Transcript 6.3 Proving Quadrilaterals are Parallelograms

6.3 Proving
Quadrilaterals are
Parallelograms
FOUR More Theorems “Converses”
A
Theorem 6.6: If both
pairs of opposite
sides of a
quadrilateral are
congruent, then the
quadrilateral is a
parallelogram.
D
B
C
ABCD is a parallelogram.
Theorems
A
Theorem 6.7: If both
pairs of opposite
angles of a
quadrilateral are
congruent, then
the quadrilateral is
a parallelogram.
D
B
C
ABCD is a parallelogram.
Theorems
A
Theorem 6.8: If an
angle of a
quadrilateral is
supplementary to
both of its
consecutive angles,
then the
quadrilateral is a
parallelogram.
B
(180 – x)°
x°
x°
D
C
ABCD is a parallelogram.
Theorems
A
Theorem 6.9: If
the diagonals of
a quadrilateral
bisect each
other, then the
quadrilateral is a
parallelogram.
D
B
C
ABCD is a parallelogram.
C
B
Ex. 1: Proof of Theorem 6.6
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
D
A
C
B
Ex. 1: Proof of Theorem 6.6
Statements:
1. AB ≅ CD, AD ≅ CB.
2.
AC ≅ AC
3.
4.
5.
6.
∆ABC ≅ ∆CDA
BAC ≅ DCA, DAC
≅ BCA
AB║CD, AD ║CB.
ABCD is a 
Reasons:
D
A
1. Given
2.
Reflexive Prop. of Congruence
C
B
Ex. 1: Proof of Theorem 6.6
Statements:
1. AB ≅ CD, AD ≅ CB.
2.
AC ≅ AC
3.
4.
5.
6.
∆ABC ≅ ∆CDA
BAC ≅ DCA, DAC
≅ BCA
AB║CD, AD ║CB.
ABCD is a 
Reasons:
D
A
1. Given
2.
Reflexive Prop. of Congruence
3.
SSS Congruence Postulate
C
B
Ex. 1: Proof of Theorem 6.6
Statements:
1. AB ≅ CD, AD ≅ CB.
2.
AC ≅ AC
3.
4.
5.
6.
∆ABC ≅ ∆CDA
BAC ≅ DCA, DAC
≅ BCA
AB║CD, AD ║CB.
ABCD is a 
Reasons:
D
A
1. Given
2.
Reflexive Prop. of Congruence
3.
4.
SSS Congruence Postulate
CPOCTAC
C
B
Ex. 1: Proof of Theorem 6.6
Statements:
1. AB ≅ CD, AD ≅ CB.
2.
AC ≅ AC
3.
4.
5.
6.
∆ABC ≅ ∆CDA
BAC ≅ DCA, DAC
≅ BCA
AB║CD, AD ║CB.
ABCD is a 
Reasons:
D
A
1. Given
2.
Reflexive Prop. of Congruence
4.
SSS Congruence Postulate
CPOCTAC
5.
Alternate Interior s Converse
3.
C
B
Ex. 1: Proof of Theorem 6.6
Statements:
1. AB ≅ CD, AD ≅ CB.
2.
AC ≅ AC
3.
4.
5.
6.
∆ABC ≅ ∆CDA
BAC ≅ DCA, DAC
≅ BCA
AB║CD, AD ║CB.
ABCD is a 
Reasons:
D
A
1. Given
2.
Reflexive Prop. of Congruence
3.
4.
5.
6.
SSS Congruence Postulate
CPOCTAC
Alternate Interior s Converse
Def. of a parallelogram.
Ex. 2: Proving Quadrilaterals are
Parallelograms

As the sewing box below is opened, the
trays are always parallel to each other.
2 in.
Why?
2.75 in.
2.75 in.
2 in.
Ex. 2: Proving Quadrilaterals are
Parallelograms

Each pair of hinges are
opposite sides of a
quadrilateral. The 2.75 inch
sides of the quadrilateral are
opposite and congruent. The
2 inch sides are also opposite
and congruent. Because
opposite sides of the
quadrilateral are congruent, it
is a parallelogram. By the
definition of a parallelogram,
opposite sides are parallel, so
the trays of the sewing box
are always parallel.
2 in.
2.75 in.
2.75 in.
2 in.
Another Theorem ~
Theorem 6.10—If one pair of opposite
sides of a quadrilateral are congruent and
parallel, then the quadrilateral is a
B
parallelogram.
 ABCD is a
parallelogram.

A
Box on bottom of page 340
D
C
Ex. 3: Proof of Theorem 6.10
Given: BC║DA, BC ≅ DA
Prove: ABCD is a 
C
B
D
Statements:
1. BC ║DA
2. DAC ≅ BCA
3. AC ≅ AC
4. BC ≅ DA
5. ∆BAC ≅ ∆DCA
6. AB ≅ CD
7. ABCD is a 
Reasons:
1. Given
A
Ex. 3: Proof of Theorem 6.10
Given: BC║DA, BC ≅ DA
Prove: ABCD is a 
C
B
D
Statements:
1. BC ║DA
2. DAC ≅ BCA
3. AC ≅ AC
4. BC ≅ DA
5. ∆BAC ≅ ∆DCA
6. AB ≅ CD
7. ABCD is a 
Reasons:
1. Given
2. Alt. Int. s Thm.
A
Ex. 3: Proof of Theorem 6.10
Given: BC║DA, BC ≅ DA
Prove: ABCD is a 
C
B
D
Statements:
1. BC ║DA
2. DAC ≅ BCA
3. AC ≅ AC
4. BC ≅ DA
5. ∆BAC ≅ ∆DCA
6. AB ≅ CD
7. ABCD is a 
Reasons:
1. Given
2. Alt. Int. s Thm.
3. Reflexive Property
A
Ex. 3: Proof of Theorem 6.10
Given: BC║DA, BC ≅ DA
Prove: ABCD is a 
C
B
D
Statements:
1. BC ║DA
2. DAC ≅ BCA
3. AC ≅ AC
4. BC ≅ DA
5. ∆BAC ≅ ∆DCA
6. AB ≅ CD
7. ABCD is a 
Reasons:
1. Given
2. Alt. Int. s Thm.
3. Reflexive Property
4. Given
A
Ex. 3: Proof of Theorem 6.10
Given: BC║DA, BC ≅ DA
Prove: ABCD is a 
C
B
D
Statements:
1. BC ║DA
2. DAC ≅ BCA
3. AC ≅ AC
4. BC ≅ DA
5. ∆BAC ≅ ∆DCA
6. AB ≅ CD
7. ABCD is a 
A
Reasons:
1. Given
2. Alt. Int. s Thm.
3. Reflexive Property
4. Given
5. SAS Congruence Post.
Ex. 3: Proof of Theorem 6.10
Given: BC║DA, BC ≅ DA
Prove: ABCD is a 
C
B
D
Statements:
1. BC ║DA
2. DAC ≅ BCA
3. AC ≅ AC
4. BC ≅ DA
5. ∆BAC ≅ ∆DCA
6. AB ≅ CD
7. ABCD is a 
A
Reasons:
1. Given
2. Alt. Int. s Thm.
3. Reflexive Property
4. Given
5. SAS Congruence Post.
6. CPOCTAC
Ex. 3: Proof of Theorem 6.10
Given: BC║DA, BC ≅ DA
Prove: ABCD is a 
C
B
D
Statements:
1. BC ║DA
2. DAC ≅ BCA
3. AC ≅ AC
4. BC ≅ DA
5. ∆BAC ≅ ∆DCA
6. AB ≅ CD
7. ABCD is a 
A
Reasons:
1. Given
2. Alt. Int. s Thm.
3. Reflexive Property
4. Given
5. SAS Congruence Post.
6. CPOCTAC
7. If opp. sides of a quad.
are ≅, then it is a .
Objective 2: 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.
Ex. 4: Using properties of parallelograms
 Show that A(2, -1), B(1,
3), C(6, 5) and D(7,1)
are the vertices of a
B(1, 3)
parallelogram.
6
C(6, 5)
4
2
D(7, 1)
5
A(2, -1)
-2
-4
1
Ex. 4: Using properties of parallelograms


Method 1—Show that opposite
sides have the same slope, so
they are parallel.
Slope of AB.


1–5=-4
7–6
B(1, 3)
2
D(7, 1)
5–3=2
6 -1 5
Slope of DA.


4
Slope of BC.


C(6, 5)
Slope of CD.


3-(-1) = - 4
1-2
6
-1–1=2
2- 7 5
AB and CD have the same
slope, so they are parallel.
Similarly, BC ║ DA.
5
A(2, -1)
-2
-4
Because opposite sides are
parallel, ABCD is a
parallelogram.
1
Ex. 4: 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
6
C(6, 5)
4
B(1, 3)
2
D(7, 1)
5

AB ≅ CD and BC ≅ DA.
Because both pairs of opposites
sides are congruent, ABCD is a
parallelogram.
A(2, -1)
-2
-4
1
Ex. 4: Using properties of parallelograms
 Method 3—Show that
one pair of opposite
sides is congruent and
B(1, 3)
parallel.
6
C(6, 5)
4


Slope of AB = Slope of CD
= -4
AB=CD = √17
2
D(7, 1)
5
A(2, -1)
-2

AB and CD are congruent
and parallel, so ABCD is a
parallelogram.
-4
1