Transcript Slide 1
Geometry Properties of Parallelogram CONFIDENTIAL 1 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° CONFIDENTIAL 2 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 parallelogram ABCD AB || CD; BC || DA □ ABCD D C CONFIDENTIAL 3 Properties of Parallelograms Theorem 1: Theorem If a quadrilateral is a parallelogram, then its opposite sides are congruent. (□ -> opp. sides ) Hypothesis A D CONFIDENTIAL B Conclusion AB CD BC DA C 4 Proof of Theorem 1: A 3 Given: ABCD is a parallelogram. Proof: AB CD, BC DA STATEMENTS 1. 2. 3. 4. 5. 6. 1 D Proof: B 4 2 C REASONS ABCD is a parallelogram AB CD, BC DA 1 2, 3 4 AC AC ABC CDA AB CD, BC DA 1. Given 2. Def. of gm 3. Alt. int. s Thm. 4. Reflex. prop. of . 5. ASA Steps 3,4 6. CPCTC CONFIDENTIAL 5 Properties of Parallelograms Theorem If a quadrilateral is a parallelogram, then its opposite angles are congruent. (□ -> opp. angles ) Hypothesis A D B Conclusion A C B D C CONFIDENTIAL 6 Properties of Parallelograms Theorem If a quadrilateral is a parallelogram, then its consecutive angles are supplementary. (□ -> cons. ∕s supp.) Hypothesis A D B C CONFIDENTIAL Conclusion mA + B 180 mB + C 180 mC + D 180 mD + A 180 7 Properties of Parallelograms Theorem If a quadrilateral is a parallelogram, then its diagonals bisect each other. (□ -> diags. bisect each other) Hypothesis A 3 Conclusion B 1 Z D CONFIDENTIAL 4 2 AZ CZ BZ DZ C 8 Racing application In □ PQRS, QR = 48 cm, RT = 30 cm, and /QPS = 73°. Find each measure. Q R A) PS PS QR PS QR PS = 48 cm gm -> opp. angles Def. of segs. P Substitute 48 for QR B) mPQR mPQR + m QPS = 180 mPQR + 73 = 180 mPQR = 107 C) PT PT RT PT RT PT = 48 cm T S gm -> opp. angles Substitute 73 for m QPS Substract 73 from both sides gm -> diag. bisect each other Def. of segs. Substitute 30 for RT CONFIDENTIAL 9 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 O N K CONFIDENTIAL 10 Using Properties of Parallelogram to Find Measures ABCD is a parallelogram. Find each measure. B 5x+19 C (6y+5)° (10y-1)° A A) AD AD BC AD BC 7x = 5x+19 2x = 19 x = 9.5 AD = 7x = 7(9.5) = 66.5 D gm -> opp. sides Def. of segs. Substitute the given values Subtract 5x from both sides Divide both sides by 2 CONFIDENTIAL 11 B 5x+19 C (6y+5)° (10y-1)° A B) mB mA + mB = 180 (10y-1) + (6y+5) = 180 16y + 4 = 180 16y = 176 y = 11 D gm -> opp. angles Substitute the given values Combine like terms Substract 4 from both sides Divide both sides by 16 B =(6y+5) = [6(11)+5] = 71 CONFIDENTIAL 12 Now you try! ABCD is a parallelogram. Find each measure: F 2a) 2b) G J JG FH E CONFIDENTIAL H 13 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. y B Since ABCD is a parallelogram, both pairs of opposite sides must be parallel. Step1: Graph the given points. C -3 5 5 6 D -2 0 -3 x A 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 14 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. B C -3 5 5 6 D -2 0 -3 x A Step4: Use the slope formula to verify that BC || AD. Slope of BC = Slope of AD = 4–3 =1 2 – (-2) 4 -1 – (-2) = 1 5–1 4 The coordinates of vertex C are (2, 4). CONFIDENTIAL 15 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. CONFIDENTIAL 16 Using properties of Parallelograms in a proof A) B Given: ABCD is a parallelogram. Proof: BAD DCB, ABC CDA C E Proof: A STATEMENTS 1. 2. 3. 4. 5. 6. 7. 8. D REASONS ABCD is a parallelogram AB CD, BC DA BD BD BAF DCB BAD DCB AC AC ABC CDA ABC CDA CONFIDENTIAL 1. Given 2. gm opp.sides 3. Reflex. prop. of . 4. SSS Steps 2,3 5. CPCTC 6. Reflex. prop. of . 7. SSS Steps 2,6 8. CPCTC 17 K H J G N B) Given: GHJN and JKLM are paralellograms. H and M are collinear. Prove: G L L M Proof: STATEMENTS REASONS 1. GHJN and JKLM are grams 2.HJN G, MJK L 3. HJN MJK 4. G L CONFIDENTIAL 1. Given 2. gm opp. s 3. Vert. s Thm. 4. Trans. prop. of . 18 Now you try! H J G N L M 4) Use the figure above to write a two-column proof. Given: GHJN and JKLM are paralellograms. H and M are collinear. H and M are collinear. Prove: N K CONFIDENTIAL 19 Now some problems for you to practice ! CONFIDENTIAL 20 Assessment C In llgm ABCD, AB = 17.5, DE = 18, and mBCD = 110. Find each measure. E B 1) BD 4) /ABC 2) CD 5) /ADC 3) 6) /DAB BE CONFIDENTIAL D A 21 Find the values of x and y for which ABCD must be a parallelogram: 7) 8) CONFIDENTIAL 22 9) Three vertices of llgm DFGH are D(-9,4), F(-1,5) and G(2,0). Find the coordinates of vertex H. CONFIDENTIAL 23 Q S P T V R 10) Write a two-column proof. Given: PSTV is a paralellogram. PQ RQ Prove: STV R CONFIDENTIAL 24 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 parallelogram ABCD AB || CD; BC || DA □ ABCD D C CONFIDENTIAL 25 Properties of Parallelograms Theorem 1: Theorem If a quadrilateral is a parallelogram, then its opposite sides are congruent. (□ -> opp. sides ) Hypothesis A D CONFIDENTIAL B Conclusion AB CD BC DA C 26 Proof of Theorem 1: A 3 Given: ABCD is a parallelogram. Proof: AB CD, BC DA STATEMENTS 1. 2. 3. 4. 5. 6. 1 D Proof: B 4 2 C REASONS ABCD is a parallelogram AB CD, BC DA 1 2, 3 4 AC AC ABC CDA AB CD, BC DA 1. Given 2. Def. of gm 3. Alt. int. s Thm. 4. Reflex. prop. of . 5. ASA Steps 3,4 6. CPCTC CONFIDENTIAL 27 Properties of Parallelograms Theorem If a quadrilateral is a parallelogram, then its opposite angles are congruent. (□ -> opp. angles ) Hypothesis A D B Conclusion A C B D C CONFIDENTIAL 28 Properties of Parallelograms Theorem If a quadrilateral is a parallelogram, then its consecutive angles are supplementary. (□ -> cons. ∕s supp.) Hypothesis A D B C CONFIDENTIAL Conclusion mA + B 180 mB + C 180 mC + D 180 mD + A 180 29 Properties of Parallelograms Theorem If a quadrilateral is a parallelogram, then its diagonals bisect each other. (□ -> diags. bisect each other) Hypothesis A 3 Conclusion B 1 Z D CONFIDENTIAL 4 2 AZ CZ BZ DZ C 30 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. y B Since ABCD is a parallelogram, both pairs of opposite sides must be parallel. Step1: Graph the given points. C -3 5 5 6 D -2 0 -3 x A 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 31 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. B C -3 5 5 6 D -2 0 -3 x A Step4: Use the slope formula to verify that BC || AD. Slope of BC = Slope of AD = 4–3 =1 2 – (-2) 4 -1 – (-2) = 1 5–1 4 The coordinates of vertex C are (2, 4). CONFIDENTIAL 32 Using properties of Parallelograms in a proof A) B Given: ABCD is a parallelogram. Proof: BAD DCB, ABC CDA C E Proof: A STATEMENTS 1. 2. 3. 4. 5. 6. 7. 8. D REASONS ABCD is a parallelogram AB CD, BC DA BD BD BAF DCB BAD DCB AC AC ABC CDA ABC CDA CONFIDENTIAL 1. Given 2. gm opp.sides 3. Reflex. prop. of . 4. SSS Steps 2,3 5. CPCTC 6. Reflex. prop. of . 7. SSS Steps 2,6 8. CPCTC 33 You did a great job today! CONFIDENTIAL 34