Download Advanced Geometry LT 7.1 – Rectangles, Rhombi, and Squares

Advanced Geometry
LT 7.1 – Rectangles, Rhombi, and Squares
What were the properties of rectangles, rhombi, and squares that you found on the quadrilateral preview
activity?
Definition of Rectangle
A rectangle is a quadrilateral with four right angles.
Properties of Rectangles Theorems
If a quadrilateral is a rectangle, then it is a parallelogram.
If a parallelogram is a rectangle, then its diagonals are congruent.
Proof:
Given: ABCD is a rectangle
Prove: 𝐴𝐶 ≅ 𝐵𝐷
Example 1:
Find each measure.
a) 𝐴𝐷 = 7.5 𝑐𝑚, 𝐷𝐶 = 10𝑐𝑚. Find BD.
b)
Definition of Rhombus
A rhombus is a quadrilateral with four congruent sides.
Properties of Rhombi Theorems
If a quadrilateral is a rhombus, then it is a parallelogram.
If a parallelogram is a rhombus, then its diagonals are perpendicular.
If a parallelogram is a rhombus, then each diagonal bisects a pair of opposite angles.
Proof:
Given: JKLM is a rhombus
Prove: 𝐽𝐿 ⊥ 𝑀𝐾
Proof:
Given: JKLM is a rhombus
Prove: 𝑀𝐾 bisects ∠𝐽𝑀𝐿 and ∠𝐽𝐾𝐿;
𝐽𝐿 bisects ∠𝑀𝐽𝐾 and ∠𝑀𝐿𝐾
Example 2:
ABCD is a rhombus.
a)
b)
Definition of Square
A square is a quadrilateral with four congruent sides and four right angles.
How does a square relate to parallelograms, rectangles and rhombi? Justify your answers.
Example 3:
Draw a Venn diagram to show how squares, rectangles, rhombi, and parallelograms are related to each other.
Example 4:
ABCD is a square. Find the measure of each numbered angle.
Example 5:
List the properties that a square “inherits” because it is each of the following quadrilaterals.
a) Parallelogram
b) Rectangle
c) Rhombus
Practice
1. Find the measure of each numbered angle in the rectangle.
2. ABCD is a rhombus, find AB and 𝑚∠𝐴𝐵𝐶.
Find the measure of each angle in the rhombus.
3.
4.
5.
Tell whether each figure must be a rectangle, rhombus or square based on the given information. Use the most
specific name possible, and justify your responses.
6.
7.
8.
Use rhombus DKLM with 𝐴𝑀 = 4𝑥, 𝐴𝐾 = 5𝑥 − 3, and 𝐷𝐿 = 10 to find each of the indicated measures.
9. x
10. AL
11. 𝑚∠𝐾𝐴𝐿
12. DM
For each given property, determine whether it applies to a parallelogram, rectangle, rhombus or square. List all
that apply.
13. The diagonals are congruent.
14. Opposite sides are congruent.
15. The diagonals are perpendicular.
16. Consecutive angles are supplementary.
17. The quadrilateral is equilateral.
18. The diagonals are perpendicular and congruent.
19.
20. Select the word that best describes when each of the following statements are true.