Maths
What are corresponding, alternate and vertically opposite angles?
Quick answer
When a line crosses two parallel lines, corresponding (F) and alternate (Z) angles are equal and co-interior (C) angles add to 180°. Vertically opposite angles are always equal.
In one sentence
When a straight line crosses two parallel lines, it makes eight angles that come in two sizes, and the names corresponding, alternate and co-interior tell you which angles are equal and which add up to 180°.
Explained
Picture a road crossing a railway: the two rails are parallel lines (they never meet), and the road is a transversal, a line that crosses both. At each rail the road makes four angles, so there are eight angles in total.
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Number them like this. The transversal leans to the right, like the stroke /. At the top crossing: 1 top left, 2 top right, 3 bottom left, 4 bottom right. At the bottom crossing, in the same way: 5 top left, 6 top right, 7 bottom left, 8 bottom right.
Because the lines are parallel, the transversal crosses both of them at the same slant. So the bottom crossing is an exact copy of the top one, and the eight angles come in just two sizes: the small (acute) angles 2, 3, 6, 7 and the big (obtuse) angles 1, 4, 5, 8. One small angle plus one big angle always makes 180°, because they sit together on a straight line.
The angle facts have names:
- Vertically opposite angles (US: vertical angles) are across from each other where two lines cross, like 1 and 4 or 2 and 3. They are always equal, even without parallel lines.
- Corresponding angles are in the same position at each crossing, like 1 and 5 or 3 and 7. They make an F shape. With parallel lines they are equal.
- Alternate angles are between the parallel lines on opposite sides of the transversal, like 3 and 6 or 4 and 5. They make a Z shape. With parallel lines they are equal. (US: alternate interior angles; 1 and 8, 2 and 7 are alternate exterior angles, also equal.)
- Co-interior angles (also called allied or, in the US, same-side interior angles) are between the parallel lines on the same side, like 3 and 5 or 4 and 6. They make a C or U shape and add up to 180°.
Why are vertically opposite angles equal? Angles 1 and 2 lie on a straight line, so 1 + 2 = 180°. Angles 2 and 4 also lie on a straight line, so 2 + 4 = 180°. Both sums contain angle 2, so angle 1 = angle 4.
The facts also work backwards (the converse): if corresponding angles are equal, or alternate angles are equal, the lines must be parallel. That is how you can prove two lines are parallel. You can revise angle types and angles on a straight line on the angles page.
Examples
If angle 2 = 65°, every other angle follows:
| Angle | Size | Reason |
|---|---|---|
| 3 | 65° | vertically opposite angle 2 |
| 6 | 65° | corresponding to angle 2 (F) |
| 7 | 65° | alternate to angle 2 across the lines, or corresponding to angle 3 |
| 1 | 115° | on a straight line with angle 2: 180° − 65° |
| 4, 5, 8 | 115° | vertically opposite, alternate or corresponding to angle 1 |
Worked example (algebra): two alternate angles are (2x + 15)° and (4x − 25)°. Alternate angles are equal, so 2x + 15 = 4x − 25. Then 40 = 2x, x = 20, and each angle is 2(20) + 15 = 55°.
A famous proof: why do the angles of a triangle add up to 180°? Draw a line through the top corner, parallel to the base. The two new angles beside the top angle are alternate to the two base angles, so they are equal to them. Together with the top angle they sit on a straight line, so the three angles of the triangle add up to 180°.
Common mistakes
- Using the rules without parallel lines. Corresponding and alternate angles are only equal if the lines are parallel (look for the arrow marks). Vertically opposite angles are the exception: always equal.
- Mixing up alternate and co-interior. Alternate angles (Z) are equal; co-interior angles (C) add to 180°. Fix: trace the letter with your finger.
- Pairing an acute angle with an obtuse one. Equal angles look the same size. If your answer says an acute angle equals an obtuse one, you have used the wrong rule: they add up to 180° instead.
- Not giving a reason. In tests you usually need to name the rule, for example "alternate angles are equal", not just the number.
Step by step
Step by step
- Check the linesMake sure the two lines are parallel (arrow marks or the question says so).
- Find the known angleMark it and decide whether it is acute or obtuse.
- Spot the shapeF means corresponding, Z means alternate, C means co-interior, X means vertically opposite.
- Apply the ruleEqual for F, Z and X; for C, subtract from 180°.
- State the reasonWrite the rule next to each angle you find.
Shortcut: on a diagram with one transversal and two parallel lines, every acute angle has the same size, every obtuse angle has the same size, and acute + obtuse = 180°.
Practice questions
Angles in parallel lines
Try each one, then check your answer.
- Two straight lines cross. One angle is 40°. What is the vertically opposite angle?
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Vertically opposite angles are equal: 40° - In question 1, what is the angle next to the 40° angle?
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Angles on a straight line: 180° − 40° = 140° - One angle is 72°. What is its corresponding angle (parallel lines)?
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Corresponding angles are equal: 72° - One angle is 118°. What is its alternate angle (parallel lines)?
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Alternate angles are equal: 118° - One co-interior angle is 75°. What is the other?
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180° − 75° = 105° - In the numbered diagram, angle 2 = 58°. Find angle 7.
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Angle 3 = 58° (vertically opposite), and angle 7 corresponds to angle 3: 58° - Angle 1 = 124°. Find angle 5.
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Corresponding angles: 124° - Angle 4 = 110°. Find angle 6.
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Co-interior angles: 180° − 110° = 70° - Angle 3 = 47°. Find angle 8.
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Angle 6 = 47° (alternate), and angle 8 is on a straight line with angle 6: 180° − 47° = 133° - Two corresponding angles are (3x + 10)° and 70°. Find x.
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3x + 10 = 70, so 3x = 60: x = 20 - Two alternate angles are (2x + 15)° and (4x − 25)°. Find x and the angle.
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2x + 15 = 4x − 25, so 2x = 40: x = 20, angle = 55° - Two co-interior angles are x and 2x. Find both angles.
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x + 2x = 180°, so x = 60°: 60° and 120° - Two co-interior angles are (x + 30)° and (2x − 15)°. Find x and both angles.
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3x + 15 = 180, so x = 55: 85° and 95° - Two vertically opposite angles are (5x − 12)° and (3x + 20)°. Find x and the angle.
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5x − 12 = 3x + 20, so 2x = 32: x = 16, angle = 68° - A transversal crosses two lines, making corresponding angles of 81° and 79°. Are the lines parallel?
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No: parallel lines would give equal corresponding angles, so they are not parallel. - One angle of a parallelogram is 65°. Find the angle next to it.
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Opposite sides are parallel, so neighbouring angles are co-interior: 180° − 65° = 115°
Word problems
Parallel lines in real life
Try each one, then check your answer.
- A straight road crosses two parallel railway tracks. It makes an angle of 62° with the first track. What angle does it make with the second track in the same position?
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Corresponding angles: 62° - Two shelves on a wall are parallel. A straight support strut makes an angle of 35° with the top shelf, on the lower side. What angle does it make with the bottom shelf on the opposite side (a Z shape)?
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Alternate angles: 35° - A slanted line crosses two parallel car park lines. Between the lines, it makes 70° with the first line. What angle does it make with the second line on the same side, also between the lines?
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Co-interior angles: 180° − 70° = 110° - Two parallel streets are crossed by a diagonal avenue. At the first junction one angle is 128°. What are the two different angle sizes at the second junction?
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Corresponding angle 128°, and on the straight line 180° − 128° = 52° - In a periscope the two mirrors are parallel. A light ray travels between them and makes 45° with the top mirror. What angle does it make with the bottom mirror?
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Alternate angles: 45°
Questions people ask
Are corresponding angles always equal?
Only when the two lines are parallel. If the lines are not parallel, the transversal crosses them at different slants and the corresponding angles are different.
What is the corresponding angles theorem (or postulate)?
It says that if a transversal crosses two parallel lines, the corresponding angles are equal (congruent). Its converse says that if corresponding angles are equal, the lines are parallel. Many US geometry courses take it as a postulate and use it to prove the other facts.
How do you prove vertically opposite angles are equal?
Each of the two angles makes a straight line (180°) with the same angle between them. So both equal 180° minus that angle, which means they are equal.
What is the difference between alternate and corresponding angles?
Corresponding angles are in the same position at each crossing (F shape). Alternate angles are on opposite sides of the transversal and between the parallel lines (Z shape). With parallel lines both pairs are equal.
What are alternate exterior angles?
They lie outside the two parallel lines, on opposite sides of the transversal, like angles 1 and 8. They are equal too.
What are corresponding angles in a triangle?
That is a different use of the word: in congruent or similar triangles, corresponding angles are the matching angles in the two shapes, and they are equal.
Quick quiz
Angles in parallel lines
Check what you know with 5 quick questions.
Question 1 of 5
Which angles make an F shape and are equal when the lines are parallel?
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Related topics
Angles
An angle measures the amount of turn between two lines that meet at a point. Angles are measured in degrees (°); a right angle is 90° and a full turn is 360°.
Parallel and perpendicular lines
Parallel lines run side by side and never meet, always the same distance apart. Perpendicular lines cross or meet at a right angle of exactly 90 degrees.
Types of triangles
Triangles are named by their sides (equilateral, isosceles, scalene) or by their angles (acute, right-angled, obtuse). The three angles of any triangle add up to 180 degrees.
Area of a circle
The area of a circle is pi times the radius squared: A = πr². A circle with radius 5 cm has an area of about 78.5 square centimetres.
Area of a trapezium (trapezoid)
Add the two parallel sides, multiply by the height and halve: A = ½(a + b)h. Parallel sides of 4 cm and 6 cm, 3 cm apart, give an area of 15 cm².
Area of a triangle
The area of a triangle is half of its base times its perpendicular height: A = ½ × b × h. A triangle with base 8 cm and height 5 cm has an area of 20 square centimetres.