CONFIDENTIAL 1
Geometry
Properties of Parallelogram
CONFIDENTIAL 2
Warm up
1) 120°
2) 135°
3) 156°
An interior angle measure of a regular polygon is given. Find the number of sides and the
measure of each exterior angle:
CONFIDENTIAL 3
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 □.
Properties of Parallelograms
parallelogram ABCD
□ ABCD
A B
CD
AB || CD; BC || DA
CONFIDENTIAL 4
Theorem Hypothesis Conclusion
If a quadrilateral is a parallelogram, then its opposite sides are congruent. (□ -> opp. sides )
Properties of Parallelograms
A B
CD
AB CDBC DA
Theorem 1:
CONFIDENTIAL 5
Proof of Theorem 1:A B
CD
1
2
3
4
Given: ABCD is a parallelogram.Proof: AB CD, BC DA
Proof:
STATEMENTS REASONS
1. ABCD is a parallelogram 1. Given2. AB CD, BC DA 2. Def. of gm3. 1 2, 3 4 3. Alt. int. s Thm.4. AC AC 4. Reflex. prop. of .5. ABC CDA 5. ASA Steps 3,46. AB CD, BC DA 6. CPCTC
CONFIDENTIAL 6
Theorem Hypothesis Conclusion
If a quadrilateral is a parallelogram, then its opposite angles are congruent. (□ -> opp. angles )
Properties of Parallelograms
A B
CD
A CB D
CONFIDENTIAL 7
Theorem Hypothesis Conclusion
If a quadrilateral is a parallelogram, then its consecutive angles are supplementary. (□ -> cons. ∕s supp.)
Properties of Parallelograms
A B
CD
mA + B 180mB + C 180mC + D 180mD + A 180
CONFIDENTIAL 8
Theorem Hypothesis Conclusion
If a quadrilateral is a parallelogram, then its diagonals bisect each other.(□ -> diags. bisect each other)
Properties of Parallelograms
A B
CD
1
2
3
4
ZAZ CZBZ DZ
CONFIDENTIAL 9
In □ PQRS, QR = 48 cm, RT = 30 cm, and /QPS = 73°. Find each measure.
Racing application
Q R
SP
TA) PS PS QR gm -> opp. angles PS QR Def. of segs. PS = 48 cm Substitute 48 for QR
B) mPQR mPQR + mQPS = 180 gm -> opp. angles mPQR + 73 = 180 Substitute 73 for mQPS mPQR = 107 Substract 73 from both sides
C) PT PT RT gm -> diag. bisect each other PT RT Def. of segs. PT = 48 cm Substitute 30 for RT
CONFIDENTIAL 10
Now you try!
In □KLMN, LM = 28 in., LN = 26 in. and m/LKN = 74°. Find each measure:
1a) KN1b) m/NML1c) LO
M L
KN
O
CONFIDENTIAL 11
ABCD is a parallelogram. Find each measure.
Using Properties of Parallelogram to Find Measures
B C
DA
5x+19
(6y+5)°
(10y-1)°
A) AD AD BC gm -> opp. sides AD BC Def. of segs. 7x = 5x+19 Substitute the given values 2x = 19 Subtract 5x from both sides x = 9.5 Divide both sides by 2 AD = 7x = 7(9.5) = 66.5
CONFIDENTIAL 12
B C
DA
5x+19
(6y+5)°
(10y-1)°
B) mB mA + mB = 180 gm -> opp. angles (10y-1) + (6y+5) = 180 Substitute the given values 16y + 4 = 180 Combine like terms 16y = 176 Substract 4 from both sides y = 11 Divide both sides by 16 B =(6y+5) = [6(11)+5] = 71
CONFIDENTIAL 13
Now you try!
ABCD is a parallelogram. Find each measure:
E
F G
H
J
4z-9
2z
w+8
3w2a) JG2b) FH
CONFIDENTIAL 14
Three vertices of □ABCD are A(1,-2), B(-2,3) and D(5,-1). Find the coordinates of vertex C.
Parallelograms in the coordinate plane
B
A
D
5
60
C
-3
-2
-3
5
x
y
Since ABCD is a parallelogram, both pairs of opposite sides must
be parallel.
Step1: Graph the given points.
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 15
B
A
D
5
60
C
-3
-2
-3
5
x
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.
Step4: Use the slope formula to verify that BC || AD.
Slope of BC = 4 – 3 = 1 2 – (-2) 4
Slope of AD = -1 – (-2) = 1 5 – 1 4
The coordinates of vertex C are (2, 4).
CONFIDENTIAL 16
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 17
Using properties of Parallelograms in a proof
Given: ABCD is a parallelogram.Proof: BAD DCB, ABC CDA
Proof:
STATEMENTS REASONS
1. ABCD is a parallelogram 1. Given2. AB CD, BC DA 2. gm opp.sides 3. BD BD 3. Reflex. prop. of .4. BAF DCB 4. SSS Steps 2,35. BAD DCB 5. CPCTC6. AC AC 6. Reflex. prop. of .7. ABC CDA 7. SSS Steps 2,68. ABC CDA 8. CPCTC
B C
DA
E
A)
CONFIDENTIAL 18
B) Given: GHJN and JKLM are paralellograms. H and M are collinear.Prove: G L
Proof:
STATEMENTS REASONS
1. GHJN and JKLM are grams 1. Given2.HJN G, MJK L 2. gm opp. s 3. HJN MJK 3. Vert. s Thm.4. G L 4. Trans. prop. of .
J
H
G
N M
L
K
CONFIDENTIAL 19
Now you try!
4) Use the figure above to write a two-column proof.
J
H
G
N M
L
Given: GHJN and JKLM are paralellograms. H and M are collinear. H and M are collinear.Prove: N K
CONFIDENTIAL 20
Now some problems for you to practice !
CONFIDENTIAL 21
1) BD
2) CD
3) BE
Assessment
I n llgm ABCD, AB = 17.5, DE = 18, and mBCD = 110 . Find each measure.
C D
AB
E
4) /ABC
5) /ADC
6) /DAB
CONFIDENTIAL 22
7) 8)
Find the values of x and y for which ABCD must be a parallelogram:
CONFIDENTIAL 23
9) Three vertices of llgm DFGH are D(-9,4), F(-1,5) and G(2,0). Find the coordinates of vertex H.
CONFIDENTIAL 24
10) Write a two-column proof.
Given: PSTV is a paralellogram. PQ RQ Prove: STV R
PV
TS
R
Q
CONFIDENTIAL 25
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 □.
Properties of Parallelograms
parallelogram ABCD
□ ABCD
A B
CD
AB || CD; BC || DA
Let’s review
CONFIDENTIAL 26
Theorem Hypothesis Conclusion
If a quadrilateral is a parallelogram, then its opposite sides are congruent. (□ -> opp. sides )
Properties of Parallelograms
A B
CD
AB CDBC DA
Theorem 1:
CONFIDENTIAL 27
Proof of Theorem 1:A B
CD
1
2
3
4
Given: ABCD is a parallelogram.Proof: AB CD, BC DA
Proof:
STATEMENTS REASONS
1. ABCD is a parallelogram 1. Given2. AB CD, BC DA 2. Def. of gm3. 1 2, 3 4 3. Alt. int. s Thm.4. AC AC 4. Reflex. prop. of .5. ABC CDA 5. ASA Steps 3,46. AB CD, BC DA 6. CPCTC
CONFIDENTIAL 28
Theorem Hypothesis Conclusion
If a quadrilateral is a parallelogram, then its opposite angles are congruent. (□ -> opp. angles )
Properties of Parallelograms
A B
CD
A CB D
CONFIDENTIAL 29
Theorem Hypothesis Conclusion
If a quadrilateral is a parallelogram, then its consecutive angles are supplementary. (□ -> cons. ∕s supp.)
Properties of Parallelograms
A B
CD
mA + B 180mB + C 180mC + D 180mD + A 180
CONFIDENTIAL 30
Theorem Hypothesis Conclusion
If a quadrilateral is a parallelogram, then its diagonals bisect each other.(□ -> diags. bisect each other)
Properties of Parallelograms
A B
CD
1
2
3
4
ZAZ CZBZ DZ
CONFIDENTIAL 31
Three vertices of □ABCD are A(1,-2), B(-2,3) and D(5,-1). Find the coordinates of vertex C.
Parallelograms in the coordinate plane
B
A
D
5
60
C
-3
-2
-3
5
x
y
Since ABCD is a parallelogram, both pairs of opposite sides must
be parallel.
Step1: Graph the given points.
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 32
B
A
D
5
60
C
-3
-2
-3
5
x
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.
Step4: Use the slope formula to verify that BC || AD.
Slope of BC = 4 – 3 = 1 2 – (-2) 4
Slope of AD = -1 – (-2) = 1 5 – 1 4
The coordinates of vertex C are (2, 4).
CONFIDENTIAL 33
Using properties of Parallelograms in a proof
Given: ABCD is a parallelogram.Proof: BAD DCB, ABC CDA
Proof:
STATEMENTS REASONS
1. ABCD is a parallelogram 1. Given2. AB CD, BC DA 2. gm opp.sides 3. BD BD 3. Reflex. prop. of .4. BAF DCB 4. SSS Steps 2,35. BAD DCB 5. CPCTC6. AC AC 6. Reflex. prop. of .7. ABC CDA 7. SSS Steps 2,68. ABC CDA 8. CPCTC
B C
DA
E
A)
CONFIDENTIAL 34
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