Review

A quadrilateral is a parallelogram if any one of the following is true.

• Both pairs of opposite sides are parallel.
• Both pairs of opposite sides are congruent.
• Consecutive angles are supplementary.
• Both pairs of opposite angles are congruent.
• The diagonals bisect each other.
• One pair of opposite sides is both congruent and parallel.

Example

Must the quadrilateral be a parallelogram?

Yes, both pairs of opposite angles are congruent.

Exercises

Determine whether the quadrilateral must be a parallelogram.

Algebra Find the values of the variables for which ABCD must be a parallelogram.

6-4 Properties of Rhombuses, Rectangles, and Squares

Quick Review

A rhombus is a parallelogram with four congruent sides.

A rectangle is a parallelogram with four right angles.

A square is a parallelogram with four congruent sides and four right angles.

The diagonals of a rhombus are perpendicular. Each diagonal bisects a pair of opposite angles.

The diagonals of a rectangle are congruent.

Example

What are the measures of the numbered angles in the rhombus?

m ∠ 1 = 60 Each diagonal of a rhombus bisects a pair of opposite angles.   m ∠ 2 = 90 The diagonals of a rhombus are  ⊥ .   60 + m ∠ 2 + m ∠ 3 = 180 Triangle Angle-Sum Thm. 60 + 90 + m ∠ 3 = 180 Substitute.   m ∠ 3 = 30 Simplify. table with 5 rows and 3 columns , row1 column 1 , m angle , 1 equals 60 , column 2 table with 2 rows and 1 column , row1 column 1 , cap eachdiagonalofarhombus , row2 column 1 , bisectsapairofoppositeangles. , end table , column 3 , row2 column 1 , m angle , 2 equals 90 , column 2 cap thediagonalsofarhombusare . up tack . , column 3 , row3 column 1 , 60 plus m angle , 2 plus , column 2 m angle 3 equals 180 , column 3 cap trianglecap angleminuscap sumcap thm. , row4 column 1 , 60 plus 90 plus , column 2 m angle 3 equals 180 , column 3 cap substitute. , row5 column 1 , , column 2 m angle 3 equals 30 , column 3 cap simplify. , end table

Exercises

Find the measures of the numbered angles in each special parallelogram.

21. 
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22. 

Determine whether each statement is always, sometimes, or never true.

23. A rhombus is a square.
24. A square is a rectangle.
25. A rhombus is a rectangle.
26. The diagonals of a parallelogram are perpendicular.
27. The diagonals of a parallelogram are congruent.
28. Opposite angles of a parallelogram are congruent.

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