1. Class Greeting

2. Properties of Parallelograms
Chapter 6 - Lesson 2
Properties of Parallelograms

3. Objective: The students will be able to prove and apply properties of parallelograms to solve problems.

4. Any polygon with 4 sides is a quadrilateral
Any polygon with 4 sides is a quadrilateral. However, some quadrilaterals have special properties. These special quadrilaterals are given their own names.
Quadrilateral

5. Helpful Hints
Opposite sides of a quadrilateral do not share a vertex.
Opposite angles do not share a side.

6. Definition Parallelogram Figure ABCD is a Parallelogram
a quadrilateral with both pairs of opposite sides parallel
Figure ABCD is a Parallelogram

9. Example 1A: Properties of Parallelograms
In CDEF, DE = 74 mm,
Find CF.
 opp. sides 
CF = DE
Def. of  segs.
CF = 74 mm
Substitute 74 for DE.

10. Check It Out! Example 1a
In KLMN, LM = 28 in.
Find KN.
 opp. sides 
LM = KN
Def. of  segs.
LM = 28 in.
Substitute 28 for DE.

11. Example 1B: Properties of Parallelograms
In CDEF, mFCD = 42°. Find mEFC.
mFCD + mEFC = 180°
 consecutive angles are supp.
42° + mEFC = 180°
Substitution
mEFC = 138°
Subtraction

12. Check It Out! Example 1b
In KLMN, mLKN = 74°. Find mNML.
NML  LKN
 opp. s 
mNML = mLKN
Def. of  s.
mNML = 74°
Substitute 74° for mLKN.

13. Example 1C: Properties of Parallelograms
In CDEF, DG = 31. Find DF.
DG = GF
 diagonals bisect each other
GF = 31
Substitution
DF = DG + GF
Segment Addition Postulate
DF =
Substitution
DF = 62
Simplify

14. Check It Out! Example 1c
In KLMN, LN = 26 in. Find LO.
LN = 2LO
 diags. bisect each other.
26 = 2LO
Substitute 26 for LN.
LO = 13 in.
Simplify.

15. Example 1D: Using Properties of Parallelograms to Find Measures
WXYZ is a parallelogram. Find YZ.
 opposite sides are 
XW = YZ 
XW = YZ
Definition of  segments
8a – 4 = 6a + 10
Substitution
YZ = 8a - 4
Given (diagram)
2a – 4 = 10
Subtraction
YZ = 8(7) - 4
Substitution
2a = 14
Addition
YZ =
Simplify
a = 7
Division
YZ = 52
Simplify

16. Check It Out! Example 1d
EFGH is a parallelogram.
Find JG.
EJ = JG
 diagonals bisect each other
3w = w + 8
Substitution
JG = w + 8
Given
2w = 8
Subtraction
JG = 4 + 8
Substitution
w = 4
Division
JG = 12
Simplify

17. RSTU is a parallelogram. Find and y.40 58
Example 2-2a

18. ABCD is a parallelogram.
Answer:
Example 2-2d

19. Example 3: 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
Step 1 Graph the given points.

20. Step 2 Find the slope of by counting the units from K to L.
Example 3 Continued
Step 2 Find the slope of by counting the units from K to L.
The rise from 2 to 6 is 4.
The run of –2 to 2 is 4. J K L
Step 3 Start at J and count the
same number of units.
A rise of 4 from –8 is –4.
A run of 4 from 3 is 7.
Label (7, –4) as vertex M. M
Answer: The coordinates of vertex M are (7, –4).

21. Answer: The coordinates of vertex R are (3, –6).
Check It Out! Example 3
Three vertices of PQRS are P(–3, –2), Q(–1, 4), and S(5, 0). Find the coordinates of vertex R. P Q S
Down 6
Down 6
Over 2
Over 2 R
Answer: The coordinates of vertex R are (3, –6).

22. Example 4: Using Properties of Parallelograms in a Proof
Write a two-column proof.
Given: ABCD is a parallelogram.
Prove: ∆AEB  ∆CED
Statements
Reasons 4.
1. ABCD is a parallelogram
1. Given
 opposite sides 
 diagonals bisect
each other ∆ ∆
4. SSS

23. Write a two-column proof.
Given: GHJN and JKLM are parallelograms. H and M are collinear. N and K are collinear.
Prove: H M
Statements
Reasons
1. GHJN and JKLM are
parallelograms.
1. Given
2. H and HJN are supp.
M and MJK are supp.
 cons. s supp.
3. HJN  MJK
3. Vert. s Theorem
4. H  M
4.  Supplements Theorem

24. Opposite sides of a parallelogram are .
Prove that if a parallelogram has two consecutive sides congruent, it has four sides congruent.
Given:
Prove:
Proof:
Reasons
Statements 1.
Given 2.
Given 3.
Opposite sides of a parallelogram are . 4.
4. Substitution Property
Example 2-1a

25. Prove that if and are the diagonals of , .
Given:
Prove:
Proof:
Reasons
Statements 1.
1. Given 2.
2. Opposite sides of a parallelogram are congruent. 3.
CBD
3. If 2 lines are cut by a transversal, alternate interior s are . 4.
4. Angle-Side-Angle
Example 2-1c

26. Kahoot!

27. Lesson Summary:
Objective: The students will be able to prove and apply properties of parallelograms to solve problems.

28. Preview of the Next Lesson:
Objective: The students will prove quadrilaterals parallelograms.

29. Stand Up Please