5-2 Parallel and Perpendicular Lines
Warm UpComplete each sentence.
1. Angles whose measures have a sum of 90° are _______________ .
2. Vertical angles have equal measures, so they are ______________.
3. Angles whose measures have a sum of 180° are ______________.
4. An angle that measures less than 90° is a(n) ____________ angle.
complementary
congruent
supplementary
acute
5-2 Parallel and Perpendicular Lines
Learning Goal: to identify parallel and perpendicular lines and the angles formed by a transversal.
5-2 Parallel and Perpendicular Lines
Vocabulary
parallel lines
perpendicular lines
transversal
5-2 Parallel and Perpendicular Lines
Parallel lines are lines in a plane that never meet, like a set of perfectly straight, infinite train tracks.
Perpendicular lines are lines that intersect at 90° angles.
5-2 Parallel and Perpendicular Lines
The sides of the windows are transversals to the top and bottom.
A transversal is a line that intersects two or more lines that lie in the same plane. Transversals to parallel lines form angles with special properties.
The top and bottom of the windows are parallel.
5-2 Parallel and Perpendicular Lines
You cannot tell if angles are congruent by measuring because measurement is not exact.
Caution!
5-2 Parallel and Perpendicular Lines
Example 1: Identifying Congruent Angles
Formed by a Transversal
Measure the angles formed by the transversal and parallel lines. Which angles seem to be congruent?
∠1, ∠3, ∠5, and ∠7 all measure 150° and appear to be congruent.
∠ 2, ∠4, ∠6, and ∠8 all measure 30° and appear to be congruent.
5-2 Parallel and Perpendicular Lines
Example 1 Continued
Angles circled in blue appear to be congruent to each other, and angles circled in red appear to be congruent to each other.
∠∠∠∠1 ≅≅≅≅ ∠∠∠∠3 ≅ ≅ ≅ ≅ ∠∠∠∠5 ≅≅≅≅ ∠∠∠∠7∠∠∠∠2 ≅≅≅≅ ∠∠∠∠4 ≅≅≅≅ ∠∠∠∠6 ≅≅≅≅ ∠∠∠∠8
5-2 Parallel and Perpendicular Lines
Check It Out: Example 2
Measure the angles formed by the transversal and parallel lines. Which angles seem to be congruent?
∠1, ∠4, ∠5, and ∠8 all measure 36° and appear to be congruent.
∠ 2, ∠3, ∠6, and ∠7 all measure 144° and appear to be congruent.
1 23 4
5 67 8
5-2 Parallel and Perpendicular Lines
Check It Out: Example 2 Continued
Angles marked in blue appear to be congruent to each other, and angles marked in red appear to be congruent to each other.
∠∠∠∠1 ≅≅≅≅ ∠∠∠∠4 ≅ ≅ ≅ ≅ ∠∠∠∠5 ≅≅≅≅ ∠∠∠∠8∠∠∠∠2 ≅≅≅≅ ∠∠∠∠3 ≅≅≅≅ ∠∠∠∠6 ≅≅≅≅ ∠∠∠∠7
23
67
145
8
5-2 Parallel and Perpendicular Lines
Some pairs of the eight angles formed by two parallel lines and a transversal have special names.
5-2 Parallel and Perpendicular Lines
5-2 Parallel and Perpendicular Lines
The symbol for parallel is ||. The symbol for perpendicular is ⊥.
Writing Math
5-2 Parallel and Perpendicular Lines
In the figure, line l || line m. Find the measure of the angle.
Example 3: Finding Angle Measures of Parallel
Lines Cut by Transversals
∠∠∠∠4
m∠∠∠∠4 = 124°
The 124° angle and ∠4 are corresponding angles.
5-2 Parallel and Perpendicular Lines
Example 4: Finding Angle Measures of Parallel
Lines Cut by Transversals Continued
∠∠∠∠2
m∠2 + 124° = 180°
∠2 is supplementary to angle 124°.
m∠2 = 56°
–124° –124°
In the figure, line l || line m. Find the measure of the angle.
5-2 Parallel and Perpendicular Lines
Example 5: Finding Angle Measures of Parallel
Lines Cut by Transversals Continued
∠6
m∠6 = 56°
In the figure, line l || line m. Find the measure of the angle.
m∠6 + 124° = 180°
∠6 is supplementary to angle ∠6.
m∠6 = 56°
–124° –124°
5-2 Parallel and Perpendicular Lines
In the figure, line n || line m. Find the measure of the angle.
Check It Out: Example 6
∠7
m∠7 = 144°
The 144° angle and ∠7 are alternate exterior angles.
1 144°
3 45 6
7 8
m
n
5-2 Parallel and Perpendicular Lines
∠1
m∠1 + 144° = 180°
∠1 is supplementary to the 144° angle.
m∠ 1 = 36°
–144° –144°
1 144°
3 45 6
7 8
m
n
In the figure, line n || line m. Find the measure of the angle.
Check It Out: Example 7
5-2 Parallel and Perpendicular Lines
∠5 and ∠1 are corresponding angles.∠5
m∠5 = 36° 1 144°
3 45 6
7 8
m
n
In the figure, line n || line m. Find the measure of the angle.
Check It Out: Example 8
5-2 Parallel and Perpendicular Lines
Examples 9-12
In the figure, a || b.
9. Name the angles congruent to ∠3.
10. Name all the angles supplementary to ∠6.
11. If m∠1 = 105° what is m∠3?
12. What is m∠6?
∠1, ∠5, ∠7
∠1, ∠3, ∠5, ∠7
105°
75°
5-2 Parallel and Perpendicular Lines
13. In the figure, x || y. Identify the angles congruent to ∠3.
A. ∠1, ∠2, ∠4
B. ∠2, ∠4, ∠6
C. ∠4, ∠5, ∠6
D. ∠1, ∠5, ∠8
Example 13
5-2 Parallel and Perpendicular Lines
2. In the figure, x || y. If m∠5 = 115°, what is m∠7?
A. 25°
B. 65°
C. 75°
D. 115°
Example 14
5-2 Parallel and Perpendicular Lines
Assignment:
Workbook Page 25 & 26
