6.5 rhombi and squares. then/now you determined whether quadrilaterals were parallelograms and/or...
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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?