6-5 Rhombi and Squares

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.

Properties of a rhombus

• A rhombus is a parallelogram with all four sides congruent.

• A rhombus has all the properties of a parallelogram:Both pairs of opposite sides are parallel.Both pairs of opposite sides are congruent.Both pairs of opposite angles are congruent.The diagonals bisect each other.A pair of opposite sides are both parallel and congruent.

AND…..

Rhombus

1. Draw a rhombus on your paper.2. Draw diagonals in your rhombus.3. Measure the diagonals. Are the diagonals

congruent?4. Are the diagonals perpendicular?

A parallelogram is a rhombus if and only if its diagonals are perpendicular.

Page 430

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

Answer: mZYX = 101

mWZY + mZYX = 180 Consecutive InteriorAngles Theorem

79 + mZYX = 180 SubstitutionmZYX = 101 Subtract 79 from both

sides.

Since WXYZ is a rhombus, diagonal ZX bisects WZY. Therefore, mWZY = 2mWZX. So, mWZY = 2(39.5) or 79.Since WXYZ is a rhombus, WZ║XY, and ZY is a transversal.

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

Answer: x = 4

WX WZ By definition, all sides of arhombus are congruent.

WX = WZ Definition of congruence8x – 5 = 6x + 3 Substitution2x – 5 = 3 Subtract 6x from each

side.2x = 8 Add 5 to each side.x = 4 Divide each side by 4.

A. mCDB = 126

B. mCDB = 63

C. mCDB = 54

D. mCDB = 27

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

Properties of Squares

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

Square1. Draw a square on your paper.2. Draw diagonals in your square.3. Measure the diagonals. Are the diagonals

congruent?4. Are the diagonals perpendicular?

A parallelogram is a square if and only if its diagonals are perpendicular and congruent.

This venn diagram summarizes the relationships among parallelograms, rhombi, rectangles, and squares.

Page 431

Page 432

Write a paragraph proof.

Given: LMNP is a parallelogram.1 2 and 2 6

Prove: LMNP is a rhombus.

Proof: Since it is given that LMNP is aparallelogram, LM║PN and 1 and 5 arealternate interior angles. Therefore, 1 5.It is also given that 1 2 and 2 6,so 1 6 by substitution and 5 6 bysubstitution.

Answer: Therefore, LN bisects L and N. By Theorem 6.18, LMNP is a rhombus.

Is there enough information given to prove that ABCD is a rhombus?Given: ABCD is a parallelogram.

AD DCProve: ADCD is a rhombus

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

B. No, you need more information.

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?

Recap…

Parallelograms:

Opp sides Opp sides Opp Diag bisect other

Rectangles:

4 rt

Opp sides Opp sides Opp Diag bisect other

Diag

Rhombi:

4 sides

Opp sides Opp sides Opp Diag bisect other

Diag Diag bisect opp

Squares:

4 sides4

Opp sides Opp sides Opp Diag bisect other

Diag

6.5 Assignment

Page 435, 7-12, 23-33