8 2 Parallelograms Objectives Recognize and apply properties

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 D B C

Parallelograms Theorems � Theorem 8. 3 – Opposite sides of � Theorem 8. 4 – Opposite s in 8. 5 – Consecutive s in supplementary. � Theorem rt. s. 8. 6 – If are ≅. are has 1 rt. , then it has 4

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

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

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

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

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: 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)? A B C D 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: 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)? A Answer: B B C D

Assignment � Pre-AP Geometry: Pg. 414 #13, 14, 16 – 33, 36, 50 � Geometry: Pg. 414 #4 – 12, 16 – 31