Where two lines cross, opposite angles are equal and adjacent ones add to 180°. A transversal across two parallel lines makes eight angles, but only two sizes.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Try it first
Two lines cross at a point. One of the four angles measures 4x, and the angle beside it along one of the straight lines measures 5x. What is the larger of those two angles, in degrees?
degrees
Stuck is fine. Let’s Learn shows the way next. Got it first time? You can jump ahead to Watch an Example.
Vertical angles
Two crossing lines make four angles. Each pair opposite each other is equal, and each pair side by side adds to 180°. If one angle is 70°, the others are 110°, 70° and 110° round the crossing.
A transversal
A line crossing two parallel lines makes two crossings, eight angles. Because the lines are parallel, the second crossing is a copy of the first, slid along the transversal.
The names
Corresponding angles sit in the same position at each crossing: equal. Alternate interior angles sit on opposite sides of the transversal, between the lines: equal. Co-interior angles sit on the same side, between the lines: they add to 180°.
Why corresponding angles are equal
Slide the first crossing along the transversal until it lands on the second. Parallel lines never change direction, so every angle lands on its corresponding angle exactly. Equal, by a translation.
The other way round
If a transversal makes equal corresponding angles, the two lines must be parallel. That is how parallel lines are proved: find one pair of equal corresponding or alternate angles.
The shortcut worth having
Find one angle, and every other is either equal to it or its supplement. Eight angles, two sizes.
Step 2: Try It Yourself
Tap and try it out.
Co-interior angles of 3x + 20 and 2x + 10 lie between parallel lines. Find x
- 1(3x + 20) + (2x + 10) = 180co-interior angles between parallel lines add to 180°
Step 0 of 4
55° is an acute angle.
Step 3: In Real Life
A railway crossing
A road crosses two parallel tracks. The angle it makes with the first track equals the angle with the second. A surveyor measures one and knows all eight.
Step 4: Watch an Example
One step at a time.
Watch Priya Fill In a Figure From One Angle
A transversal crosses parallel lines p and q. At the crossing on p: ∠1 above p and left of the transversal, ∠2 above p and right, ∠3 below p and left, ∠4 below p and right. The same four positions at the crossing on q are ∠5, ∠6, ∠7 and ∠8. Priya is told ∠1 = 124° and no other angle.
- Step 1
The one given is ∠1 = 124°, at the crossing on p. Everything else on the page has to be argued out of it, with a reason on every move, the way the last lesson set out.
Watch Dev Find x Between Two Parallel Lines
The same figure and the same numbering. This time p and q are parallel, ∠3 = 2x + 15 and ∠6 = 3x − 24. Dev writes the first two lines; you supply the next three.
- Step 1
The givens go down first: p is parallel to q, ∠3 = 2x + 15 and ∠6 = 3x − 24. ∠3 lies below p, left of the transversal, and ∠6 lies above q, right of it, so both are between the lines and they are on opposite sides.
Two Roads to ∠8
The same figure and numbering once more, p parallel to q, and this time ∠1 = 76°. Maya and Ravi both find ∠8, the angle below q and right of the transversal, and they take different routes.
- Step 1
Maya's road turns at the first crossing. ∠4 lies opposite ∠1, so ∠4 = 76° because vertical angles are equal; then ∠8 sits at the second crossing in the same position as ∠4, so ∠8 = 76° because corresponding angles at parallel lines are equal.
Step 5: Your Turn
Practice makes it stick.
The Gate Brace
Problem 1 of 2
A gate has two parallel rails with one diagonal brace crossing both. Where the brace meets the top rail, the angle on one side measures 118°. At each rail the joiner cuts the acute angle the brace makes, and the shop charges $1.50 for every degree that a cut is set away from its standard 45°. Both cuts are made. What does the pair of cuts cost, in dollars?
dollars
The Window Wedge
Problem 2 of 2
A window frame has two parallel bars crossed by one glazing strip. The two angles that lie between the bars on the same side of the strip measure 3x − 5 and 2x + 15 degrees. A glass wedge is cut to fill each of those two angles, and the glazier charges $0.80 for every degree of angle cut. What do the two wedges cost together, in dollars?
dollars
Find the Angle
Name the pair first. Once the relationship is named, the arithmetic is one line.
1 of 5
A transversal crosses two parallel lines. ∠a = 47°, and ∠b is the angle at the other crossing in the same position as ∠a. What is ∠b, in degrees?
degrees
2 of 5
A transversal crosses two parallel lines. ∠a = 132°, and ∠b lies between the two lines on the same side of the transversal as ∠a. What is ∠b, in degrees?
degrees
3 of 5
A transversal crosses two parallel lines. ∠1 = 71° at the first crossing, ∠5 is the angle at the second crossing in the same position as ∠1, and ∠6 sits beside ∠5 along that second line. What is ∠6, in degrees?
degrees
4 of 5
A transversal crosses two parallel lines. Two corresponding angles measure 5x − 12 and 3x + 18. What is x?
5 of 5
Two co-interior angles between a pair of parallel lines measure 6x and 2x + 20. What is the larger of the two angles, in degrees?
degrees
Whose Step?
One line in each piece of working is a slip. Find it, then put it right.
1 of 6
A transversal crosses two lines. Nothing in the figure says the two lines are parallel. ∠1 = 68° at the first crossing, and ∠5 sits at the second crossing in the same position. Maya concluded ∠5 = 68°. One line is a slip. Which?
Maya's working
- 1∠1 = 68°
- 2∠1 and ∠5 are corresponding angles
- 3∠1 = ∠5
- 4∠5 = 68°
2 of 6
Suppose the figure is corrected so that the two lines are parallel. Maya's chain then holds all the way down. ∠6 sits beside ∠5 along that second line. What is ∠6, in degrees?
degrees
3 of 6
Two parallel lines are crossed by a transversal. ∠4 and ∠6 are co-interior: between the lines, both on the right of the transversal. ∠4 = 3x + 10 and ∠6 = 2x + 40. Leo concluded x = 30. One line is a slip. Which?
Leo's working
- 1∠4 = 3x + 10, ∠6 = 2x + 40
- 23x + 10 = 2x + 40
- 3x + 10 = 40− 2x
- 4x = 30− 10
4 of 6
Put Leo's second line right and finish the working. What is x?
5 of 6
Two parallel lines are crossed by a transversal. ∠3 = 54°, and ∠6 lies between the lines on the opposite side of the transversal from ∠3. Ana concluded ∠6 = 54°. Her three statements are below, and the options carry the reason she wrote beside each one. Every statement is true, but one reason is not a reason. Which line?
Ana's proof
- 1∠3 = 54°
- 2∠3 = ∠6
- 3∠6 = 54°
6 of 6
What reason should Ana write beside ∠3 = ∠6?
Pick the Move
Decide the first line after the givens, before any arithmetic.
1 of 2
Parallel lines p and q are crossed by a transversal. Two angles, one at each crossing, lie between the lines on opposite sides of the transversal; they measure 4x + 6 and 6x − 24. Which line comes first, after the givens?
2 of 2
A transversal crosses two lines, and nobody has said whether they are parallel. An angle at the first crossing measures 118°, and the angle at the second crossing in the same position measures 118° too. A student wants to prove the two lines are parallel. Which line does it?
Same Size?
Which of these must match, and which only match sometimes? Decide without measuring anything.
1 of 2
A transversal crosses two parallel lines. Select every pair below that must be equal.
2 of 2
This time the two lines the transversal crosses are not parallel. Select every pair that must still be the same size.
Step 6: Quick Check
Show what you know.
Question 1 of 1
Co-interior with 125°, lines parallel. What is it?
Ready for more?Optional. Past the lesson, for anyone who wants it.
Stretch 1 of 2
A transversal crosses two parallel lines, and one of the eight angles measures k degrees. For which value of k do all eight angles come out the same size?
degrees
Stretch 2 of 2
A transversal crosses two parallel lines. Two of the eight angles are picked out, one at each crossing, and they measure 5x − 20 and 3x + 8. You are not told which named pair they form — only that both readings are correct, and that the two angles are not the same size. What is x?
What You Learned
- Vertical angles are equal wherever two lines cross; neighbors add to 180°.
- With parallel lines, corresponding and alternate angles are equal and co-interior ones add to 180°, because one crossing is the other slid along.
- Equal corresponding angles prove lines parallel.
- Eight angles, but only two sizes: find one and the rest follow.
For the grown-up
The misconception this lesson targets is that the angle-pair names carry their rules around with them. Corresponding, alternate interior and co-interior are names for positions, and those positions exist at any two lines a transversal crosses; what makes the first two equal, and the third supplementary, is the lines being parallel. Only vertical angles and angles on a straight line hold with or without it. Its cousin is treating every named pair as an equal pair, which turns a co-interior pair into an equation whose two angles then come out equal — and a pair of equal angles totals 180° only when both are right angles. Adding the pair up is the check worth teaching: a co-interior pair totalling anything but 180° belongs to two lines that meet somewhere, not to parallel ones.