1. Properties of Rhombuses, Rectangles, and Squares
Skill 31

2. Objective
HSG-CO.11: Students are responsible for defining and classifying special types of parallelograms, and using properties of rhombuses and rectangles.

3. Definitions
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.

4. Thm. 50: Rhombus – Diagonals Perpendicular
If a parallelogram is a rhombus,
then its diagonals are perpendicular. A B C D

5. Thm. 51: Rhombus – Diagonals Bisect Opp. Angles
If a parallelogram is a rhombus,
then its diagonals bisect the opposite angles. A B C D

6. Thm. 52: Rectangle – Diagonals Congruent
If a parallelogram is a rectangle,
then its diagonals are congruent. A B C D

7. Example 1; Finding Angle Measures
a) What are the measures of the numbered angles in rhombus ABCD, if 𝑚∠𝐴𝐵𝐷=58°?
𝒎∠𝟏=𝟗𝟎ᵒ
Rhombus – Diagonals Perpendicular Thm. C D A B 1 2 3 4
𝒎∠𝟐=𝟓𝟖ᵒ
AIA Theorem
𝒎∠𝟑=𝟓𝟖ᵒ
Rhombus – Diagonals Bisect Opp. Angles Thm.
𝒎∠𝟏+𝒎∠𝟑+𝒎∠𝟒=𝟏𝟖𝟎
Triangle Angle-Sum Thm.
𝟗𝟎+𝟓𝟖+𝒎∠𝟒=𝟏𝟖𝟎
𝒎∠𝟒=𝟑𝟐°

8. Example 1; Finding Angle Measures
a) What are the measures of the numbered angles in rhombus PQRS, if 𝑚∠𝑃𝑄𝑅=104°?
Rhombus – Diagonals Bisect Opp. Angles Thm.
𝒎∠𝟏=𝒎∠𝟐 and 𝒎∠𝟑=𝒎∠𝟒
𝒎∠𝟑=𝒎∠𝟐 and 𝒎∠𝟏=𝒎∠𝟒
AIA Theorem
𝒎∠𝟏=𝒎∠𝟐=𝒎∠𝟑=𝒎∠𝟒
Transitive Property
𝟏𝟎𝟒+𝟐 𝒎∠𝟏 =𝟏𝟖𝟎
Triangle Angle-Sum Thm.
𝟐 𝒎∠𝟏 =𝟕𝟔 R S P Q 1 2 3 4
𝒎∠𝟏=𝟑𝟖°
𝒎∠𝟏=𝒎∠𝟐=𝒎∠𝟑=𝒎∠𝟒=𝟑𝟖°

9. Example 2; Finding Diagonal Length
a) In rectangle RSBF, 𝑆𝐹=2𝑥+15 and 𝑅𝐵=5𝑥−12, what is the length of a diagonal?
𝟐𝒙+𝟏𝟓=𝟓𝒙−𝟏𝟐 B S R F
−𝟑𝒙=−𝟐𝟕
𝒙=𝟗
𝑫𝒊𝒂𝒈𝒐𝒏𝒂𝒍=𝟐𝒙+𝟏𝟓
𝑫𝒊𝒂𝒈𝒐𝒏𝒂𝒍=𝟐 𝟗 +𝟏𝟓
𝑫𝒊𝒂𝒈𝒐𝒏𝒂𝒍=𝟏𝟖+𝟏𝟓
𝑫𝒊𝒂𝒈𝒐𝒏𝒂𝒍=𝟑𝟑

10. Example 2; Finding Diagonal Length
b) In rectangle LMNO, 𝐿𝑁=4𝑥−17 & 𝑀𝑂=2𝑥+13, what is the length of a diagonal?
𝟒𝒙−𝟏𝟕=𝟐𝒙+𝟏𝟑 B S R F
𝟐𝒙=𝟑𝟎
𝒙=𝟏𝟓
𝑫𝒊𝒂𝒈𝒐𝒏𝒂𝒍=𝟐𝒙+𝟏𝟑
𝑫𝒊𝒂𝒈𝒐𝒏𝒂𝒍=𝟐 𝟏𝟓 +𝟏𝟑
𝑫𝒊𝒂𝒈𝒐𝒏𝒂𝒍=𝟑𝟎+𝟏𝟑
𝑫𝒊𝒂𝒈𝒐𝒏𝒂𝒍=𝟒𝟑

11. Thm. 53: Converse of Rhombus – Diagonals Perpendicular
If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus. A B C D

12. Thm. 54: Converse of Rhombus – Diagonals Bisect Opp. ∠’s
If one diagonal of a parallelogram bisects a pair of opposite angles,
then the parallelogram is a rhombus. A B C D

13. Thm. 55: Rectangle – Diagonals Congruent
If the diagonals of a parallelogram are congruent,
then the parallelogram is a rectangle. A B C D

14. Example 1; Identifying Special Parallelograms
Can you conclude the parallelogram is a rhombus, a rectangle, or a square? Explain.
a) Given: Parallelogram ABCD, ∠𝐴𝐵𝐷≅∠𝐶𝐵𝐷, and ∠𝐴𝐷𝐵≅∠𝐶𝐷𝐵
The diagonals bisect the opposite angles C D A B
By Theorem 54, ABCD is a Rhombus.

15. Example 1; Identifying Special Parallelograms
Can you conclude the parallelogram is a rhombus, a rectangle, or a square? Explain.
b) Given: Parallelogram ABCD, 𝐵𝐷 ⏊ 𝐴𝐶 , and 𝐵𝐷=𝐴𝐶
ABCD is a rhombus because the diagonals are perpendicular. C D A B
ABCD is a rectangle because the diagonals are congruent.
ABCD is a square because it is bot a rhombus and a rectangle.

16. Example 1; Identifying Special Parallelograms
Can you conclude the parallelogram is a rhombus, a rectangle, or a square? Explain.
c) Given: Parallelogram ABCD, 𝑚∠𝐴𝐵𝐶=20=𝑚∠𝐶𝐷𝐴, and 𝑚∠𝐵𝐴𝐷=160=𝑚∠𝐵𝐶𝐷
Cannot be a rectangle or square because there are no right angles.
Could be a rhombus if we knew the sides were congruent. C D A B
20ᵒ
160ᵒ
Not enough information to say for sure.

17. Example 1; Identifying Special Parallelograms
Can you conclude the parallelogram is a rhombus, a rectangle, or a square? Explain.
d) Given: Parallelogram ABCD, and 𝐵𝐷 bisects 𝐴𝐶
All parallelograms have diagonals that bisect each other. C D A B
Not enough information to say for sure.

18. Example 2; Using Properties of Special Parallelograms
a) For what value of x is parallelogram ABCD a rhombus?
Want ABCD to be a rhombus.
So the diagonals bisect opposite angles. (Thm. 54) D C B A
𝟔𝒙−𝟐=𝟒𝒙+𝟖
𝟐𝒙=𝟏𝟎
𝟔𝒙−𝟐 °
𝒙=𝟓
𝟒𝒙+𝟖 °

19. Example 2; Using Properties of Special Parallelograms
b) For what value of y is parallelogram DEFG a rectangle?
Want DEFG to be a rectangle. G D E F
So the diagonals bisect and are congruent.
𝟓𝒚+𝟑=𝟕𝒚−𝟓
𝟓𝒚+𝟑
𝟕𝒚−𝟓
−𝟐𝒚=−𝟖
𝒚=𝟒

20. #31: Properties of Rhombuses, Rectangles, and Squares
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