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Geometry Lesson 6.3 Proving Quadrilaterals are Parallelograms Parallelogram Properties If a quadrilateral is a parallelogram, then… • Both pairs of opposite sides are… Parallel (||) • Both pairs of opposite sides are… Congruent () • Both pairs of opposite angles are… Congruent () • Consecutive angles are… Supplementary • Diagonals have this property… They bisect each other • If both pairs of opposite sides of a quadrilateral are congruent, then it is a parallelogram B C A D • If both pairs of opposite angles of a quadrilateral are congruent, then it is a parallelogram • If an angle of a quadrilateral is supplementary to both of its consecutive angles, then it’s a parallelogram B (180-x)° A x° (180-x)° D C B A C D • If the diagonals of a quadrilateral bisect each other, then it’s a parallelogram • If one pair of opposite sides of a quadrilateral are congruent and parallel, then it’s a parallelogram B A C D Example 1 • Can you prove each quad. is a give a reason in if-then form a. b. c. Yes! Yes! if opp. sides & ||, then if diags. bisect each other, then ? If so, Yes! if opp. sides , then Practice 1 • Can you prove each quad. is a give a reason in if-then form a. b. No! sides not || Parallel sides would need to also be congruent or congruent sides would need to also be parallel Yes! ? If so, c. Yes! if alt. int. s , if supp. to then lines || both its consec. s, then if 2 pairs || sides, then Example / Practice 2 • What additional information do you need to prove that ABCD is a parallelogram? • AB || CD AD || BC • AB CD AD BC • DCA BAC DAC BCA AE CE mCBA + mDAB = 180° • mCDA + mDAB = 180° • DE BE Example 3 • What values of x and y will make each quadrilateral into a parallelogram (a) A B (b) S D C if opp. sides , then : 3x = 6 x = 2 and (x+2) = (y-1) y = 5 P Q R if opp. s , then 2x° = 70° x = 35 and (3x+5)° = (x+3y)° 110°= (35+3y)° y = 25 Practice 3 • What values of x and y will make each quadrilateral into a parallelogram (a) A B (b) S D C P Q R rise over _____ run • The slope of a line is ______ y • The slope of a line is the change in ____ x divided by the change in ____ • Two lines are parallel if they have the same slope ______. • Two line segments are congruent if they length have the same ________. 3 Methods to Prove a Quadrilateral is a Parallelogram 1. Opposite sides are parallel Show same slope 2. Opposite sides are congruent. Show same length 3.One pair of opposite sides and || Show same slope & same length Example 4: Method 1 • Prove that A(2, -1), B(1, 3), C(6, 5), and D(7, 1) are the vertices of a parallelogram y Method 1: Slope of AB = -4 Slope of CD = -4 Slope of BC = 2/5 Slope of AD = 2/5 C B D x A Because opposite sides have same slope, they are ||. By definition, ABCD is a Example 5: Method 2 • Prove that A(2, -1), B(1, 3), C(6, 5), and D(7, 1) are the vertices of a parallelogram Method 2: Length of AB = Length of CD = Length of BC = Length of AD = y C B D A x 17 17 29 29 Because opposite sides have same length, they are . By theorem 6.6, ABCD is a Method 3: Opposite Sides are & || • Prove that A(2, -1), B(1, 3), C(6, 5), and D(7, 1) are the vertices of a parallelogram Method 3: Slope of AB = -4 Slope of CD = -4 Length of AB = 17 Length of CD = 17 Because opposite sides have same slope and length, they are || & . By theorem 6.10, ABCD is a Practice 1 • Use methods 1 & 2 to prove that M(-4, 7), A(3, 0), T(2, -5), and H(-5, 2) are the vertices of a parallelogram M H Slopes: y A x T Lengths: MA = -1 MA = 7 2 HT = -1 HT = 7 2 MH = 5 MH = 26 AT = 5 AT = 26 Six ways to show a quadrilateral is a parallelogram: 1. parallel Opposite sides are _________ congruent 2. Opposite sides are _________ congruent 3. Opposite angles are _________ supplementary to two 4. One angle is _____________ consecutive angles bisect each other 5. Diagonals _______ || and ___ 6. One pair of opposite sides is ___ (symbols) Assignment • Ch 6.3 (pg. 342-343) #10-26, #40-46 EVEN