This document defines and provides properties of parallelograms, rectangles, squares, and rhombuses. It states that a parallelogram is a quadrilateral with both pairs of opposite sides parallel. A rectangle is a parallelogram with right angles, and a square is a rectangle with all sides congruent. A rhombus is a parallelogram with all sides equal. It then provides 5 theorems about parallelograms, including that diagonals bisect each other and opposite angles are congruent. An example problem applies these properties to find missing angle and side measures.

1. PARALLELOGRAM

2. ■ Definitions
A parallelogram is a quadrilateral in which both pairs of opposite sides are
parallel.
A rectangle is a parallelogram all of whose angles are right angles.
A square is a rectangle all whose sides are congruent.
A rhombus is a parallelogram of whose sides are equal.

3. BC ∥ AD and BA ∥ CD
BADC is a parallelogram

4. Theorem 1.
Each diagonal separates a parallelogram into two congruent triangles.
∆𝐴𝐷𝐶 ≅ ∆ABC
Theorem 2.
In a parallelogram, any two opposite angles are congruent.
∠𝐴𝐷𝐶 ≅ ∠𝐴𝐵𝐶, and ∠𝐷𝐴𝐵 ≅ ∠𝐵𝐶𝐷
Theorem 3.
In a parallelogram, any two opposite sides are congruent.
DA ≅ CB, and DC ≅ AB
Corollary 3.1
Parallel lines are everywhere equidistant
DA = CB, and DC = AB

5. Theorem 4.
Any two consecutive interior angles of a
parallelogram are supplementary
Theorem 5.
The diagonals of a parallelogram bisect
each other.

6. Let’s Apply
A. Find the following:
1. x =
2. BC =
3. AD =
Solution:
1. 𝐵𝐶 = 𝐴𝐷
4x – 10 = 3x + 5
4x – 3x = 5 + 10
x = 15
2. 𝐵𝐶 = = 4x – 10
= 4(15) – 10
= 60 – 10
= 50
3. 𝐴𝐷 = 3x + 5
= 3(15) + 5
= 45 + 5
= 50

7. Solution:
a. ∠𝐴𝐷𝐶
3x° + 120° = 180°
3x = 180 – 120
3x = 60
x = 20
∠𝐴𝐷𝐶 =
= 3(20)°
= 60°
B. Given ∠𝐴𝐷𝐶 = 3x° and ∠𝐷𝐶𝐵 = 120°.
a. Find ∠𝐴𝐷𝐶.
b. Find ∠𝐷𝐴𝐵.
c. Find ∠𝐴𝐵𝐶.
b. ∠𝐷𝐴𝐵 = 120°
Opposite angles of a quadrilateral are
congruent. Thus, ∠𝐷𝐴𝐵 = ∠𝐷𝐶𝐵 )
c. ∠𝐴𝐵𝐶 = 60°
Opposite angles of a quadrilateral are congruent.
Thus, ∠𝐴𝐵𝐶 = ∠𝐴𝐷𝐶 )