1. 6-3 Proving That a Quadrilateral is a Parallelogram
OBJECTIVE: To prove that a quadrilateral is a parallelogram.
Each section of glass in the exterior of a building in Macau, China, forms an equilateral triangle.
Do you think the window washer’s feet stay parallel to the ground as he lands at each level of windows?
Explain. (Assume that the bases of the lowest triangles are parallel to the ground.)
Yes,
the angles could either be corresponding s
or alternate interior  and they are all congruent,
therefore the lines are parallel.
𝟔𝟎
𝟔𝟎
𝟔𝟎
𝟔𝟎
𝟔𝟎
𝟔𝟎

2. KEY CONCEPTS 6-3 Proving That a Quadrilateral is a Parallelogram
OBJECTIVE: To prove that a quadrilateral
is a parallelogram.
You can decide whether a quadrilateral is a parallelogram
if its sides, angles, and diagonals have certain properties.
KEY CONCEPTS
Theorem 6-8
If both pairs of opposite sides are congruent,
then the quadrilateral is a parallelogram.
Theorem 6-9
If an angle of a quadrilateral is supplementary to both of its consecutive angles, then the quadrilateral is a parallelogram.
Theorem 6-10
If both pairs of opposite angles are congruent,
then the quadrilateral is a parallelogram.
Theorem 6-11
If diagonals of a quadrilateral bisect each other,
then the quadrilateral is a parallelogram.
Theorem 6-12
If one pair of opposite sides of a quadrilateral is both congruent and parallel, then the quadrilateral is a parallelogram.

3. 6-3 Proving That a Quadrilateral is a Parallelogram
OBJECTIVE: To prove that a quadrilateral is a parallelogram.
In Problem 1, you proved Theorem 6-8. Use one of the theorems in this lesson to complete the following proof.
Given: S ≅ U, STV ≅UVT
Prove: STUV is a parallelogram.
Statements
Reasons
1. S ≅ U, STV ≅UVT
1. Given
𝟐. 𝑉𝑇 ≅ 𝑉𝑇
2. Reflexive Property of ≅
3. 𝑆𝑇𝑉≅ UVT
3. AAS
𝑆𝑉 ≅ 𝑈𝑇
𝟒. 𝑆𝑇 ≅ 𝑈𝑉 ,
4. CPCTC
5. ∴𝑆𝑇𝑈𝑉 is a parallelogram.
If both pairs of sides of a quadrilateral are ≅,
then the quadrilateral is a

4. 6-3 Proving That a Quadrilateral is a Parallelogram
OBJECTIVE: To prove that a quadrilateral is a parallelogram.
Given: J ≅ NKL, MLK ≅ NKL,
M ≅KLP, JKL≅ KLP
Prove: JKLM is a parallelogram.
Statements
Reasons
1. J ≅ NKL, MLK ≅ NKL
1. Given
2. J ≅ MLK
2. Transitive Property of ≅
3. Given
3. M ≅ KLP, JKL ≅ KLP
4. Transitive Property of ≅
4. M ≅ JKL
If both pairs of s of a quadrilateral are ≅,
then the quadrilateral is a
5. ∴𝑆𝑇𝑈𝑉 is a parallelogram.

5. 6-3 Proving That a Quadrilateral is a Parallelogram
OBJECTIVE: To prove that a quadrilateral is a parallelogram.
Statements
Reasons
1. Given
1. 𝑇𝑄 ≅ 𝑅𝑆 , TQX ≅ RSX
2. 𝑸𝑻 ‖ 𝑹𝑺
2. Converse of the Alt. Interior s Theorem
3. If a pair of sides of a quadrilateral is both ≅𝐚𝐧𝐝 ‖,
then the quadrilateral is a
3. ∴𝑺𝑻𝑼𝑽 is a parallelogram.

6. 6-3 Proving That a Quadrilateral is a Parallelogram
OBJECTIVE: To prove that a quadrilateral is a parallelogram.
Answer: 𝒙=𝟏𝟎, 𝒚=𝟒𝟑
Answer: 𝒎=𝟑, 𝒏=𝟑𝟕

7. 6-3 Proving That a Quadrilateral is a Parallelogram
No,
opposite angles are not congruent.
Yes.
By the Pythagorean Theorem, VY=3 and XY=3.
Diagonals bisect each other, therefore UVWX is a

8. 6-3 Proving That a Quadrilateral is a Parallelogram
Yes.
Since all sides are , opposite sides are .
Therefore, QRST is a
Yes.
Alternate int. s are , so 𝑀𝐽 ‖ 𝐾𝐿 .
Since 𝑀𝐽 ≅ 𝐾𝐿 , JKLM is a by Theorem 6-12.

9. 6-3 Proving That a Quadrilateral is a Parallelogram
OBJECTIVE: To prove that a quadrilateral is a parallelogram.
A truck sits on the platform of a vehicle lift. Two moving arms raise the platform until a mechanic can fit underneath. What is the maximum height that the vehicle lift can elevate the truck? Explain.
The maximum height is 6 ft and it occurs when 𝑸𝑷 is vertical (or 𝑸𝑷 ⏊ 𝑷𝑺 .
Study the illustration both before and after you read Problem 6. Then reread Theorem 6-8 and solve the problem. Use the terms congruent angles, congruent sides, parallelogram, and opposite sides in your explanation. Draw additional illustrations as you work through the problem.

10. 6-3 Proving That a Quadrilateral is a Parallelogram
OBJECTIVE: To prove that a quadrilateral is a parallelogram.
Answer: 𝒙=𝟏𝟎, 𝒚=𝟒𝟑
Answer: 𝒙=𝟐

11. 6-3 Proving That a Quadrilateral is a Parallelogram
OBJECTIVE: To prove that a quadrilateral is a parallelogram.
Do you understand?
4. Vocabulary Explain why you can now write a biconditional statement regarding
opposite sides of a parallelogram.
5. Select Techniques to Solve Problems (1)(C) How is Theorem 6-11 in this lesson different from
Theorem 6-6 in the previous lesson? In what situations should you use each theorem? Explain.
6. Explain Mathematical Ideas (1)(G) Your friend says, “If a quadrilateral has a pair of opposite sides
that are congruent and a pair of opposite sides that are parallel, then it is a parallelogram.”
Is your friend correct? Explain.