Topic 2 of 7
Parallel lines and angle chasing
Step 12 · Geometry topics
Name the angle pairs a transversal makes with parallel lines, use them to find unknown angles step by step, and use the converses to prove lines parallel.
Angles made by a transversal
A transversal is a line that crosses two other lines. When those two lines are parallel, the angles it makes come in matching sets.
| Pair | Where they sit | With parallel lines |
|---|---|---|
| corresponding | the same position at each crossing | equal |
| alternate interior | between the lines, on opposite sides of the transversal | equal |
| alternate exterior | outside the lines, on opposite sides | equal |
| same-side interior | between the lines, on the same side | add to 180° |
Example: Using the pairs
Parallel lines are cut by a transversal. One angle is 72°. What is its same-side interior partner?
- Same-side interior angles add to 180°.
- 180 − 72 = 108.
Answer108°
Chasing angles
Angle chasing means finding one angle after another, with a reason for each: vertical angles, linear pairs, parallel-line pairs and the triangle sum.
Example: A chase with algebra
Two parallel lines are cut by a transversal. One angle is 3x + 10 and its alternate interior partner is 5x − 30. Find x and the angle.
- Alternate interior angles are equal: 3x + 10 = 5x − 30.
- 40 = 2x, so x = 20.
- 3 × 20 + 10 = 70.
Answerx = 20, and the angle is 70°
Example: Why a triangle makes 180°
A triangle has its base on one line and its top corner on a parallel line. At the top corner, its sides make angles of 50° and 60° with the top line. Find the triangle's angles.
- Each base angle is an alternate interior angle with one of the top angles: 50° and 60°.
- The three angles at the top make a straight line, so the top angle is 180 − 50 − 60 = 70.
Answer50°, 60° and 70°
Proving lines parallel
The converses work too: if a transversal makes corresponding angles equal, or alternate interior angles equal, or same-side interior angles that add to 180°, then the lines are parallel.
Example: Are they parallel?
A transversal makes same-side interior angles of 115° and 65°. Are the lines parallel?
- 115 + 65 = 180.
AnswerYes: same-side interior angles add to 180°
Example: Finding x for parallel lines
Corresponding angles measure 4x − 8 and 2x + 20. For what x are the lines parallel?
- They must be equal: 4x − 8 = 2x + 20.
- 2x = 28, so x = 14.
Answerx = 14
Try it yourself
Work each one on paper first, then open the answer to check.
Parallel lines are cut by a transversal. A corresponding angle is 118°. What is its partner?
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118°
Alternate interior angles are 2x + 15 and 85°. What is x?
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35
Same-side interior angles are x and 3x. What is x?
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45
One angle where two lines cross is 40°. What are its vertical angle and its linear-pair partner?
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40° and 140°
Alternate interior angles measure 52° and 54°. Are the lines parallel?
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No: they would have to be equal.
Why do a triangle's angles add to 180°?
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Draw a line through the top corner parallel to the base. The base angles reappear there as alternate interior angles, and with the top angle they make a straight line.