8.2 parallelograms. objectives recognize and apply properties of the sides and angles of...
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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
CD
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.
2.
3. 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 ItemSince 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 verticesL(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