1. 6.3 Tests for Parallelograms
Geometry

2. TARGETS:
Recognize the conditions that ensure a quadrilateral is a parallelogram.
Prove that a set of points forms a parallelogram in the coordinate plane.

3. Content Standards G-CO.11 Prove geometric theorems.
G-CO.12 Make geometric constructions.
G-GPE.4 Use coordinates to prove simple geometric theorems algebraically.
Mathematical Practices
1 Make sense of problems and persevere in solving them
2 Reason abstractly and quantitatively.
6 Attend to precision.

4. You recognized and applied properties of parallelograms. (Lesson 6-2)
Recognize the conditions that ensure a quadrilateral is a parallelogram.
Prove that a set of points forms a parallelogram in the coordinate plane.

5. Theorems
Theorem 6.9: If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.
ABCD is a parallelogram.

6. Theorems
Theorem 6.10: If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.
ABCD is a parallelogram.

7. Theorems
Theorem 6.11: If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.
ABCD is a parallelogram.

8. Theorems
Theorem 6.12—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

9. Another Theorem~
If an angle of a quadrilateral is supplementary to both of its consecutive angles, then the quadrilateral is a parallelogram.
(180 – x)° x° x°
ABCD is a parallelogram.

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

11. Ex. 1: Proof of Theorem 6.6 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

12. Ex. 1: Proof of Theorem 6.6 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

13. Ex. 1: Proof of Theorem 6.6 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

14. Ex. 1: Proof of Theorem 6.6 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

15. Ex. 1: Proof of Theorem 6.6 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.

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

17. Ex. 2: Proving Quadrilaterals are Parallelograms
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.

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

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

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

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

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

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

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

25. 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.

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

27. Ex. 4: Using properties of parallelograms
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
Slope of DA.
- 1 – 1 = 2
AB and CD have the same slope, so they are parallel. Similarly, BC ║ DA.
Because opposite sides are parallel, ABCD is a parallelogram.

28. Ex. 4: Using properties of parallelograms
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.

29. Ex. 4: Using properties of parallelograms
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.