Theorem

1. Theorems about parallelograms
Theorems about parallelograms
• If a quadrilateral is
a parallelogram,
then its opposite
sides are
congruent.
►PQ RS and
≅
SP QR
≅
P
Q R
S

2. Theorems about parallelograms
Theorems about parallelograms
• If a quadrilateral is
a parallelogram,
then its opposite
angles are
congruent.
P ≅ R and
Q ≅ S P
Q R
S

3. Theorems about parallelograms
Theorems about parallelograms
• If a quadrilateral is a
parallelogram, then its
consecutive angles are
supplementary (add
up to 180°).
mP +mQ = 180°,
mQ +mR = 180°,
mR + mS = 180°,
mS + mP = 180°
P
Q R
S

4. Theorems about parallelograms
Theorems about parallelograms
• If a quadrilateral is
a parallelogram,
then its diagonals
bisect each other.
QM SM and
≅
PM RM
≅
P
Q R
S

5. Ex. 1: Using properties of
Ex. 1: Using properties of
Parallelograms
Parallelograms
• FGHJ is a
parallelogram. Find
the unknown length.
Explain your
reasoning.
a. JH
b. JK
F G
J H
K
5
3
b.

6. Ex. 1: Using properties of
Ex. 1: Using properties of
Parallelograms
Parallelograms
• FGHJ is a parallelogram.
Find the unknown
length. Explain your
reasoning.
a. JH
b. JK
SOLUTION:
a. JH = FG Opposite sides
of a are .
≅
JH = 5 Substitute 5 for
FG.
F G
J H
K
5
3
b.

7. Ex. 1: Using properties of
Ex. 1: Using properties of
Parallelograms
Parallelograms
• FGHJ is a parallelogram.
Find the unknown
length. Explain your
reasoning.
a. JH
b. JK
SOLUTION:
a. JH = FG Opposite sides
of a are .
≅
JH = 5 Substitute 5 for
FG.
F G
J H
K
5
3
b.
b. JK = GK Diagonals of a
bisect each other.
JK = 3 Substitute 3 for GK

8. TRY YOURSELF : 9/29/15
TRY YOURSELF : 9/29/15
PQRS is a parallelogram.
Find the angle measure.
a. mR
b. mQ P
R
Q
70°
S

9. PQRS is a parallelogram.
Find the angle measure.
a. mR
b. mQ
a. mR = mP Opposite angles of a are .
≅
mR = 70° Substitute 70° for mP.
P
R
Q
70°
S

10. PQRS is a parallelogram.
Find the angle measure.
a. mR
b. mQ
a. mR = mP Opposite angles of a are .
≅
mR = 70° Substitute 70° for mP.
b. mQ + mP = 180° Consecutive s of a are supplementary.
mQ + 70° = 180° Substitute 70° for mP.
mQ = 110° Subtract 70° from each side.
P
R
Q
70°
S

11. Ex. 3: Using Algebra with Parallelograms
Ex. 3: Using Algebra with Parallelograms
PQRS is a parallelogram.
Find the value of x.
mS + mR = 180°
3x + 120 = 180
3x = 60
x = 20
Consecutive s of a □ are supplementary.
Substitute 3x for mS and 120 for mR.
Subtract 120 from each side.
Divide each side by 3.
S
Q
P
R
3x° 120°

12. TOTD
TOTD

13. Ex. 5: Proving Theorem 6.2
Ex. 5: Proving Theorem 6.2
Given: ABCD is a parallelogram.
Prove AB CD, AD CB.
≅ ≅
1. ABCD is a .
2. Draw BD.
3. AB ║CD, AD ║ CB.
4. ABD ≅ CDB, ADB ≅
 CBD
5. DB DB
≅
6. ∆ADB ≅ ∆CBD
7. AB CD, AD CB
≅ ≅
1. Given
A
D
B
C

14. Ex. 5: Proving Theorem 6.2
Ex. 5: Proving Theorem 6.2
Given: ABCD is a parallelogram.
Prove AB CD, AD CB.
≅ ≅
1. ABCD is a .
2. Draw BD.
3. AB ║CD, AD ║ CB.
4. ABD ≅ CDB, ADB ≅ 
CBD
5. DB DB
≅
6. ∆ADB ≅ ∆CBD
7. AB CD, AD CB
≅ ≅
1. Given
2. Through any two points, there
exists exactly one line.
A
D
B
C

15. Ex. 5: Proving Theorem 6.2
Ex. 5: Proving Theorem 6.2
Given: ABCD is a parallelogram.
Prove AB CD, AD CB.
≅ ≅
1. ABCD is a .
2. Draw BD.
3. AB ║CD, AD ║ CB.
4. ABD ≅ CDB, ADB ≅ 
CBD
5. DB DB
≅
6. ∆ADB ≅ ∆CBD
7. AB CD, AD CB
≅ ≅
1. Given
2. Through any two points, there
exists exactly one line.
3. Definition of a parallelogram
A
D
B
C

16. Ex. 5: Proving Theorem 6.2
Ex. 5: Proving Theorem 6.2
Given: ABCD is a parallelogram.
Prove AB CD, AD CB.
≅ ≅
1. ABCD is a .
2. Draw BD.
3. AB ║CD, AD ║ CB.
4. ABD ≅ CDB, ADB ≅ 
CBD
5. DB DB
≅
6. ∆ADB ≅ ∆CBD
7. AB CD, AD CB
≅ ≅
1. Given
2. Through any two points, there
exists exactly one line.
3. Definition of a parallelogram
4. Alternate Interior s Thm.
A
D
B
C

17. Ex. 5: Proving Theorem 6.2
Ex. 5: Proving Theorem 6.2
Given: ABCD is a parallelogram.
Prove AB CD, AD CB.
≅ ≅
1. ABCD is a .
2. Draw BD.
3. AB ║CD, AD ║ CB.
4. ABD ≅ CDB, ADB ≅ 
CBD
5. DB DB
≅
6. ∆ADB ≅ ∆CBD
7. AB CD, AD CB
≅ ≅
1. Given
2. Through any two points, there
exists exactly one line.
3. Definition of a parallelogram
4. Alternate Interior s Thm.
5. Reflexive property of congruence
A
D
B
C

18. Ex. 5: Proving Theorem 6.2
Ex. 5: Proving Theorem 6.2
Given: ABCD is a parallelogram.
Prove AB CD, AD CB.
≅ ≅
1. ABCD is a .
2. Draw BD.
3. AB ║CD, AD ║ CB.
4. ABD ≅ CDB, ADB ≅ 
CBD
5. DB DB
≅
6. ∆ADB ≅ ∆CBD
7. AB CD, AD CB
≅ ≅
1. Given
2. Through any two points, there
exists exactly one line.
3. Definition of a parallelogram
4. Alternate Interior s Thm.
5. Reflexive property of congruence
6. ASA Congruence Postulate
A
D
B
C

19. Ex. 5: Proving Theorem 6.2
Ex. 5: Proving Theorem 6.2
Given: ABCD is a parallelogram.
Prove AB CD, AD CB.
≅ ≅
1. ABCD is a .
2. Draw BD.
3. AB ║CD, AD ║ CB.
4. ABD ≅ CDB, ADB ≅ 
CBD
5. DB DB
≅
6. ∆ADB ≅ ∆CBD
7. AB CD, AD CB
≅ ≅
1. Given
2. Through any two points, there
exists exactly one line.
3. Definition of a parallelogram
4. Alternate Interior s Thm.
5. Reflexive property of congruence
6. ASA Congruence Postulate
7. CPCTC
A
D
B
C