1. Properties of Parallelograms
6-2
Properties of Parallelograms
Warm Up
Lesson Presentation
Lesson Quiz
Holt Geometry

2. Are you ready?
Find the value of each variable.
1. x 2. y 3. z

3. Objectives TSW prove and apply properties of parallelograms.
TSW use properties of parallelograms to solve problems.

4. Vocabulary
parallelogram

5. Any polygon with four sides is a quadrilateral
Any polygon with four sides is a quadrilateral. However, some quadrilaterals have special properties. These special quadrilaterals are given their own names.

6. Opposite sides of a quadrilateral do not share a vertex
Opposite sides of a quadrilateral do not share a vertex. Opposite angles do not share a side.
Helpful Hint

7. A quadrilateral with two pairs of parallel sides is a parallelogram
A quadrilateral with two pairs of parallel sides is a parallelogram. To write the name of a parallelogram, you use the symbol .

10. Example 1: Properties of Parallelograms
In CDEF, DE = 74 mm,
DG = 31 mm, and mFCD = 42°. Find CF.

11. Example 2: Properties of Parallelograms
In CDEF, DE = 74 mm,
DG = 31 mm, and mFCD = 42°.

12. Example 3: Properties of Parallelograms
In CDEF, DE = 74 mm,
DG = 31 mm, and mFCD = 42°. Find DF.

13. Example 4
In KLMN, LM = 28 in.,
LN = 26 in., and mLKN = 74°. Find KN.

14. Example 5
In KLMN, LM = 28 in.,
LN = 26 in., and mLKN = 74°. Find mNML.
Def. of angles.

15. Example 6
In KLMN, LM = 28 in.,
LN = 26 in., and mLKN = 74°. Find LO.

16. Example 7: Using Properties of Parallelograms to Find Measures
WXYZ is a parallelogram. Find YZ.

17. Example 8: Using Properties of Parallelograms to Find Measures
WXYZ is a parallelogram. Find mZ .

18. Example 9
EFGH is a parallelogram.
Find JG.

19. Example 10
EFGH is a parallelogram.
Find FH.

20. When you are drawing a figure in the coordinate plane, the name ABCD gives the order of the vertices.
Remember!

21. Example 11: Parallelograms in the Coordinate Plane
Three vertices of JKLM are J(3, –8), K(–2, 2), and L(2, 6). Find the coordinates of vertex M.
Since JKLM is a parallelogram, both pairs of opposite sides must be parallel. J K L

22. Example 12
Three vertices of PQRS are P(–3, –2), Q(–1, 4), and S(5, 0). Find the coordinates of vertex R.
Since PQRS is a parallelogram, both pairs of opposite sides must be parallel. P Q S

23. 6.3 Are you ready? 1. 2. 3. 2x + 7 4. 16x – 9 5. (8y + 5)°
Justify each statement. 1. 2.
Evaluate each expression for x = 12 and
y = 8.5.
3. 2x + 7
4. 16x – 9
5. (8y + 5)°

24. Objective
TSW prove that a given quadrilateral is a parallelogram.

25. You have learned to identify the properties of a parallelogram
You have learned to identify the properties of a parallelogram. Now you will be given the properties of a quadrilateral and will have to tell if the quadrilateral is a parallelogram. To do this, you can
use the definition of a parallelogram or the conditions below.

27. The two theorems below can also be used to show that a given quadrilateral is a parallelogram.

28. Example 1: Verifying Figures are Parallelograms
Show that JKLM is a parallelogram for a = 3 and b = 9.

29. Example 2: Verifying Figures are Parallelograms
Show that PQRS is a parallelogram for x = 10 and y = 6.5.

30. Example 3
Show that PQRS is a parallelogram for a = 2.4 and b = 9.

31. Example 4: Applying Conditions for Parallelograms
Determine if the quadrilateral must be a parallelogram. Justify your answer.

32. Example 5: Applying Conditions for Parallelograms
Determine if the quadrilateral must be a parallelogram. Justify your answer.

33. Example 6
Determine if the quadrilateral must be a parallelogram. Justify your answer.

34. Example 7
Determine if each quadrilateral must be a parallelogram. Justify your answer.

35. To say that a quadrilateral is a parallelogram by
definition, you must show that both pairs of opposite sides are parallel.
Helpful Hint

36. Example 8: Proving Parallelograms in the Coordinate Plane
Show that quadrilateral JKLM is a parallelogram by using the definition of parallelogram. J(–1, –6), K(–4, –1), L(4, 5), M(7, 0).

37. Example 9: Proving Parallelograms in the Coordinate Plane
Show that quadrilateral ABCD is a parallelogram by using Theorem A(2, 3), B(6, 2), C(5, 0), D(1, 1).

38. Example 10
Use the definition of a parallelogram to show that the quadrilateral with vertices K(–3, 0), L(–5, 7), M(3, 5), and N(5, –2) is a parallelogram.

39. You have learned several ways to determine whether a quadrilateral is a parallelogram. You can use the given information about a figure to decide which condition is best to apply.

40. To show that a quadrilateral is a parallelogram, you only have to show that it satisfies one of these sets of conditions.
Helpful Hint

41. Example 11: Application
The legs of a keyboard tray are connected by a bolt at their midpoints, which allows the tray to be raised or lowered. Why is PQRS always a parallelogram?

42. Example 12
The frame is attached to the tripod at points A and B such that AB = RS and BR = SA. So ABRS is also a parallelogram. How does this ensure that the angle of the binoculars stays the same?

44. Lesson Quiz: Part I
1. Show that JKLM is a parallelogram for a = 4 and b = 5.
2. Determine if QWRT must be a parallelogram. Justify your answer.
JN = LN = 22; KN = MN = 10; so JKLM is a parallelogram by Theorem
No; One pair of consecutive s are , and one pair of opposite sides are ||. The conditions for a parallelogram are not met.

45. Lesson Quiz: Part II
3. Show that the quadrilateral with vertices E(–1, 5), F(2, 4), G(0, –3), and H(–3, –2) is a parallelogram.
Since one pair of opposite sides are || and , EFGH is a parallelogram by Theorem

47. Check It Out! Example 3 Continued
Step 4 Use the slope formula to verify that P Q S R
The coordinates of vertex R are (7, 6).

48. Lesson Quiz: Part I
In PNWL, NW = 12, PM = 9, and
mWLP = 144°. Find each measure.
1. PW mPNW 18
144°

49. Lesson Quiz: Part II
QRST is a parallelogram. Find each measure.
2. TQ mT
71° 28

50. Lesson Quiz: Part III
5. Three vertices of ABCD are A (2, –6), B (–1, 2), and C(5, 3). Find the coordinates of vertex D.
(8, –5)

51. Lesson Quiz: Part IV
6. Write a two-column proof.
Given: RSTU is a parallelogram.
Prove: ∆RSU  ∆TUS
Statements
Reasons
1. RSTU is a parallelogram.
1. Given
4. SAS
4. ∆RSU  ∆TUS
3. R  T
 cons. s 
 opp. s 