6.3 Proving Quadrilaterals are Parallelograms

1. 6.3 Proving Quadrilaterals are Parallelograms Geometry NCSCOS: 2.02; 2.03

2. E.Q: How do we prove that quadrilaterals are ll’ograms? How can we use coordinate geometry with ll’orgrams?

3. Warmup • Find the slope of AB. • A(2,1), B(6,9) m=2 • A(-4,2), B(2, -1) m= - ½ • A(-8, -4), B(-1, -3) m= 1/7

4. Objectives: • Prove that a quadrilateral is a parallelogram. • Use coordinate geometry with parallelograms.

5. Theorem 6.6: If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. Theorems ABCD is a parallelogram.

6. Theorem 6.7: If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram. Theorems ABCD is a parallelogram.

7. Theorem 6.8: If an angle of a quadrilateral is supplementary to both of its consecutive angles, then the quadrilateral is a parallelogram. Theorems (180 – x)° x° x° ABCD is a parallelogram.

8. Theorem 6.9: If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. Theorems ABCD is a parallelogram.

9. Statements: AB ≅ CD, AD ≅ CB. AC ≅ AC ∆ABC ≅ ∆CDA BAC ≅ DCA, DAC ≅ BCA AB║CD, AD ║CB. ABCD is a  Reasons: Given Ex. 1: Proof of Theorem 6.6

10. Statements: AB ≅ CD, AD ≅ CB. AC ≅ AC ∆ABC ≅ ∆CDA BAC ≅ DCA, DAC ≅ BCA AB║CD, AD ║CB. ABCD is a  Reasons: Given Reflexive Prop. of Congruence Ex. 1: Proof of Theorem 6.6

11. Statements: AB ≅ CD, AD ≅ CB. AC ≅ AC ∆ABC ≅ ∆CDA BAC ≅ DCA, DAC ≅ BCA AB║CD, AD ║CB. ABCD is a  Reasons: Given Reflexive Prop. of Congruence SSS Congruence Postulate Ex. 1: Proof of Theorem 6.6

12. Statements: AB ≅ CD, AD ≅ CB. AC ≅ AC ∆ABC ≅ ∆CDA BAC ≅ DCA, DAC ≅ BCA AB║CD, AD ║CB. ABCD is a  Reasons: Given Reflexive Prop. of Congruence SSS Congruence Postulate CPCTC Ex. 1: Proof of Theorem 6.6

13. Statements: AB ≅ CD, AD ≅ CB. AC ≅ AC ∆ABC ≅ ∆CDA BAC ≅ DCA, DAC ≅ BCA AB║CD, AD ║CB. ABCD is a  Reasons: Given Reflexive Prop. of Congruence SSS Congruence Postulate CPCTC Alternate Interior s Converse Ex. 1: Proof of Theorem 6.6

14. Statements: AB ≅ CD, AD ≅ CB. AC ≅ AC ∆ABC ≅ ∆CDA BAC ≅ DCA, DAC ≅ BCA AB║CD, AD ║CB. ABCD is a  Reasons: Given Reflexive Prop. of Congruence SSS Congruence Postulate CPCTC Alternate Interior s Converse Def. of a parallelogram. Ex. 1: Proof of Theorem 6.6

15. Using properties of parallelograms. • Method 1 Use the slope formula to show that opposite sides have the same slope, so they are parallel. • Method 2 Use the distance formula to show that the opposite sides have the same length. • Method 3 Use both slope and distance formula to show one pair of opposite side is congruent and parallel.

16. Ex. 2: Proving Quadrilaterals are Parallelograms • As the sewing box below is opened, the trays are always parallel to each other. Why?

17. Each pair of hinges are opposite sides of a quadrilateral. The 2.75 inch sides of the quadrilateral are opposite and congruent. The 2 inch sides are also opposite and congruent. Because opposite sides of the quadrilateral are congruent, it is a parallelogram. By the definition of a parallelogram, opposite sides are parallel, so the trays of the sewing box are always parallel. Ex. 2: Proving Quadrilaterals are Parallelograms

18. Another Theorem ~ • Theorem 6.10—If one pair of opposite sides of a quadrilateral are congruent and parallel, then the quadrilateral is a parallelogram. • ABCD is a parallelogram. B C A D

19. Statements: BC ║DA DAC ≅ BCA AC ≅ AC BC ≅ DA ∆BAC ≅ ∆DCA AB ≅ CD ABCD is a  Reasons: 1. Given Ex. 3: Proof of Theorem 6.10Given: BC║DA, BC ≅ DAProve: ABCD is a 

20. Statements: BC ║DA DAC ≅ BCA AC ≅ AC BC ≅ DA ∆BAC ≅ ∆DCA AB ≅ CD ABCD is a  Reasons: Given Alt. Int. s Thm. Ex. 3: Proof of Theorem 6.10Given: BC║DA, BC ≅ DAProve: ABCD is a 

21. Statements: BC ║DA DAC ≅ BCA AC ≅ AC BC ≅ DA ∆BAC ≅ ∆DCA AB ≅ CD ABCD is a  Reasons: Given Alt. Int. s Thm. Reflexive Property Ex. 3: Proof of Theorem 6.10Given: BC║DA, BC ≅ DAProve: ABCD is a 

22. Statements: BC ║DA DAC ≅ BCA AC ≅ AC BC ≅ DA ∆BAC ≅ ∆DCA AB ≅ CD ABCD is a  Reasons: Given Alt. Int. s Thm. Reflexive Property Given Ex. 3: Proof of Theorem 6.10Given: BC║DA, BC ≅ DAProve: ABCD is a 

23. Statements: BC ║DA DAC ≅ BCA AC ≅ AC BC ≅ DA ∆BAC ≅ ∆DCA AB ≅ CD ABCD is a  Reasons: Given Alt. Int. s Thm. Reflexive Property Given SAS Congruence Post. Ex. 3: Proof of Theorem 6.10Given: BC║DA, BC ≅ DAProve: ABCD is a 

24. Statements: BC ║DA DAC ≅ BCA AC ≅ AC BC ≅ DA ∆BAC ≅ ∆DCA AB ≅ CD ABCD is a  Reasons: Given Alt. Int. s Thm. Reflexive Property Given SAS Congruence Post. CPCTC Ex. 3: Proof of Theorem 6.10Given: BC║DA, BC ≅ DAProve: ABCD is a 

25. Statements: BC ║DA DAC ≅ BCA AC ≅ AC BC ≅ DA ∆BAC ≅ ∆DCA AB ≅ CD ABCD is a  Reasons: Given Alt. Int. s Thm. Reflexive Property Given SAS Congruence Post. CPCTC If opp. sides of a quad. are ≅, then it is a . Ex. 3: Proof of Theorem 6.10Given: BC║DA, BC ≅ DAProve: ABCD is a 

26. Objective 2: Using Coordinate Geometry • When a figure is in the coordinate plane, you can use the Distance Formula (see—it never goes away) to prove that sides are congruent and you can use the slope formula (see how you use this again?) to prove sides are parallel.

27. Show that A(2, -1), B(1, 3), C(6, 5) and D(7,1) are the vertices of a parallelogram. Ex. 4: Using properties of parallelograms

28. Method 1—Show that opposite sides have the same slope, so they are parallel. Slope of AB. 3-(-1) = - 4 1 - 2 Slope of CD. 1 – 5 = - 4 7 – 6 Slope of BC. 5 – 3 = 2 6 - 1 5 Slope of DA. - 1 – 1 = 2 2 - 7 5 AB and CD have the same slope, so they are parallel. Similarly, BC ║ DA. Ex. 4: Using properties of parallelograms Because opposite sides are parallel, ABCD is a parallelogram.

29. Method 2—Show that opposite sides have the same length. AB=√(1 – 2)2 + [3 – (- 1)2] = √17 CD=√(7 – 6)2 + (1 - 5)2 = √17 BC=√(6 – 1)2 + (5 - 3)2 = √29 DA= √(2 – 7)2 + (-1 - 1)2 = √29 AB ≅ CD and BC ≅ DA. Because both pairs of opposites sides are congruent, ABCD is a parallelogram. Ex. 4: Using properties of parallelograms

30. Method 3—Show that one pair of opposite sides is congruent and parallel. Slope of AB = Slope of CD = -4 AB=CD = √17 AB and CD are congruent and parallel, so ABCD is a parallelogram. Ex. 4: Using properties of parallelograms

31. Proving quadrilaterals are parallelograms: • Show that both pairs of opposite sides are parallel. • Show that both pairs of opposite sides are congruent. • Show that both pairs of opposite angles are congruent. • Show that one angle is supplementary to both consecutive angles.

32. .. continued.. • Show that the diagonals bisect each other • Show that one pair of opposite sides are congruent and parallel.

33. Engineering Deshon uses an expandable gate to keep his new puppy in the kitchen. As the gate expands or collapses, the shapes that form the gate always remain parallelograms. Explain why this is true.

34. Drafting Before computer drawing program become available, blueprints for buildings or mechanical parts were drawn by hand. One of the tools drafters used, is a parallel ruler. Holding one of the bars in the place and moving the other allowed the drafter to draw a line ll to the first in many position on the page. Why does the parallel ruler guarantee that the second line will be ll to the first?

35. Arts The Navoja people are well known for their skills in weaving. Eye- Dazzler rugs became popular with Navoja weavers in the 1880s.What types of shapes do you see most?