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 C D

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.

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

1.1.Given Proof: ReasonsStatements 4.Transitive Property4. 2.Given2. 3.Opposite sides of a parallelogram are . 3. Example 1:

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

Proof: ReasonsStatements 1. Given1. 4. Angle-Side-Angle4. 2. Opposite sides of a parallelogram are congruent If 2 lines are cut by a transversal, alternate interior s are . 3. Your Turn:

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

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

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

ABCD is a parallelogram. Answer: Your Turn:

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.

Read the Test Item Since the diagonals of a parallelogram bisect each other, the intersection point is the midpoint of 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)? Example 3:

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

Answer: B 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)? Your Turn:

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