8.2 Parallelograms

Objectives Recognize and apply properties of the sides and angles of parallelograms. Recognize and apply properties of the diagonals of parallelograms.

Parallelograms A quadrilateral with parallel opposite sides is called a parallelogram ( ABCD). A B D C

Parallelograms Theorems Theorem 8.3 – Opposite sides of are ≅. Theorem 8.4 – Opposite s in are ≅. Theorem 8.5 – Consecutive s in are supplementary. Theorem 8.6 – If has 1 rt. , then it has 4 rt. s.

Example 1: Prove that if a parallelogram has two consecutive sides congruent, it has four sides congruent. Given: Prove:

Example 1: Proof: Reasons Statements 1. 1. Given 2. Given 2. 3. Opposite sides of a parallelogram are . 3. 4. Transitive Property 4.

Your Turn: Prove that if and are the diagonals of , and Given: Prove:

Your Turn: Proof: Reasons Statements 1. Given 1. 2. Opposite sides of a parallelogram are congruent. 2. 3. If 2 lines are cut by a transversal, alternate interior s are . 3. 4. Angle-Side-Angle 4.

Example 2: RSTU is a parallelogram. Find and y. If lines are cut by a transversal, alt. int. Definition of congruent angles Substitution

Example 2: Angle Addition Theorem Substitution Subtract 58 from each side.

Example 2: Definition of congruent segments Substitution Divide each side by 3. Answer:

Your Turn: ABCD is a parallelogram. Answer:

Diagonals of Parallelograms Theorem 8.7 – The diagonals of a bisect each other. Theorem 8.8 – Each diagonal of a separates the into two ≅ ∆s.

Example 3: A B C D MULTIPLE-CHOICE TEST ITEM What are the coordinates of the intersection of the diagonals of parallelogram MNPR, with vertices M(–3, 0), N(–1, 3), P(5, 4), and R(3, 1)? Read the Test Item Since the diagonals of a parallelogram bisect each other, the intersection point is the midpoint of

Example 3: Solve the Test Item Find the midpoint of Midpoint Formula The coordinates of the intersection of the diagonals of parallelogram MNPR are (1, 2). Answer: C

Your Turn: A B C D MULTIPLE-CHOICE TEST ITEM What are the coordinates of the intersection of the diagonals of parallelogram LMNO, with vertices L(0, –3), M(–2, 1), N(1, 5), O(3, 1)? Answer: B

Assignment Pre-AP Geometry: Pg. 414 #13, 14, 16 – 33, 36, 50