University:
Missouri State University
Course:
MTH 460 | College Geometry
Academic year:

2021
Views:

116

Pages:

8
Author:

Nevaeh Bauer

Definition: A parallelogram is a quadrilateral with both pairs of opposite sides parallel. There are several properties of parallelograms in the form of theorems. We will look at these properties and their converses today. Theorem 5-1: Opposite sides of a parallelogram are congruent. H 1 E 3 2 1. HGEF is a parallelogram 1. Given 2. GH ll EF; EH ll FG 3. 3 @ 2 2. Definition of a parallelogram. 3. If two parallel lines are cut by a 1 @ 4 G 4 F transversal, then alternate interior angles are congruent. 4. GE @ GE 4. Reflexive 5. DGHE @ DEFG 5. ASA 6. HG @ EF; HE @ GF 6. CPCTC Converse (Theorem 5-4): If both pairs of opposite sides are congruent, then the quadrilateral is a parallelogram. Theorem 5-2: Opposite angles of a parallelogram are congruent. A 1 B 3 2 1. ABCD is a parallelogram 1. Given 2. AD ll BC; AB ll DC 3. 3 @ 2 2. Definition of a parallelogram. 3. If two parallel lines are cut by a 1 @ 4 D 4 C transversal, then alternate interior angles are congruent. 4. BD @ BD 4. Reflexive 5. DABD @ DCDB 5. ASA 6. A @ C 6. CPCTC Converse (Theorem 5-6): If both pairs of opposite angles are congruent, then the quadrilateral is a parallelogram. Example 1: Example 2: A 142° B 2x + 5 5y 105° C D mB = ______ 38 38 mC = ______ 142° mD = ______ 180 mA + mB = ______ mA + mB + mC + mD = ______ 360 2x + 5 = 105 2x = 100 x = 50 105 + 5y = 180 5y = 75 y = 15 Theorem 5-3: Diagonals of a parallelogram bisect one another. T Q S M R M is the midpoint of TR and QS Therefore, TM = MR and QM = MS. Converse (Theorem 5-7): If diagonals of a quadrilateral bisect one another, then the quadrilateral is a parallelogram. Examples (all figures are parallelograms): A 3 5 8 2 6 7 9 1 4 B AC = 3x - 1 C D CE = ½x + 9 E CD = ¾y CB = y - 4 m1 = _____ m2 = _____ m3 = 80 m4 = _____ x = ____ AC = ____ m5 = _____ m6 = 80 CE = ____ AE = ____ m7 = _____ m8 = 40 y = ____ CD = ____ m9 = _____ CB = ____ DB = ____ Theorem 5-5: If one pair of opposite sides of a quadrilateral are parallel and congruent, then the quadrilateral is a parallelogram. T Given: TS @ QR; TS ll QR Prove: TSRQ is a parallelogram. 4 Q 1. TS @ RQ; TS ll RQ 2. 1 @ 3 3 1 2 R 1. Given 2. If two parallel lines are cut by a transversal, then alternate interior angles are congruent 3. SQ @ SQ 3. Reflexive 4. DTSQ @ DRQS 4. SAS 5. 4 @ 2 5. CPCTC 6. TQ ll SR 6. If two lines are cut by a transversal and alternate interior angles are congruent, then the lines are parallel. 7. TSRQ is a parallelogram 7. Definition of a parallelogram. S RECAP: There are 5 ways to prove that a quadrilateral is a parallelogram. They are… 1. If both pairs of opposite sides are parallel, then the quadrilateral is a parallelogram. (Definition of a Parallelogram). 2. If both pairs of opposite sides are congruent, then the quadrilateral is a parallelogram. 3. If one pair of opposite sides are congruent AND parallel, then the quadrilateral is a parallelogram. 4. If both pairs of opposite angles are congruent, then the quadrilateral is a parallelogram. 5. If the diagonals of a quadrilateral bisect one another, then the quadrilateral is a parallelogram.