Download geometry module 1 lesson 28 properties of parallelograms

GEOMETRY
MODULE 1 LESSON 28
PROPERTIES OF PARALLELOGRAMS
OPENING EXERCISE
Consider the figure to the right.
1. If the triangles are congruent, state the congruence. Take care
to list vertices in a corresponding order.
∆𝐴𝐺𝑇 ≅ ∆𝑀𝑌𝐽
2. Which triangle congruence criterion guarantees part 1?
AAS
̅̅̅̅ corresponds with 𝐽𝑌
̅̅̅
3. 𝑇𝐺
HOMEWORK QUESTIONS
Problem Set Module 1 Lesson 25 is due today.
DISCUSSION
How can we use our knowledge of triangle congruence criteria to establish other geometry facts?
For instance, what can we now prove about properties of parallelograms?
What we know about parallelograms

A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel.
What we can assume about parallelograms

The opposite sides are congruent (equal in measure).

The opposite angles are congruent (equal in measure).

The diagonals bisect each other.
MOD1 L28
1
PRACTICE
If a quadrilateral is a parallelogram, then its opposite sides and angles are equal in measure.
Given: Parallelogram ABCD
Prove: 𝐴𝐷 = 𝐶𝐵, 𝐴𝐵 = 𝐶𝐷
𝑚∠𝐴 = 𝑚∠𝐶, 𝑚∠𝐵 = 𝑚∠𝐷
STEP
JUSTIFICATION
1
Parallelogram ABCD
Given
2
𝑚∠𝐴𝐵𝐷 = 𝑚∠𝐶𝐷𝐵
Alternate interior angles from ̅̅̅̅
𝐴𝐵 ∥ ̅̅̅̅
𝐷𝐶
3
𝐵𝐷 = 𝐷𝐵
Reflexive Property
4
𝑚∠𝐶𝐵𝐷 = 𝑚∠𝐴𝐷𝐵
Alternate interior angles from ̅̅̅̅
𝐴𝐵 ∥ ̅̅̅̅
𝐷𝐶
5
∆𝐴𝐵𝐷 ≅ ∆𝐶𝐷𝐵
ASA
6
𝐴𝐷 = 𝐶𝐵, 𝐴𝐵 = 𝐶𝐷
Property of congruent triangles
7
𝑚∠𝐴 = 𝑚∠𝐶
Property of congruent triangles
𝑚∠𝐴𝐵𝐷 + 𝑚∠𝐶𝐵𝐷 = 𝑚∠𝐴𝐵𝐶
8
Angle addition
𝑚∠𝐶𝐷𝐵 + 𝑚∠𝐴𝐷𝐵 = 𝑚∠𝐴𝐷𝐶
9
𝑚∠𝐴𝐵𝐷 + 𝑚∠𝐶𝐵𝐷 = 𝑚∠𝐶𝐷𝐵 + 𝑚∠𝐴𝐷𝐵
Addition property of equality
10
𝑚∠𝐵 = 𝑚∠𝐷
Substitution
MOD1 L28
2
If a quadrilateral is a parallelogram, then the diagonals bisect each other.
Given: Parallelogram ABCD
Prove: 𝐴𝐸 = 𝐶𝐸, 𝐷𝐸 = 𝐵𝐸
STEP
JUSTIFICATION
1
Parallelogram ABCD
Given
2
𝑚∠𝐵𝐴𝐶 = 𝑚∠𝐷𝐶𝐴
Alternate interior angles from ̅̅̅̅
𝐴𝐵 ∥ ̅̅̅̅
𝐷𝐶
3
𝑚∠𝐴𝐸𝐵 = 𝑚∠𝐶𝐸𝐷
Vertical Angles
4
𝐴𝐵 = 𝐶𝐷
Opposite sides of a parallelogram are equal in length.
5
∆𝐴𝐸𝐵 ≅ ∆𝐶𝐸𝐷
AAS
6
𝐴𝐸 = 𝐶𝐸, 𝐷𝐸 = 𝐵𝐸
Property of congruent triangles
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: rectangles,
squares, and rhombuses (diamond shape with four equal sides).
Given: Rectangle GHIJ
Prove: 𝐺𝐼 = 𝐻𝐽 (The diagonals are equal in length.)
STEP
JUSTIFICATION
1
Rectangle GHIJ
Given
2
𝐺𝐽 = 𝐼𝐻
Opposite sides of a parallelogram are equal in length.
3
𝐺𝐻 = 𝐻𝐺
Reflexive Property
4
∠𝐽𝐺𝐻 = ∠𝐼𝐻𝐺 = 90°
Definition of Rectangle
5
∆𝐺𝐻𝐽 ≅ ∆𝐻𝐺𝐼
SAS
6
𝐺𝐼 = 𝐻𝐽
Property of congruent triangles
MOD1 L28
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CONVERSE PROPERTIES
Given: 𝐴𝐵 = 𝐶𝐷, 𝐴𝐷 = 𝐵𝐶
Prove: The quadrilateral ABCD is a parallelogram
STEP
JUSTIFICATION
1
𝐴𝐵 = 𝐶𝐷, 𝐴𝐷 = 𝐵𝐶
Given
2
𝐵𝐷 = 𝐷𝐵
Reflexive Property
3
∆𝐴𝐵𝐷 ≅ ∆𝐶𝐷𝐵
SSS
∠𝐴𝐵𝐷 ≅ ∠𝐶𝐷𝐵
4
Property of congruent triangles
∠𝐴𝐷𝐵 ≅ ∠𝐶𝐵𝐷
̅̅̅̅
𝐴𝐵 ∥ ̅̅̅̅
𝐶𝐷
5
Alternate Interior Angles Converse
̅̅̅̅ ∥ 𝐶𝐵
̅̅̅̅
𝐴𝐷
6
The quadrilateral ABCD is a parallelogram.
Definition of Parallelogram
ON YOUR OWN
Given: Diagonals ̅̅̅̅
𝐴𝐶 and ̅̅̅̅
𝐵𝐷 bisect each other.
Prove: The quadrilateral ABCD is a parallelogram
1
STEP
JUSTIFICATION
Diagonals ̅̅̅̅
𝐴𝐶 and ̅̅̅̅
𝐵𝐷 bisect each other.
Given
2
𝐴𝐸 = 𝐸𝐶
3
𝑚∠𝐴𝐸𝐵 = 𝑚∠𝐷𝐸𝐶
4
∆𝐴𝐸𝐵 ≅ ∆𝐷𝐸𝐶
5
𝑚∠𝐴𝐵𝐷 = 𝑚∠𝐶𝐷𝐵
6
7
MOD1 L28
̅̅̅̅ ∥ 𝐶𝐷
̅̅̅̅
𝐴𝐵
𝐷𝐸 = 𝐸𝐵
𝑚∠𝐴𝐸𝐷 = 𝑚∠𝐵𝐸𝐶
∆𝐴𝐸𝐷 ≅ ∆𝐵𝐸𝐶
𝑚∠𝐴𝐷𝐵 = 𝑚∠𝐶𝐵𝐷
̅̅̅̅ ∥ 𝐶𝐵
̅̅̅̅
𝐴𝐷
The quadrilateral ABCD is a parallelogram
Definition of Segment Bisector
Vertical Angles
SAS
Property of congruent triangles
Alternate Interior Angles Converse
Definition of Parallelogram
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