WARM UP Statements Reasons 1. WXYX is a 1. Given 2. WX  ZY, WZ  YX 1) Complete the Proof. Z Y Given: WXYZ is a parallelogram Prove: ΔYZX  ΔWXZ W X Statements Reasons 1. WXYX is a 1. Given 2. Opposite sides of a parallelogram are congruent. 2. WX  ZY, WZ  YX 3. ZX  ZX 3. Reflexive Property 4. ΔYZX  ΔWXZ 4. SSS Postulate

Statements Reasons 1. BCDE is a 1. Given 2. EB  DC 3. AE  CD 2) Complete the Proof. Given: BCDE is a parallelogram AE  CD Prove: EAB  EBA A B C Statements Reasons 1. BCDE is a 1. Given 2. Opposite sides of a parallelogram are congruent. 2. EB  DC 3. AE  CD 3. Given 4. EB  AE 4. Substitution 5. If two sides of a triangle are , then the angles opposite those sides are . 5. EAB  EBA

HW ANSWERS Pg.168 Def. of Parallelogram If lines are ||, alternate interior angles are congruent. Opposite angles of a parallelogram are congruent. Opposite sides of a parallelogram are congruent. Diagonals of a parallelogram bisect each other. Pg.169 a = 8, b = 10, x = 118, y = 62 a = 8, b = 15, x = 80, y = 70 a = 5, b = 3, x = 120, y = 22 a = 9, b = 11, x = 33, y = 27 a = 8, b = 8, x = 56, y = 68 a = 10, b = 4, x = 90, y = 45

HW ANSWERS Pg.169 Perimeter = 60 ST = 14, SP = 13 Pg.170 x = 3, y = 5

1) Name all the properties of a parallelogram. REVIEW 1) Name all the properties of a parallelogram. • 2 pairs of opposite sides are parallel • 2 pairs of opposite sides are congruent • 2 pairs of opposite angles are congruent • Diagonals bisect each other • Consecutive angles are supplementary

Given the below parallelogram, complete the statements. D M Given the below parallelogram, complete the statements. DAC  _____ AB = _____ mBCD = _____ CM = _____ BCA AD || _____ DA = _____ BAC  _____ BM = _____ BC CD BC mDAB DCA AM DM

Proving Quadrilaterals Section 5-2 Proving Quadrilaterals are Parallelograms

Theorem 5-4: If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. E F G H

1. EF  GH; FG  EH 1. Given 2. FH  FH 2. Reflexive Property 3 4 PROOF OF THEOREM 5-4: Given: EF  GH, FG  EH Prove: EFGH is a 1. EF  GH; FG  EH 1. Given 2. FH  FH 2. Reflexive Property 3. ΔEFH  ΔGHF 3. SSS Postulate 4. 1  4, 2  3 4. CPCTC 5. If alternate interior angles are congruent, then lines are parallel. 5. EF || GH, FG || HE 6. EFGH is a 6. Def. of parallelogram

Theorem 5-5: If one pair of opposite sides of a quadrilateral are both congruent and parallel, then the quadrilateral is a parallelogram. E F G H

1. EF  GH; EF || GH 1. Given 2. 1  4 3. FH  FH PROOF OF THEOREM 5-5: Given: EF  GH, EF || GH Prove: EFGH is a 1. EF  GH; EF || GH 1. Given 2. If lines are parallel, alternate interior angles are congruent. 2. 1  4 3. FH  FH 3. Reflexive Property 4. ΔEFH  ΔGHF 4. SAS Postulate 5. FG  HE 5. CPCTC 6. If both pairs of opposite sides of a quadrilateral are congruent, then it is a parallelogram. 6. EFGH is a

Theorem 5-6: If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram. D C B A

1. mA = mC = y; 1. Given 2. 2x + 2y = 360 3. x + y = 180 D PROOF OF THEOREM 5-6: x y Given: mA = mC = y; mB = mD = x Prove: ABCD is a y x B A 1. mA = mC = y; mB = mD = x 1. Given 2. The sum of the interior angles of a quadrilateral is 360. 2. 2x + 2y = 360 3. x + y = 180 3. Division Property A and D are supp. 4. Definition of supp. angles A and B are supp. 5. If same-side interior angles are supplementary, then lines are parallel. 5. AB || CD, AD || BC 6. ABCD is a 6. Def. of parallelogram

Theorem 5-7: If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. Q R S T M

Based on the markings on each figure, A. Decide if each figure is a parallelogram (YES or NO). B. If yes, justify your answer. State the theorem that is supported by the figure. If no, identify which theorem is not justified or is not met by the diagram.

NO YES Opposite sides are not congruent. EXAMPLE 1: NO Opposite sides are not congruent. EXAMPLE 2: YES Both pairs of opposite sides are parallel.

YES YES Diagonals bisect each other. EXAMPLE 3: YES Diagonals bisect each other. EXAMPLE 4: YES Both pairs of opposite angles are congruent.

NO YES Pair of congruent/parallel sides is not the same pair of sides. EXAMPLE 5: NO Pair of congruent/parallel sides is not the same pair of sides. EXAMPLE 6: YES One pair of sides is both congruent and parallel.