1. 1 2 1 2 PROPERTIES OF PARALLEL LINES POSTULATE
POSTULATE 15 Corresponding Angles Postulate
If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent. 1 2

2. 3 4 3 4 PROPERTIES OF PARALLEL LINES THEOREMS ABOUT PARALLEL LINES
THEOREM 3.4 Alternate Interior Angles
If two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent. 3 4

3. 5 6 m 5 + m 6 = 180° PROPERTIES OF PARALLEL LINES
THEOREMS ABOUT PARALLEL LINES
THEOREM 3.5 Consecutive Interior Angles
If two parallel lines are cut by a transversal, then the pairs of consecutive interior angles are supplementary.
m m 6 = 180° 5 6

4. 7 8 7 8 PROPERTIES OF PARALLEL LINES THEOREMS ABOUT PARALLEL LINES
THEOREM 3.6 Alternate Exterior Angles
If two parallel lines are cut by a transversal, then the pairs of alternate exterior angles are congruent. 7 8

5. j k PROPERTIES OF PARALLEL LINES THEOREMS ABOUT PARALLEL LINES
THEOREM 3.7 Perpendicular Transversal
If a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other.
j k

6. 1  3 Corresponding Angles Postulate
Proving the Alternate Interior Angles Theorem
Prove the Alternate Interior Angles Theorem.
SOLUTION
GIVEN
p || q
PROVE
Statements
Reasons
p || q Given 1
1  Corresponding Angles Postulate 2 3
3  Vertical Angles Theorem
1  Transitive property of Congruence 4

7. which postulate or theorem you use.
Using Properties of Parallel Lines
Given that m = 65°,
find each measure. Tell
which postulate or theorem
you use.
SOLUTION
m = m = 65°
Vertical Angles Theorem
Linear Pair Postulate
m = 180° – m = 115°
Corresponding Angles Postulate
m = m = 65°
Alternate Exterior Angles Theorem
m = m = 115°

8. parallel lines to find the value of x.
PROPERTIES OF SPECIAL PAIRS OF ANGLES
Using Properties of Parallel Lines
Use properties of
parallel lines to find the value of x.
SOLUTION
Corresponding Angles Postulate
m = 125°
Linear Pair Postulate
m (x + 15)° = 180°
Substitute.
125° + (x + 15)° = 180°
Subtract.
x = 40°

9. Estimating Earth’s Circumference: History Connection
Over 2000 years ago Eratosthenes estimated Earth’s circumference by using the fact that the Sun’s rays are parallel.
When the Sun shone exactly down a vertical well in Syene, he measured the angle the Sun’s rays made with a vertical stick in Alexandria. He discovered that
m 2 1 50
of a circle

10. 1 m 2 of a circle 50 m 1 = m 2 1 m 1 of a circle 50
Estimating Earth’s Circumference: History Connection
m 2 1 50
of a circle
Using properties of parallel lines, he knew that
m = m 2
He reasoned that
m 1 1 50
of a circle

11. 1 m 1 of a circle 50 1 of a circle 50 50(575 miles) 29,000 miles
Estimating Earth’s Circumference: History Connection
m 1 1 50
of a circle
The distance from Syene to Alexandria was believed to be 575 miles
Earth’s circumference 1 50
of a circle
575 miles
Earth’s circumference
50(575 miles)
Use cross product property
29,000 miles
How did Eratosthenes know that m = m ?

12. How did Eratosthenes know that m 1 = m 2 ?
Estimating Earth’s Circumference: History Connection
How did Eratosthenes know that m = m ?
SOLUTION
Because the Sun’s rays are parallel,
Angles 1 and 2 are alternate interior angles, so
1 
By the definition of congruent angles,
m = m 2