Properties of Rhombuses, Rectangles, and Squares Skill 31

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

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.

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

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

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

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°

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°

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

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

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

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

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

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.

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.

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.

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.

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 °

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

#31: Properties of Rhombuses, Rectangles, and Squares Questions? Summarize Notes Homework Video Quiz