Properties of Rhombuses Rectangles and Squares Skill 32
Properties of Rhombuses, Rectangles, and Squares Skill 32 a
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. B A C D
Thm. 51: Rhombus – Diagonals Bisect Opp. Angles If a parallelogram is a rhombus, then its diagonals bisect the opposite angles. B A C D
Thm. 52: Rectangle – Diagonals Congruent If a parallelogram is a rectangle, then its diagonals are congruent. B C A D
Example 1; Finding Angle Measures Rhombus – Diagonals Perpendicular Thm. C B AIA Theorem Rhombus – Diagonals Bisect Opp. Angles Thm. A 4 1 2 3 D Triangle Angle-Sum Thm.
Example 1; Finding Angle Measures Rhombus – Diagonals Bisect Opp. Angles Thm. AIA Theorem Transitive Property Triangle Angle-Sum Thm. Q 1 P 2 3 4 S R
Example 2; Finding Diagonal Length S B R F
Example 2; Finding Diagonal Length S B R F
Conditions of Rhombuses, Rectangles, and Squares Skill 32 b
Objective HSG-CO. 11: Students are responsible for determining if a parallelogram is a rhombus, a rectangle, or a square.
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. 53: Converse of Rhombus – Diagonals Perpendicular If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus. B A 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. B C A D
Thm. 55: Rectangle – Diagonals Congruent If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle. B C A D
Example 1; Identifying Special Parallelograms Can you conclude the parallelogram is a rhombus, a rectangle, or a square? Explain. The diagonals bisect the opposite angles By Theorem 54, ABCD is a Rhombus. B C A D
Example 1; Identifying Special Parallelograms Can you conclude the parallelogram is a rhombus, a rectangle, or a square? Explain. ABCD is a rhombus because the B diagonals are perpendicular. ABCD is a rectangle because the A diagonals are congruent. ABCD is a square because it is bot a rhombus and a rectangle. C D
Example 1; Identifying Special Parallelograms Can you conclude the parallelogram is a rhombus, a rectangle, or a square? Explain. Cannot be a rectangle or square because there are no right angles. Could be a rhombus if we knew B 20ᵒ the sides were congruent. Not enough information to say for sure. A 160ᵒ 20ᵒ C D
Example 1; Identifying Special Parallelograms Can you conclude the parallelogram is a rhombus, a rectangle, or a square? Explain. All parallelograms have diagonals that bisect each other. Not enough information to say for sure. B C A D
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 A angles. (Thm. 54) B D C
Example 2; Using Properties of Special Parallelograms b) For what value of y is parallelogram DEFG a rectangle? Want DEFG to be a rectangle. D So the diagonals bisect and are congruent. G E F
#33 b: Conditions of Rhombuses, Rectangles, and Squares Ø Questions? Ø Summarize Notes Ø Homework Ø Video Ø Quiz
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