Parallel Lines and Transversals
When a line crosses two parallel lines, it makes 8 angles - but only two different sizes. Learn to spot which angles match, and you can find all 8 from just one.
Know these first
- Master List skill 7.G.5
This unlocks
- Triangle Angle Sum 8.G.5
What you'll be able to do
- Name the angle pairs a transversal makes: corresponding, alternate interior, alternate exterior, same-side interior.
- Find every angle in the figure from one given angle.
- Write and solve an equation when angles are given as expressions like (3x + 12)°.
- Use angle measures to decide whether two lines are parallel.
- Solve bent-path (zigzag) problems by drawing an extra parallel line.
- Solve bent-path (zigzag) problems by drawing an extra parallel line.
The big idea
A transversal is a line that crosses two other lines. Where it crosses parallel lines, it crosses both at the same tilt.
Same tilt means the two crossings are copies of each other. Slide the top crossing down the transversal and it lands exactly on the bottom one. So every angle at the top has a twin at the bottom.
At each crossing the angles come in only two sizes, a small one and a big one, and they always add to 180°. That gives the one rule behind this topic: two angles in the figure are either equal, or they add to 180°.
Words you need
- Interior
- Between the two parallel lines.
- Exterior
- Outside the two parallel lines.
- Alternate
- On opposite sides of the transversal.
The angle pairs
| Pair | Where they sit | Example | Rule |
|---|---|---|---|
| Corresponding | Same position at each crossing | 1 & 5 | Equal |
| Alternate interior | Between the lines, opposite sides | 3 & 6 | Equal |
| Alternate exterior | Outside the lines, opposite sides | 1 & 8 | Equal |
| Same-side interior | Between the lines, same side | 3 & 5 | Add to 180° |
| Vertical | Across from each other at one crossing | 1 & 4 | Equal |
| Linear pair | Side by side on a straight line | 1 & 2 | Add to 180° |
Try it: explore the angles
Drag the slider to tilt the transversal. Pick a pair to highlight it. Watch which measures stay equal.
Worked examples
Example 1
Angle 1 measures 110°. Find angle 6.
- Angle 1 and angle 5 are corresponding, so angle 5 = 110°.
- Angles 5 and 6 form a linear pair, so angle 6 = 180° − 110° = 70°.
Answer: 70°
Example 2
Angle 3 = (2x + 14)° and angle 6 = (4x − 30)°. Find x and both angles.
- Angles 3 and 6 are alternate interior, so they are equal: 2x + 14 = 4x − 30.
- Subtract 2x and add 30: 44 = 2x, so x = 22.
- Angle 3 = 2(22) + 14 = 58°, and angle 6 = 4(22) − 30 = 58°.
Answer: x = 22; each angle is 58°
Example 3
Angle 4 = 63° and angle 6 = 117°. Are lines m and n parallel?
- Angles 4 and 6 are same-side interior. For parallel lines they must add to 180°.
- 63° + 117° = 180°, so the lines are parallel.
Answer: Yes, parallel
Common mistakes
"Same-side interior angles are equal." They add to 180°. They're only equal when both are 90°.
Using the rules when the lines aren't parallel. The equal and add-to-180° rules only hold for parallel lines. If you aren't told the lines are parallel, don't assume it.
Answering with x. When a question asks for the angle, find x, then put it back in.
Mixing up interior and alternate. Interior is about the parallel lines (between them). Alternate is about the transversal (opposite sides).
Adding the outer angles of a zigzag to 180°. Draw a parallel line through the bend: the bend angle EQUALS the two outer angles added together.
Adding the outer angles of a zigzag to 180°. Draw a parallel line through the bend: the bend angle EQUALS the two outer angles added together.
Practice: 20 questions
12 of 20 are word problems. For a word problem, sketch it first and mark what you know.
Tier 1 Foundations Use the rule directly
- Angle 1 = 115°. Find angle 5.
- Angle 3 = 72°. Find angle 6.
- A road crosses two parallel railroad tracks. At the north track, the angle on the west side of the road, north of the track, is 64°. At the south track, what is the angle on the west side of the road, north of the track? Word problem
- Parking stripes are painted parallel, and the curb line crosses two stripes. Between the stripes, left of the curb at the first stripe, the angle is 118°. What is the angle between the stripes, right of the curb, at the second stripe? Word problem
- Elm Street and Oak Street are parallel, and Main Avenue crosses both. The northeast corner at Elm and Main measures 75°. What does the southwest corner at Oak and Main measure? (North is above the street.) Word problem
Tier 2 Intermediate One more step to think through
- Angle 4 = 58°. Find angle 6.
- What kind of angle pair are angles 3 and 5, and how are they related?
- Angle 1 = 68° and angle 5 = 70°. Are lines m and n parallel?
- A slanted brace runs between two parallel shelves. Just under the top shelf, right of the brace, it makes a 52° angle. What angle does it make just above the bottom shelf, right of the brace? Word problem
- A bike path crosses two parallel lane lines and makes a 47° angle at one corner. Of the 8 angles formed, how many measure 47°, and what do the others measure? Word problem
- Kayla measures the angle above the top rail of a ladder-style bookcase, right of a diagonal support, and the angle below the bottom rail, left of the support. What kind of pair did she measure? Word problem
- A carpenter checks two shelves with a slanted board crossing both. Between the shelves, on opposite sides of the board, she measures 87° and 93°. Are the shelves parallel? Word problem
Tier 3 Advanced Several steps - find the hidden structure
- Angle 3 = (3x + 12)° and angle 6 = (5x − 28)°. Find x and angle 3.
- Angle 4 = (2x + 20)° and angle 6 = (3x + 10)°. Find x, angle 4 and angle 6.
- A fence has two parallel rails and a diagonal brace. Under the top rail, left of the brace, the angle is (4x + 6)°. Above the bottom rail, right of the brace, it is (6x − 30)°. Find x and the angle. Word problem
- On a bridge, the deck and top beam are parallel, and a strut crosses both. Between them, left of the strut, the angle at the top beam is (x + 40)° and the angle at the deck is (2x + 20)°. Find x and both angles. Word problem
- Two parallel streets are crossed by a third. Jordan marks the angle below the first street, right of the crossing street, as 70°, and the angle above the second street, right of the crossing street, also as 70°. What is his mistake, and what is the correct angle? Word problem
Tier 4 Challenge Stretch problems, some above grade level
- Lines m and n are parallel. A path leaves line m at a 40° angle, bends at a point between the lines, and meets line n at a 35° angle, as shown. Find the angle at the bend.
- A hiking trail runs between two parallel fences. It leaves the north fence at 45°, bends once, then meets the south fence. The bend measures 100°. What angle does the trail make with the south fence? Word problem
- A carpenter wants two shelves to be parallel. A brace crosses both, and between the shelves on the same side it makes angles of (3x + 10)° and (2x + 20)°. What must x be? Word problem
Answers
Tier 1 Foundations
- 115°. Angles 1 and 5 are corresponding angles, so they are equal: 115°.
- 72°. Angles 3 and 6 are alternate interior angles, so they are equal: 72°.
- 64°. Angles 1 and 5 are corresponding angles, so they are equal: 64°.
- 118°. Angles 3 and 6 are alternate interior angles, so they are equal: 118°.
- 75°. Angles 2 and 7 are alternate exterior angles, so they are equal: 75°.
Tier 2 Intermediate
- 122°. Angles 4 and 6 are same-side interior angles, so they are supplementary: 180° − 58° = 122°.
- Same-side interior angles (supplementary). Both sit between the parallel lines, on the same side of the transversal.
- No, not parallel. These are corresponding angles, which must be equal for parallel lines. 68° and 70° are not equal.
- 128°. Angles 4 and 6 are same-side interior angles, so they are supplementary: 180° − 52° = 128°.
- 4 measure 47°; the other 4 measure 133°. Every angle either matches the 47° angle or forms a straight line with it: 180° − 47° = 133°.
- Alternate exterior angles (equal). Both sit outside the parallel lines, on opposite sides of the transversal.
- No, not parallel. These are alternate interior angles, which must be equal for parallel lines. 87° and 93° are not equal.
Tier 3 Advanced
- x = 20; each angle is 72°. Angles 3 and 6 are alternate interior angles (equal), so set them equal and solve.
- x = 30; angle 4 = 80°, angle 6 = 100°. Angles 4 and 6 are same-side interior angles (supplementary), so set their sum to 180° and solve.
- x = 18; each angle is 78°. Angles 3 and 6 are alternate interior angles (equal), so set them equal and solve.
- x = 40; angle 3 = 80°, angle 5 = 100°. Angles 3 and 5 are same-side interior angles (supplementary), so set their sum to 180° and solve.
- It should be 110°, not 70°. They are same-side interior angles, which are supplementary, not equal: 180° − 70° = 110°.
Tier 4 Challenge
- 75°. Draw a line through the bend parallel to the other two. It splits the bend into two angles that are alternate interior angles with the outer ones, so the bend equals their sum: 40° + 35° = 75°.
- 55°. Draw a line through the bend parallel to the other two. It splits the bend into two angles that are alternate interior angles with the outer ones, so the bend equals their sum: 100° − 45° = 55°.
- x = 30; angle 3 = 100°, angle 5 = 80°. Angles 3 and 5 are same-side interior angles (supplementary), so set their sum to 180° and solve.
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