Download Unit 4 Lesson 3

Unit 4 Lesson 3: Properties of Parallelograms
Name: _________________________________________
Discussion
How can we use our knowledge of triangle congruence criteria to establish other geometry facts? For instance, what can we now
prove about the properties of parallelograms?
To date, we have defined a parallelogram to be a quadrilateral in which both pairs of opposite sides are parallel. However, we have
assumed other details about parallelograms to be true, too. We assume that:

Opposite sides are congruent.

Opposite angles are congruent.

Diagonals bisect each other.
Let us examine why each of these properties is true.
Example 1
If a quadrilateral is a parallelogram, then its opposite sides and angles are equal in measure. Complete the diagram, and develop
an appropriate Given and Prove for this case. Use triangle congruence criteria to demonstrate why opposite sides and angles of a
parallelogram are congruent.
Given:
Prove:
Construction:
Label the quadrilateral 𝐴𝐵𝐶𝐷, and mark opposite sides as parallel. Draw diagonal ̅̅̅̅
𝐵𝐷 .
Evidence
1. _______________________________________
2. _______________________________________
3. _______________________________________
4. _______________________________________
5. _______________________________________
6. _______________________________________
7. _______________________________________
8. _______________________________________
9. _______________________________________
Justifications
1. Given
2. Alternate Interior Angles
3. Reflexive P.O.E.
4. Alternate Interior Angles
5. ASA
6. CPCTC (3)
7. Angle Addition Postulate
8. CPCTC
9. Substitution P.O.E.
Example 2 If a quadrilateral is a parallelogram, then the diagonals bisect each other. Complete the diagram, and develop an
appropriate Given and Prove for this case. Use triangle congruence criteria to demonstrate why diagonals of a parallelogram bisect
each other. Remember, now that we have proved opposite sides and angles of a parallelogram to be congruent, we are free to use
these facts as needed (i.e., 𝐴𝐷 = 𝐶𝐵, 𝐴𝐵 = 𝐶𝐷, ∠𝐴 ≅ ∠𝐶, ∠𝐵 ≅ ∠𝐷).
Given:
Prove:
Construction:
Label the quadrilateral 𝐴𝐵𝐶𝐷. Mark opposite sides as parallel. Draw diagonals ̅̅̅̅
𝐴𝐶 and ̅̅̅̅
𝐵𝐷 .
Evidence
1. _______________________________________
2. _______________________________________
3. _______________________________________
4. _______________________________________
5. _______________________________________
6. _______________________________________
Justifications
1. Given
2. Alternate Interior Angles
3. Vertical Angles
4. Properties of a parallelogram
5. AAS
6. CPCTC
Now we have established why the properties of parallelograms that we have assumed to be true are in fact true. By
extension, these facts hold for any type of parallelogram, including rectangles, squares, and rhombuses. Let us look at
one last fact concerning rectangles. We established that the diagonals of general parallelograms bisect each other. Let
us now demonstrate that a rectangle has congruent diagonals.
Example 3
If the parallelogram is a rectangle, then the diagonals are equal in length. Complete the diagram, and develop an appropriate
Given and Prove for this case. Use triangle congruence criteria to demonstrate why diagonals of a rectangle are congruent. As in the
last proof, remember to use any already proven facts as needed.
Given:
Prove:
Construction:
Label the rectangle 𝐺𝐻𝐼𝐽. Mark opposite sides as parallel, and add small squares at the vertices to indicate 90°
̅̅̅.
angles. Draw diagonals ̅̅̅
𝐺𝐼 and ̅𝐻𝐽
Evidence
1. _______________________________________
2. _______________________________________
3. _______________________________________
4. _______________________________________
5. _______________________________________
Justifications
1. Given
2. Properties of a parallelogram (2)
3. Definition of a rectangle
4. SAS
5. CPCTC
Converse Properties: Now we examine the converse of each of the properties we proved.
Example 4 If both pairs of opposite angles of a quadrilateral are equal, then the quadrilateral is a parallelogram.
Given:
Quadrilateral ABCD,
Prove:
Quadrilateral ABCD is a parallelogram
Construction:
̅̅̅̅ . Label the measures of ∠𝐴
Label the quadrilateral 𝐴𝐵𝐶𝐷. Mark opposite angles as congruent. Draw diagonal 𝐵𝐷
̅̅̅̅ as 𝑟°, 𝑠°, 𝑡°, and 𝑢°.
and ∠𝐶 as 𝑥°. Label the measures of the four angles created by 𝐵𝐷
Evidence
1. _______________________________________
2. _______________________________________
3. _______________________________________
4. _______________________________________
5. _______________________________________
6. _______________________________________
7.
Justifications
1. Given
2. Angle Addition Postulate (2)
3. Substitution P.O.E.
4. Triangle Sum Theorem (2)
5. Substitution P.O.E.
6. Subtraction P.O.E.
7.
8. _______________________________________ 8. Substitution P.O.E.
9. _______________________________________ 9. Alternate Interior Angles Converse
10. Quadrilateral ABCD is a parallelogram
10. __________________________________________
Example 5 If the opposite sides of a quadrilateral are equal, then the quadrilateral is a parallelogram.
Given:
Quadrilateral ABCD,
Prove:
Quadrilateral ABCD is a parallelogram
Construction:
Label the quadrilateral 𝐴𝐵𝐶𝐷, and mark opposite sides as equal. Draw diagonal ̅̅̅̅
𝐵𝐷 .
Evidence
1. _______________________________________
2. _______________________________________
3. _______________________________________
4. _______________________________________
Justifications
1. Given
2. Reflexive P.O.E.
3. SSS
4. CPCTC
5. _______________________________________ 5. Alternate Interior Angles Converse (2)
6. Quadrilateral ABCD is a parallelogram
6. __________________________________________
Example 6
If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.
Given:
Quadrilateral ABCD,
Prove:
Quadrilateral ABCD is a parallelogram
Construction:
̅̅̅̅ and ̅̅̅̅
Label the quadrilateral 𝐴𝐵𝐶𝐷, and mark opposite sides as equal. Draw diagonals 𝐴𝐶
𝐵𝐷 .
Evidence
1. _______________________________________
2. _______________________________________
3. _______________________________________
4. _______________________________________
Justifications
1. Given
2. Definition of bisect (2)
3. Vertical Angles (2)
4. SAS (2)
5. _______________________________________ 5. Alternate Interior Angles Converse (2)
6. Quadrilateral ABCD is a parallelogram
6. __________________________________________
Example 7 If the diagonals of a parallelogram are equal in length, then the parallelogram is a rectangle. Complete the diagram,
and develop an appropriate Given and Prove for this case.
Given:
Parallelogram GHIJ with
Prove:
GHIJ is a rectangle
Construction:
̅̅̅̅.
Label the quadrilateral 𝐺𝐻𝐼𝐽. Draw diagonals ̅̅̅
𝐺𝐼 and 𝐻𝐽
Evidence
1. _______________________________________
2. _______________________________________
3. _______________________________________
4. _______________________________________
5. _______________________________________
6. _______________________________________
7. _______________________________________
8. GHIJ is a rectangle
Justifications
1. Given
2. Reflexive P.O.E. (2)
3. Opposite sides of a parallelogram are congruent
4. SSS (2)
5. CPCTC (2)
6. SSI angles converse
7. Division P.O.E.
8. __________________________________________
Homework:
Use the facts you have established to complete exercises involving different types of parallelograms.
1.
Given: ̅̅̅̅
𝐴𝐵 ∥ ̅̅̅̅
𝐶𝐷 , 𝐴𝐷 = 𝐴𝐵, 𝐶𝐷 = 𝐶𝐵
Prove: 𝐴𝐵𝐶𝐷 is a rhombus.
2.
Evidence
Justification
1. 𝑨𝑫 = 𝑨𝑩, 𝑪𝑫 = 𝑪𝑩
1.
2. 𝑨𝑪 = 𝑪𝑨
2.
3. △ 𝑨𝑫𝑪 ≅ △ 𝑪𝑩𝑨
3.
4. 𝑨𝑫 = 𝑪𝑩, 𝑨𝑩 = 𝑪𝑫
4.
5. 𝑨𝑩 = 𝑩𝑪 = 𝑪𝑫 = 𝑨𝑫
5.
6. 𝑨𝑩𝑪𝑫 is a rhombus.
6.
Given: 𝐴𝐵𝐶𝐷 is a parallelogram, 𝑅𝐷 bisects ∠𝐴𝐷𝐶, 𝑆𝐵 bisects ∠𝐶𝐵𝐴.
Prove: 𝐷𝑅𝐵𝑆 is a parallelogram.
Evidence
Justification
1.
1. Givens
2.
congruent. (2)
2. Opposite sides of a parallelogram are
3.
3. Opposite angles of a parallelogram are congruent.
(2)
4.
4. Definition of angle bisector (2)
5. 𝒎∠𝑹𝑫𝑨 + 𝒎∠𝑹𝑫𝑺 = 𝒎∠𝑫,
5.
𝒎∠𝑺𝑩𝑪 + 𝒎∠𝑺𝑩𝑹 = 𝒎∠𝑩
6. 𝒎∠𝑹𝑫𝑨 + 𝒎∠𝑹𝑫𝑨 = 𝒎∠𝑫
6.
𝒎∠𝑺𝑩𝑪 + 𝒎∠𝑺𝑩𝑪 = 𝒎∠𝑩
7. 𝟐(𝒎∠𝑹𝑫𝑨) = 𝒎∠𝑫, 𝟐(𝒎∠𝑺𝑩𝑪) = 𝒎∠𝑩
𝟏
𝟐
𝟏
𝟐
7.
8. 𝒎∠𝑹𝑫𝑨 = 𝒎∠𝑫, 𝒎∠𝑺𝑩𝑪 = 𝒎∠𝑩
8.
9. ∠𝑹𝑫𝑨 ≅ ∠𝑺𝑩𝑪
9.
10.
10. ASA
11.
11. Corresponding angles of congruent triangles are
congruent.
12
12. Supplements of congruent angles are congruent.
13.
13. Opposite angles of quadrilateral 𝑫𝑹𝑩𝑺 are
congruent.
3.
Given: 𝐷𝐸𝐹𝐺 is a rectangle, 𝑊𝐸 = 𝑌𝐺, 𝑊𝑋 = 𝑌𝑍
Prove: 𝑊𝑋𝑌𝑍 is a parallelogram.
Evidence
Justification
1. 𝑫𝑬 = 𝑭𝑮, 𝑫𝑮 = 𝑭𝑬
1. Opposite sides of a rectangle are congruent.
2. 𝑫𝑬𝑭𝑮 is a rectangle;
2. Given
𝑾𝑬 = 𝒀𝑮, 𝑾𝑿 = 𝒀𝒁
4.
3. 𝑫𝑬 = 𝑫𝑾 + 𝑾𝑬, 𝑭𝑮 = 𝒀𝑮 + 𝑭𝒀
3.
4. 𝑫𝑾 + 𝑾𝑬 = 𝒀𝑮 + 𝑭𝒀
4.
5. 𝑫𝑾 + 𝒀𝑮 = 𝒀𝑮 + 𝑭𝒀
5.
6. 𝑫𝑾 = 𝑭𝒀
6.
7. 𝒎∠𝑫 = 𝒎∠𝑬 = 𝒎∠𝑭 = 𝒎∠𝑮 = 𝟗𝟎°
7.
8. △ 𝒁𝑮𝒀, △ 𝑿𝑬𝑾 are right triangles.
8.
9. △ 𝒁𝑮𝒀 ≅ △ 𝑿𝑬𝑾
9.
10. 𝒁𝑮 = 𝑿𝑬
10.
11. 𝑫𝑮 = 𝒁𝑮 + 𝑫𝒁; 𝑭𝑬 = 𝑿𝑬 + 𝑭𝑿
11.
12. 𝑫𝒁 = 𝑭𝑿
12.
13. △ 𝑫𝒁𝑾 ≅ △ 𝑭𝑿𝒀
13.
14. 𝒁𝑾 = 𝑿𝒀
14.
15. 𝑾𝑿𝒀𝒁 is a parallelogram.
15.
Given: Rectangle 𝑅𝑆𝑇𝑈, 𝑀 is the midpoint of 𝑅𝑆.
Prove: △ 𝑈𝑀𝑇 is isosceles.
Evidence
Justification
1. Rectangle 𝑹𝑺𝑻𝑼
1. Given
2.
2. Opposite sides of a rectangle are congruent.
3. ∠𝑹, ∠𝑺 are right angles.
3.
4. 𝑴 is the midpoint of ̅̅̅̅
𝑹𝑺.
4. Given
5. 𝑹𝑴 = 𝑺𝑴
5. Definition of a midpoint
6.
6. SAS
7.
7.
8.
8. Definition of an isosceles triangle