4/13/2015
7.3: Proving
Theorems about all
Parallelograms
Homework: Page 325 (#12-13)
Page 326 (#20-21)
Focus Learning Target:
Given a diagram containing
parallelograms, students will be able to
prove congruency statements by applying
theorems and properties of parallelograms
to construct a 2-column proof.
1
4/13/2015
Warm-up
1. Find the values of x and y.
2. For what value of x is
quadrilateral MNPQ a
parallelogram?
Explain your reasoning.
Theorems
A
Theorem 6.5: If both
pairs of opposite
sides of a
quadrilateral are
D
C
congruent, then the
quadrilateral is a
ABCD is a parallelogram.
parallelogram.
B
2
4/13/2015
Theorems
A
Theorem 6.6: If both
pairs of opposite
angles of a
quadrilateral are
D
C
congruent, then the
quadrilateral is a
ABCD is a parallelogram.
parallelogram.
B
Theorems
A
B
Theorem 6.7: If
the diagonals of a
quadrilateral
bisect each other, D
C
then the
quadrilateral is a
ABCD is a parallelogram.
parallelogram.
3
4/13/2015
Another Theorem ~
Theorem 6.8—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
C
D
C
B
Example 1:Proof of Theorem 6.5
Statements:
Reasons:
D
A
1. AB ≅ CD, AD ≅ CB. 1. Given
2. AC ≅ AC
2. Reflexive Prop. of
Congruence
3. ∆ABC ≅ ∆CDA
3. SSS Congruence Postulate
4. CPCTC
4. ∠BAC ≅ ∠DCA,
5. Alternate Interior ∠s
∠DAC ≅ ∠BCA
Converse
5. AB║CD, AD ║CB.
6. Def. of a parallelogram.
6. ABCD is a 4
4/13/2015
Ex. 2: Proof of Theorem 6.8
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. CPCTC
7. If opp. sides of a quad.
are ≅, then it is a .
Example 3:
(a) One pair of opposite side both parallel and congruent
(b) Both pairs of opposite sides congruent
(c) Both pairs of opposite angles congruent
(d) Both pairs of opposite sides parallel
(e) Diagonals bisect each other
5
4/13/2015
4.
D
Reasons:
Statements:
1. ABCD is a quadrilateral
1. Given
C
1
2
A
B
2. AB ≅ CD
2. Given
3. ∠1 ≅ ∠2
3. Given
4. AB CD
4. Two lines cut by a transversal
that form congruent alternate interior
angles are parallel
5. A quadrilateral with one pair of
opposite sides that are both parallel
and congruent is a parallelogram
5. ABCD is a parallelogram
5.
Statement
Reason
1. PQRS is a quadrilateral
1. Given
2. ∠1 ≅ ∠2
2. Given
3. ∠3 ≅ ∠4
3. Given
4. SP RQ
4. Two lines cut by a transversal
that form congruent alternate
interior angles are parallel
PQ SR
5. PQRS is a parallelogram
5. A quadrilateral with both pairs of
opposite sides parallel is a
parallelogram
6
4/13/2015
6.
Statement
Reason
1. ABCD is a rectangle
1. Given
2. DA ≅ CB
2. Opposite sides of a
rectangle are congruent
3. AB ≅ AB
3. Reflexive postulate
4. ∠DAB ≅ ∠ABC
4. All angles of a rectangle are
congruent
5. ∆DAB ≅ ∆CBA
5. SAS ≅ SAS
6. ∠1 ≅ ∠ 2
6. CPCTC
7. ∆AEB is isosceles
7. A triangle with two congruent
base angles is isosceles
7