6.5 Rhombi and Squares

Then/Now

You determined whether quadrilaterals were parallelograms and/or rectangles.

• Recognize and apply the properties of rhombi and squares.

• Determine whether quadrilaterals are rectangles, rhombi, or squares.

Vocabulary

• Rhombus:

a parallelogram with all four sides congruent

Vocabulary

• Square:

a parallelogram with four congruent sides and four right angles

VocabularyTheorem 6.15: Diagonals of a Rhombus #1If a parallelogram is a rhombus, then its

diagonals are perpendicular.

VocabularyTheorem 6.16: Diagonals of a Rhombus #2If a parallelogram is a rhombus, then each

diagonal bisects a pair of opposite angles.

VocabularyTheorem 6.17: Condition #1 for a RhombusIf the diagonals of a parallelogram are

perpendicular, then the parallelogram is a rhombus.

*Converse of 6.15

VocabularyTheorem 6.18: Condition #2 for a RhombusIf one diagonal of a parallelogram bisects a

pair of opposite angles, then the parallelogram is a rhombus.

*Converse of 6.16

VocabularyTheorem 6.19: Condition #3 for a RhombusIf one pair of consecutive sides of a

parallelogram are congruent, then the parallelogram is a rhombus.

VocabularyTheorem 6.20: Square Conditions If a quadrilateral is both a rectangle and a

rhombus, then it is a square.

Example 1AUse Properties of a Rhombus

A. The diagonals of rhombus WXYZ intersect at V.If mWZX = 39.5, find mZYX.

Answer: mZYX = 101

Example 1BUse Properties of a Rhombus

B. ALGEBRA The diagonals of rhombus WXYZ intersect at V. If WX = 8x – 5 and WZ = 6x + 3, find x.

Answer: x = 4

Example 1A

A. mCDB = 126

B. mCDB = 63

C. mCDB = 54

D. mCDB = 27

A. ABCD is a rhombus. Find mCDB if mABC = 126.

Example 1B

A. x = 1

B. x = 3

C. x = 4

D. x = 6

B. ABCD is a rhombus. If BC = 4x – 5 and CD = 2x + 7, find x.

Rectangle Rhombi

Example 2Is there enough information given to prove that ABCD is a rhombus?

Given: ABCD is a parallelogram.AD DC

Prove: ABCD is a rhombus

Proofs Using Properties of Rhombi and Squares

A. Yes, if one pair of consecutive sides of a parallelogram are congruent, the parallelogram is a rhombus.

B. No, you need more information

Example 3

A. The diagonal bisects a pair of opposite angles.

B. The diagonals bisect each other.

C. The diagonals are perpendicular.

D. The diagonals are congruent.

Sachin has a shape he knows to be a parallelogram and all four sides are congruent. Which information does he need to know to determine whether it is also a square?