Properties of Parallelogram Geometry Properties of Parallelogram CONFIDENTIAL

Warm up An interior angle measure of a regular polygon is given. Find the number of sides and the measure of each exterior angle: 1) 120° 2) 135° 3) 156° hexagon, 60° ; 2) heptagon, 45° ; 3) 15 - gon, 24° CONFIDENTIAL

Properties of Parallelograms Any polygon with four sides is a quadrilateral. However, some quadrilaterals have special properties. These special quadrilaterals have their own names. A quadrilaterals with two pairs of parallel sides is a parallelogram. To write the name of a parallelogram, you use the symbol □. A B C D parallelogram ABCD □ ABCD AB || CD; BC || DA CONFIDENTIAL

Properties of Parallelograms Theorem 1: Theorem Hypothesis Conclusion If a quadrilateral is a parallelogram, then its opposite sides are congruent. (□ -> opp. sides ) A B C D CONFIDENTIAL

Proof of Theorem 1: A B C D 1 2 3 4 CONFIDENTIAL

Properties of Parallelograms Theorem Hypothesis Conclusion If a quadrilateral is a parallelogram, then its opposite angles are congruent. (□ -> opp. angles ) A B C D CONFIDENTIAL

Properties of Parallelograms Theorem Hypothesis Conclusion If a quadrilateral is a parallelogram, then its consecutive angles are supplementary. (□ -> cons. ∕s supp.) A B C D CONFIDENTIAL

Properties of Parallelograms Theorem Hypothesis Conclusion If a quadrilateral is a parallelogram, then its diagonals bisect each other. (□ -> diags. bisect each other) A B C D 1 2 3 4 Z CONFIDENTIAL

In □ PQRS, QR = 48 cm, RT = 30 cm, and /QPS = 73°. Find each measure. Racing application In □ PQRS, QR = 48 cm, RT = 30 cm, and /QPS = 73°. Find each measure. Q R S P T CONFIDENTIAL

In □KLMN, LM = 28 in., LN = 26 in. and m/LKN = 74°. Find each measure: Now you try! In □KLMN, LM = 28 in., LN = 26 in. and m/LKN = 74°. Find each measure: 1a) KN 1b) m/NML 1c) LO M L K N O ANSWER: 1a) 28 in.; 1b) 74° ; 1c) 13 in. CONFIDENTIAL

Using Properties of Parallelogram to Find Measures ABCD is a parallelogram. Find each measure. B C D A 5x+19 (6y+5)° (10y-1)° CONFIDENTIAL

B C D A 5x+19 (6y+5)° (10y-1)° CONFIDENTIAL

ABCD is a parallelogram. Find each measure: Now you try! ABCD is a parallelogram. Find each measure: E F G H J 4z-9 2z w+8 3w 2a) JG 2b) FH 2a) 12; 2b) 18 CONFIDENTIAL

Parallelograms in the coordinate plane Three vertices of □ABCD are A(1,-2), B(-2,3) and D(5,-1). Find the coordinates of vertex C. B A D 5 6 C -3 -2 x y Since ABCD is a parallelogram, both pairs of opposite sides must be parallel. Step1: Graph the given points. Step2: Find the slope of AB by counting the units from A to B. The rise from -2 to 3 is 5. The run from 1 to -2 is -3. CONFIDENTIAL

Step3: Start at D and count the same number of units. 5 6 C -3 -2 x y Step3: Start at D and count the same number of units. The rise from -1 to 4 is 5. The run from 5 to 2 is -3. Label (2,4) as vertex C. Step4: Use the slope formula to verify that BC || AD. Slope of BC = 4 – 3 = 1 2 – (-2) 4 Slope of AD = -1 – (-2) = 1 5 – 1 4 The coordinates of vertex C are (2, 4). CONFIDENTIAL

Now you try! 3) Three vertices of □PQRS are P(-3,-2), Q(-1,4) and S(5,0). Find the coordinates of vertex R. 3) (7,6) CONFIDENTIAL

Using properties of Parallelograms in a proof B C D A E CONFIDENTIAL

K J H G N M L CONFIDENTIAL

4) Use the figure above to write a two-column proof. Now you try! J H G N M L 4) Use the figure above to write a two-column proof. 5) 45° CONFIDENTIAL

Now some problems for you to practice ! CONFIDENTIAL

Assessment C D A B E BD CD 3) BE 4) /ABC 5) /ADC 6) /DAB CONFIDENTIAL 1) 36; 2) 17.5; 3) 18; 4) 70°; 5) 70°; 6) 110° CONFIDENTIAL

Find the values of x and y for which ABCD must be a parallelogram: 7) 8) 7) x = 5; 8) x = 1.6; y = 1 CONFIDENTIAL

9) Three vertices of llgm DFGH are D(-9,4), F(-1,5) and G(2,0) 9) Three vertices of llgm DFGH are D(-9,4), F(-1,5) and G(2,0). Find the coordinates of vertex H. 9) (-6,-1) CONFIDENTIAL

P V T S R Q CONFIDENTIAL

Properties of Parallelograms Let’s review Properties of Parallelograms Any polygon with four sides is a quadrilateral. However, some quadrilaterals have special properties. These special quadrilaterals have their own names. A quadrilaterals with two pairs of parallel sides is a parallelogram. To write the name of a parallelogram, you use the symbol □. A B C D parallelogram ABCD □ ABCD AB || CD; BC || DA CONFIDENTIAL

Properties of Parallelograms Theorem 1: Theorem Hypothesis Conclusion If a quadrilateral is a parallelogram, then its opposite sides are congruent. (□ -> opp. sides ) A B C D CONFIDENTIAL

Proof of Theorem 1: A B C D 1 2 3 4 CONFIDENTIAL

Properties of Parallelograms Theorem Hypothesis Conclusion If a quadrilateral is a parallelogram, then its opposite angles are congruent. (□ -> opp. angles ) A B C D CONFIDENTIAL

Properties of Parallelograms Theorem Hypothesis Conclusion If a quadrilateral is a parallelogram, then its consecutive angles are supplementary. (□ -> cons. ∕s supp.) A B C D CONFIDENTIAL

Properties of Parallelograms Theorem Hypothesis Conclusion If a quadrilateral is a parallelogram, then its diagonals bisect each other. (□ -> diags. bisect each other) A B C D 1 2 3 4 Z CONFIDENTIAL

Parallelograms in the coordinate plane Three vertices of □ABCD are A(1,-2), B(-2,3) and D(5,-1). Find the coordinates of vertex C. B A D 5 6 C -3 -2 x y Since ABCD is a parallelogram, both pairs of opposite sides must be parallel. Step1: Graph the given points. Step2: Find the slope of AB by counting the units from A to B. The rise from -2 to 3 is 5. The run from 1 to -2 is -3. CONFIDENTIAL

Step3: Start at D and count the same number of units. 5 6 C -3 -2 x y Step3: Start at D and count the same number of units. The rise from -1 to 4 is 5. The run from 5 to 2 is -3. Label (2,4) as vertex C. Step4: Use the slope formula to verify that BC || AD. Slope of BC = 4 – 3 = 1 2 – (-2) 4 Slope of AD = -1 – (-2) = 1 5 – 1 4 The coordinates of vertex C are (2, 4). CONFIDENTIAL

Using properties of Parallelograms in a proof B C D A E CONFIDENTIAL

You did a great job today! CONFIDENTIAL