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Ch. 8.3 Proving
Quadrilaterals are
Parallelograms
Geometry B
Theorem 8.7
If both pairs of
opposite sides of a
quadrilateral are
congruent, then the
quadrilateral is a
parallelogram.
A
D
B
C
ABCD is a parallelogram.
Proof of Theorem 8.7 Given: AB ≅ CD, AD ≅ CB
Prove: ABCD is a parallelogram
Statements:
1. AB ≅ CD, AD ≅ CB.
2. AC ≅ AC
3. ∆ABC ≅ ∆CDA
4. BAC ≅ DCA,
DAC ≅ BCA
5. AB║CD, AD ║CB.
6. ABCD is a
Reasons:
1. Given
C
D
B
A
Proof of Theorem 8.7 Statements:
1. AB ≅ CD, AD ≅ CB.
2. AC ≅ AC
3. ∆ABC ≅ ∆CDA
4. BAC ≅ DCA, DAC ≅ BCA
5. AB║CD, AD ║CB.
6. ABCD is a
Reasons:
1. Given
2. Reflexive Prop. of Congruence
C
D
B
A
Proof of Theorem 8.7 Statements:
1. AB ≅ CD, AD ≅ CB.
2. AC ≅ AC
3. ∆ABC ≅ ∆CDA
4. BAC ≅ DCA, DAC ≅ BCA
5. AB║CD, AD ║CB.
6. ABCD is a
Reasons:
1. Given
2. Reflexive Prop. of Congruence
3. SSS Congruence Postulate
C
D
B
A
Proof of Theorem 8.7 Statements:
1. AB ≅ CD, AD ≅ CB.
2. AC ≅ AC
3. ∆ABC ≅ ∆CDA
4. BAC ≅ DCA, DAC ≅ BCA
5. AB║CD, AD ║CB.
6. ABCD is a
Reasons:
1. Given
2. Reflexive Prop. of Congruence
3. SSS Congruence Postulate
4. CPCTC
C
D
B
A
Proof of Theorem 8.7 Statements:
1. AB ≅ CD, AD ≅ CB.
2. AC ≅ AC
3. ∆ABC ≅ ∆CDA
4. BAC ≅ DCA, DAC ≅ BCA
5. AB║CD, AD ║CB.
6. ABCD is a
Reasons:
1. Given
2. Reflexive Prop. of Congruence
3. SSS Congruence Postulate
4. CPCTC(Corresponding parts of congruent triangles are congruent)
5. Alternate Interior s Converse
C
D
B
A
Proof of Theorem 8.7 Statements:
1. AB ≅ CD, AD ≅ CB.
2. AC ≅ AC
3. ∆ABC ≅ ∆CDA
4. BAC ≅ DCA, DAC ≅ BCA
5. AB║CD, AD ║CB.
6. ABCD is a
Reasons:
1. Given
2. Reflexive Prop. of Congruence
3. SSS Congruence Postulate
4. CPCTC
5. Alternate Interior s Converse
6. Def. of a parallelogram.
C
D
B
A
Theorem 8.8
If both pairs of
opposite angles
of a quadrilateral
are congruent,
then the
quadrilateral is a
parallelogram.
A
D
B
C
ABCD is a parallelogram.
Theorem 8.9
If an angle of a
quadrilateral is
supplementary to
both of its
consecutive
angles, then the
quadrilateral is a
parallelogram.
A
D
B
C
ABCD is a parallelogram.
x°
(180 – x)° x°
Theorem 8.10
If the diagonals
of a quadrilateral
bisect each
other, then the
quadrilateral is a
parallelogram. If BD and AC bisect each
other, then ABCD is a
parallelogram.
A
D
B
C
Example: Proving Quadrilaterals are
Parallelograms Using Algebra
For what value of x is quadrilateral HIJK a
parallelogram?
Example: Proving Quadrilaterals are
Parallelograms Using Algebra
By Theorem 8.10, if the diagonals of HIJK bisect each other then it is a parallelogram.
You are told that HO and JO are congruent. Find x.
X+40 = 2x +18
X=22
HO= (22) + 40 = 62
JO = 2(22) + 18 = 62
HO ≅ JO
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.
Example: 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.
6
4
2
-2
-4
5 10 15
D(7, 1)
C(6, 5)
B(1, 3)
A(2, -1)
Example: 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
6 - 1 5
Slope of DA. - 1 – 1 = 2
2 - 7 5
AB and CD have the same slope, so they are parallel. Similarly, BC ║ DA.
6
4
2
-2
-4
5 10 15
D(7, 1)
C(6, 5)
B(1, 3)
A(2, -1)
Because opposite sides are
parallel, ABCD is a
parallelogram.
Example: 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.
6
4
2
-2
-4
5 10 15
D(7, 1)
C(6, 5)
B(1, 3)
A(2, -1)
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