Geometry
Guided Notes Section 3.5
Four Theorems for the
For each of the special angle types, we have a formal postulate or theorem.
Four Angle Types
Each of them is written like the following:
Corresponding Angles Postulate – “If two parallel lines are cut by a
transversal, then the pairs of Corresponding Angles angles are
congruent.”
Alternate Interior Angles Theorem – “If …, then the pairs of
alternate interior
angles are congruent.”
Alternate Exterior Angles Theorem – “If …, then the pairs of
alternate exterior
angles are congruent.”
Consecutive Interior Angles Theorem – “If …, then the pairs of
consecutive interior
angles are supplementary.”
How Do We Use These
Theorems?
We use these theorems as the reasons in proofs. For example:
1
3
5
7
2
4
6
l1
Given: l1 || l2 (means the lines are parallel)
l2
8
Sample Proof Steps
Reason
1. 1  5
Corresponding Angles Postulate
2. 1  8
Alternate Exterior Angles Theorem
3. 3  6
Alternate Interior Angles Theorem
4. 3 and 5 are supplementary
Consecutive Interior Angle Theorem
San Marcos High School
Geometry
Guided Notes Section 3.5
We also commonly use some of our past theorems and definitions
in proofs involving parallel line.:
Other Common Proof
Reasons
1
3
5
7
2
l1
4
6
Given: l1 || l2 (means the lines are parallel)
l2
8
More Sample Proof Steps
Reason
1. 1  5
1. Corresponding Angles Postulate
2. m1 = m5
2. Definition of Congruence
3. 1  4
3. Vertical Angle Theorem
4. 5 and 7 are supplementary
4. Linear Pair Postulate
5. m5 + 7 = 180
5. Definition of Supplementary
6. m1 + 7 = 180
6. Substitution Property of Equality
We Can Also Use These
Theorems to Find the
Measurement of Angles
m1 = 4x + 20
3
m5 = 2x + 80°
7
2
4
1. Find x
2. Find the measure of all 8 angles
6
8
1 and 5: Type of Angles = Corresponding Angles
1. Find x:
4x + 20 = 2x + 80
m1 = 4(40) + 20 = 140
x = 30
m2 = 180 – 140 = 40
m3 = 180 – 140 = 40
…
2.
m1 = 140
m5 = 140
m2 = 40_
m6 = 40
m3 = 40
m7 = 40
m4 = 140
m8 = 140
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