GCSE Maths: Parallel lines (Foundation)
Part of the GCSE Maths angles, polygons and symmetry chapter (Foundation). Chapter overview · Higher version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Parallel lines
Parallel lines are lines that stay the same distance apart for ever. We mark them with matching arrows. A line that crosses a pair of parallel lines is called a transversal. It makes eight angles, and they come in sets of equal angles or angles that add up to 180°. You do not need to learn eight facts: learn the three shapes below.
Alternate angles are equal (a Z shape). They lie between the parallel lines, on opposite sides of the transversal. Trace a Z (or an S) and you will find the two equal angles in its corners.
Corresponding angles are equal (an F shape). They are in the same position at the two crossings, like a copy of the same angle slid along the transversal. Trace an F (it may be backwards or upside down).
Co-interior angles add up to 180° (a C or U shape). They lie between the parallel lines on the same side of the transversal. They are not equal (unless both are 90°); instead they make a straight-line total of 180°. They are also called allied angles, or “interior angles on the same side”.
Why do these facts work? Corresponding angles are equal because sliding the top crossing down the transversal along the parallel lines moves one angle exactly onto the other. Alternate angles are equal because one of them is vertically opposite to the angle corresponding to the other. And co-interior angles add to 180° because one of them is next to an angle that is alternate to the other, on a straight line. These facts are only true when the lines are parallel. If the lines are not parallel, the angles are not equal.
How to answer. (1) Check for arrows. (2) Find a pair of angles making a Z, F or C shape (or an X or a straight line). (3) Write the reason in words. Often you need two steps: write the reason for each.
Key ideas
- Alternate angles are equal (Z shape).
- Corresponding angles are equal (F shape).
- Co-interior angles add up to 180° (C or U shape).
- These only work for parallel lines (look for the arrows).
- You can use vertically opposite angles and angles on a line as well, to reach the angle you need.
- Always write the reason in words: “alternate angles are equal”.
Key words. Parallel lines always stay the same distance apart and never meet (they are marked with arrows). A transversal is a line that crosses them. It makes alternate angles (a Z shape), corresponding angles (an F shape) and co-interior angles (a C or U shape).
Method. Find which angle fact links the two angles, then use it.
- Look at the diagram. The arrows on the two lines show that they are parallel, so the parallel-line angle facts can be used. We know one angle and want another.Known: the 65° angle = 65°
Wanted: x - Find which angle fact links the 65° angle and x. The two angles are in the same position at the two crossing points: an F shape (parallel lines).They make an F shape, so the fact is: Corresponding angles are equal.
- Use the fact to find x. The two angles are equal, so the new angle is the same size as the known one.the 65° angle = 65°
x = 65°
Check. Does the answer look right on the diagram? The angle marked x looks acute (less than 90°), and 65° is acute (less than 90°). ✓
Watch out. Alternate (Z), corresponding (F) and co-interior (C) angles only work between parallel lines. Check the arrows on the lines first.
Key words. Parallel lines always stay the same distance apart and never meet (they are marked with arrows). A transversal is a line that crosses them. It makes alternate angles (a Z shape), corresponding angles (an F shape) and co-interior angles (a C or U shape).
Method. Find which angle fact links the two angles, then use it.
- Look at the diagram. The arrows on the two lines show that they are parallel, so the parallel-line angle facts can be used. We know one angle and want another.Known: the 48° angle = 48°
Wanted: y - Find which angle fact links the 48° angle and y. The two angles are between the parallel lines on opposite sides of the transversal: a Z shape (parallel lines).They make a Z shape, so the fact is: Alternate angles are equal.
- Use the fact to find y. The two angles are equal, so the new angle is the same size as the known one.the 48° angle = 48°
y = 48°
Check. Does the answer look right on the diagram? The angle marked y looks acute (less than 90°), and 48° is acute (less than 90°). ✓
Watch out. Alternate (Z), corresponding (F) and co-interior (C) angles only work between parallel lines. Check the arrows on the lines first.
Key words. Parallel lines always stay the same distance apart and never meet (they are marked with arrows). A transversal is a line that crosses them. It makes alternate angles (a Z shape), corresponding angles (an F shape) and co-interior angles (a C or U shape).
Method. Find which angle fact links the two angles, then use it.
- Look at the diagram. The arrows on the two lines show that they are parallel, so the parallel-line angle facts can be used. We know one angle and want another.Known: the 70° angle = 70°
Wanted: z - Find which angle fact links the 70° angle and z. The two angles are between the parallel lines on the same side of the transversal: a C or U shape (parallel lines).They make a C or U shape, so the fact is: Co-interior angles add up to 180°.
- Use the fact to find z. The two angles add up to 180°, so subtract the known angle from 180°.the 70° angle = 70°
180 − 70 = 110°
Check. Does the answer look right on the diagram? The angle marked z looks obtuse (between 90° and 180°), and 110° is obtuse (between 90° and 180°). ✓
Watch out. Alternate (Z), corresponding (F) and co-interior (C) angles only work between parallel lines. Check the arrows on the lines first.
Key words. Parallel lines always stay the same distance apart and never meet (they are marked with arrows). A transversal is a line that crosses them. It makes alternate angles (a Z shape), corresponding angles (an F shape) and co-interior angles (a C or U shape).
Method. Use one angle fact at a time, starting from the angle we are given, and write down the reason for each step.
- Look at the diagram. The arrows on the two lines show that they are parallel, so the parallel-line angle facts can be used. We know one angle and want another.Known: the 128° angle = 128°
Wanted: x
(this needs 2 steps) - Find which angle fact links the 128° angle and the angle at the top crossing, below the line and to the left of the transversal. The two angles sit side by side on one straight line, so together they make a half turn.They make a straight line, so the fact is: Angles on a straight line add up to 180°.
- Use the fact to find the angle at the top crossing, below the line and to the left of the transversal. The two angles add up to 180°, so subtract the known angle from 180°.the 128° angle = 128°
180 − 128 = 52° - Find which angle fact links the angle at the top crossing, below the line and to the left of the transversal and x. The two angles are in the same position at the two crossing points: an F shape (parallel lines).They make an F shape, so the fact is: Corresponding angles are equal.
- Use the fact to find x. The two angles are equal, so the new angle is the same size as the known one.the angle at the top crossing, below the line and to the left of the transversal = 52°
x = 52°
Check. Does the answer look right on the diagram? The angle marked x looks acute (less than 90°), and 52° is acute (less than 90°). ✓
Watch out. Alternate (Z), corresponding (F) and co-interior (C) angles only work between parallel lines. Check the arrows on the lines first.
Watch out
Never use these facts unless the lines are parallel. Also, do not mix up the shapes: Z (alternate) and F (corresponding) angles are equal, but a C shape (co-interior) adds to 180°. A quick check: if both angles look the same size on the diagram, they are equal; if one is acute and the other obtuse, they add to 180°.Exercise 17D Parallel lines: alternate, corresponding and co-interior angles
Non-calculatorKey words. Parallel lines always stay the same distance apart and never meet (they are marked with arrows). A transversal is a line that crosses them. It makes alternate angles (a Z shape), corresponding angles (an F shape) and co-interior angles (a C or U shape).
Method. Find which angle fact links the two angles, then use it.
- Look at the diagram. The arrows on the two lines show that they are parallel, so the parallel-line angle facts can be used. We know one angle and want another.Known: the 65° angle = 65°
Wanted: x - Find which angle fact links the 65° angle and x. The two angles are in the same position at the two crossing points: an F shape (parallel lines).They make an F shape, so the fact is: Corresponding angles are equal.
- Use the fact to find x. The two angles are equal, so the new angle is the same size as the known one.the 65° angle = 65°
x = 65°
Check. Does the answer look right on the diagram? The angle marked x looks acute (less than 90°), and 65° is acute (less than 90°). ✓
Watch out. Alternate (Z), corresponding (F) and co-interior (C) angles only work between parallel lines. Check the arrows on the lines first.
Key words. Parallel lines always stay the same distance apart and never meet (they are marked with arrows). A transversal is a line that crosses them. It makes alternate angles (a Z shape), corresponding angles (an F shape) and co-interior angles (a C or U shape).
Method. Find which angle fact links the two angles, then use it.
- Look at the diagram. The arrows on the two lines show that they are parallel, so the parallel-line angle facts can be used. We know one angle and want another.Known: the 72° angle = 72°
Wanted: y - Find which angle fact links the 72° angle and y. The two angles are between the parallel lines on opposite sides of the transversal: a Z shape (parallel lines).They make a Z shape, so the fact is: Alternate angles are equal.
- Use the fact to find y. The two angles are equal, so the new angle is the same size as the known one.the 72° angle = 72°
y = 72°
Check. Does the answer look right on the diagram? The angle marked y looks acute (less than 90°), and 72° is acute (less than 90°). ✓
Watch out. Alternate (Z), corresponding (F) and co-interior (C) angles only work between parallel lines. Check the arrows on the lines first.
Key words. Parallel lines always stay the same distance apart and never meet (they are marked with arrows). A transversal is a line that crosses them. It makes alternate angles (a Z shape), corresponding angles (an F shape) and co-interior angles (a C or U shape).
Method. Find which angle fact links the two angles, then use it.
- Look at the diagram. The arrows on the two lines show that they are parallel, so the parallel-line angle facts can be used. We know one angle and want another.Known: the 65° angle = 65°
Wanted: z - Find which angle fact links the 65° angle and z. The two angles are between the parallel lines on the same side of the transversal: a C or U shape (parallel lines).They make a C or U shape, so the fact is: Co-interior angles add up to 180°.
- Use the fact to find z. The two angles add up to 180°, so subtract the known angle from 180°.the 65° angle = 65°
180 − 65 = 115°
Check. Does the answer look right on the diagram? The angle marked z looks obtuse (between 90° and 180°), and 115° is obtuse (between 90° and 180°). ✓
Watch out. Alternate (Z), corresponding (F) and co-interior (C) angles only work between parallel lines. Check the arrows on the lines first.
Key words. Parallel lines always stay the same distance apart and never meet (they are marked with arrows). A transversal is a line that crosses them. It makes alternate angles (a Z shape), corresponding angles (an F shape) and co-interior angles (a C or U shape).
Method. Find which angle fact links the two angles, then use it.
- Look at the diagram. The arrows on the two lines show that they are parallel, so the parallel-line angle facts can be used. We know one angle and want another.Known: the 48° angle = 48°
Wanted: a - Find which angle fact links the 48° angle and a. The two angles are in the same position at the two crossing points: an F shape (parallel lines).They make an F shape, so the fact is: Corresponding angles are equal.
- Use the fact to find a. The two angles are equal, so the new angle is the same size as the known one.the 48° angle = 48°
a = 48°
Check. Does the answer look right on the diagram? The angle marked a looks acute (less than 90°), and 48° is acute (less than 90°). ✓
Watch out. Alternate (Z), corresponding (F) and co-interior (C) angles only work between parallel lines. Check the arrows on the lines first.
Key words. Parallel lines always stay the same distance apart and never meet (they are marked with arrows). A transversal is a line that crosses them. It makes alternate angles (a Z shape), corresponding angles (an F shape) and co-interior angles (a C or U shape).
Method. Find which angle fact links the two angles, then use it.
- Look at the diagram. The arrows on the two lines show that they are parallel, so the parallel-line angle facts can be used. We know one angle and want another.Known: the 133° angle = 133°
Wanted: b - Find which angle fact links the 133° angle and b. The two angles are between the parallel lines on opposite sides of the transversal: a Z shape (parallel lines).They make a Z shape, so the fact is: Alternate angles are equal.
- Use the fact to find b. The two angles are equal, so the new angle is the same size as the known one.the 133° angle = 133°
b = 133°
Check. Does the answer look right on the diagram? The angle marked b looks obtuse (between 90° and 180°), and 133° is obtuse (between 90° and 180°). ✓
Watch out. Alternate (Z), corresponding (F) and co-interior (C) angles only work between parallel lines. Check the arrows on the lines first.
Key words. Parallel lines always stay the same distance apart and never meet (they are marked with arrows). A transversal is a line that crosses them. It makes alternate angles (a Z shape), corresponding angles (an F shape) and co-interior angles (a C or U shape).
Method. Find which angle fact links the two angles, then use it.
- Look at the diagram. The arrows on the two lines show that they are parallel, so the parallel-line angle facts can be used. We know one angle and want another.Known: the 56° angle = 56°
Wanted: c - Find which angle fact links the 56° angle and c. The two angles are between the parallel lines on the same side of the transversal: a C or U shape (parallel lines).They make a C or U shape, so the fact is: Co-interior angles add up to 180°.
- Use the fact to find c. The two angles add up to 180°, so subtract the known angle from 180°.the 56° angle = 56°
180 − 56 = 124°
Check. Does the answer look right on the diagram? The angle marked c looks obtuse (between 90° and 180°), and 124° is obtuse (between 90° and 180°). ✓
Watch out. Alternate (Z), corresponding (F) and co-interior (C) angles only work between parallel lines. Check the arrows on the lines first.
Key words. Parallel lines always stay the same distance apart and never meet (they are marked with arrows). A transversal is a line that crosses them. It makes alternate angles (a Z shape), corresponding angles (an F shape) and co-interior angles (a C or U shape).
Method. Find which angle fact links the two angles, then use it.
- Look at the diagram. The arrows on the two lines show that they are parallel, so the parallel-line angle facts can be used. We know one angle and want another.Known: the 139° angle = 139°
Wanted: x - Find which angle fact links the 139° angle and x. The two angles are between the parallel lines on opposite sides of the transversal: a Z shape (parallel lines).They make a Z shape, so the fact is: Alternate angles are equal.
- Use the fact to find x. The two angles are equal, so the new angle is the same size as the known one.the 139° angle = 139°
x = 139°
Check. Does the answer look right on the diagram? The angle marked x looks obtuse (between 90° and 180°), and 139° is obtuse (between 90° and 180°). ✓
Watch out. Alternate (Z), corresponding (F) and co-interior (C) angles only work between parallel lines. Check the arrows on the lines first.
Key words. Parallel lines always stay the same distance apart and never meet (they are marked with arrows). A transversal is a line that crosses them. It makes alternate angles (a Z shape), corresponding angles (an F shape) and co-interior angles (a C or U shape).
Method. Find which angle fact links the two angles, then use it.
- Look at the diagram. The arrows on the two lines show that they are parallel, so the parallel-line angle facts can be used. We know one angle and want another.Known: the 103° angle = 103°
Wanted: p - Find which angle fact links the 103° angle and p. The two angles are in the same position at the two crossing points: an F shape (parallel lines).They make an F shape, so the fact is: Corresponding angles are equal.
- Use the fact to find p. The two angles are equal, so the new angle is the same size as the known one.the 103° angle = 103°
p = 103°
Check. Does the answer look right on the diagram? The angle marked p looks obtuse (between 90° and 180°), and 103° is obtuse (between 90° and 180°). ✓
Watch out. Alternate (Z), corresponding (F) and co-interior (C) angles only work between parallel lines. Check the arrows on the lines first.
Key words. Parallel lines always stay the same distance apart and never meet (they are marked with arrows). A transversal is a line that crosses them. It makes alternate angles (a Z shape), corresponding angles (an F shape) and co-interior angles (a C or U shape).
Method. Find which angle fact links the two angles, then use it.
- Look at the diagram. The arrows on the two lines show that they are parallel, so the parallel-line angle facts can be used. We know one angle and want another.Known: the 104° angle = 104°
Wanted: q - Find which angle fact links the 104° angle and q. The two angles are in the same position at the two crossing points: an F shape (parallel lines).They make an F shape, so the fact is: Corresponding angles are equal.
- Use the fact to find q. The two angles are equal, so the new angle is the same size as the known one.the 104° angle = 104°
q = 104°
Check. Does the answer look right on the diagram? The angle marked q looks obtuse (between 90° and 180°), and 104° is obtuse (between 90° and 180°). ✓
Watch out. Alternate (Z), corresponding (F) and co-interior (C) angles only work between parallel lines. Check the arrows on the lines first.
Key words. Parallel lines always stay the same distance apart and never meet (they are marked with arrows). A transversal is a line that crosses them. It makes alternate angles (a Z shape), corresponding angles (an F shape) and co-interior angles (a C or U shape).
Method. Find which angle fact links the two angles, then use it.
- Look at the diagram. The arrows on the two lines show that they are parallel, so the parallel-line angle facts can be used. We know one angle and want another.Known: the 123° angle = 123°
Wanted: m - Find which angle fact links the 123° angle and m. The two angles are between the parallel lines on opposite sides of the transversal: a Z shape (parallel lines).They make a Z shape, so the fact is: Alternate angles are equal.
- Use the fact to find m. The two angles are equal, so the new angle is the same size as the known one.the 123° angle = 123°
m = 123°
Check. Does the answer look right on the diagram? The angle marked m looks obtuse (between 90° and 180°), and 123° is obtuse (between 90° and 180°). ✓
Watch out. Alternate (Z), corresponding (F) and co-interior (C) angles only work between parallel lines. Check the arrows on the lines first.
Key words. Parallel lines always stay the same distance apart and never meet (they are marked with arrows). A transversal is a line that crosses them. It makes alternate angles (a Z shape), corresponding angles (an F shape) and co-interior angles (a C or U shape).
Method. Look at where angle a is, then find the angle that makes the right shape with it.
- Find angle a on the diagram. Say where it is: the angle at the top crossing, above the line and to the right of the transversal.angle a = the angle at the top crossing, above the line and to the right of the transversal
- Look for the corresponding angle. Corresponding angles are in the same position at the other crossing point.The angles a and e make an F shape.
- Write the answer. angle e
Check. Both angles are marked on the diagram, and a and e are corresponding. ✓
Watch out. Alternate and corresponding angles are equal; co-interior angles add up to 180°.
Key words. Parallel lines always stay the same distance apart and never meet (they are marked with arrows). A transversal is a line that crosses them. It makes alternate angles (a Z shape), corresponding angles (an F shape) and co-interior angles (a C or U shape).
Method. Look at where angle c is, then find the angle that makes the right shape with it.
- Find angle c on the diagram. Say where it is: the angle at the top crossing, below the line and to the left of the transversal.angle c = the angle at the top crossing, below the line and to the left of the transversal
- Look for the alternate angle. Alternate angles are between the parallel lines on the opposite side of the transversal.The angles c and e make a Z shape.
- Write the answer. angle e
Check. Both angles are marked on the diagram, and c and e are alternate. ✓
Watch out. Alternate and corresponding angles are equal; co-interior angles add up to 180°.
Key words. Parallel lines always stay the same distance apart and never meet (they are marked with arrows). A transversal is a line that crosses them. It makes alternate angles (a Z shape), corresponding angles (an F shape) and co-interior angles (a C or U shape).
Method. Look at where angle d is, then find the angle that makes the right shape with it.
- Find angle d on the diagram. Say where it is: the angle at the top crossing, below the line and to the right of the transversal.angle d = the angle at the top crossing, below the line and to the right of the transversal
- Look for the co-interior angle. Co-interior angles are between the parallel lines on the same side of the transversal.The angles d and e make a C or U shape.
- Write the answer. angle e
Check. Both angles are marked on the diagram, and d and e are co-interior. ✓
Watch out. Alternate and corresponding angles are equal; co-interior angles add up to 180°.
Key words. Parallel lines always stay the same distance apart and never meet (they are marked with arrows). A transversal is a line that crosses them. It makes alternate angles (a Z shape), corresponding angles (an F shape) and co-interior angles (a C or U shape).
Method. Look at where angle d is, then find the angle that makes the right shape with it.
- Find angle d on the diagram. Say where it is: the angle at the top crossing, below the line and to the right of the transversal.angle d = the angle at the top crossing, below the line and to the right of the transversal
- Look for the corresponding angle. Corresponding angles are in the same position at the other crossing point.The angles d and h make an F shape.
- Write the answer. angle h
Check. Both angles are marked on the diagram, and d and h are corresponding. ✓
Watch out. Alternate and corresponding angles are equal; co-interior angles add up to 180°.
Key words. Parallel lines always stay the same distance apart and never meet (they are marked with arrows). A transversal is a line that crosses them. It makes alternate angles (a Z shape), corresponding angles (an F shape) and co-interior angles (a C or U shape).
Method. Look at where angle d is, then find the angle that makes the right shape with it.
- Find angle d on the diagram. Say where it is: the angle at the top crossing, below the line and to the right of the transversal.angle d = the angle at the top crossing, below the line and to the right of the transversal
- Look for the alternate angle. Alternate angles are between the parallel lines on the opposite side of the transversal.The angles d and f make a Z shape.
- Write the answer. angle f
Check. Both angles are marked on the diagram, and d and f are alternate. ✓
Watch out. Alternate and corresponding angles are equal; co-interior angles add up to 180°.
Key words. Parallel lines always stay the same distance apart and never meet (they are marked with arrows). A transversal is a line that crosses them. It makes alternate angles (a Z shape), corresponding angles (an F shape) and co-interior angles (a C or U shape).
Method. Look at where angle c is, then find the angle that makes the right shape with it.
- Find angle c on the diagram. Say where it is: the angle at the top crossing, below the line and to the left of the transversal.angle c = the angle at the top crossing, below the line and to the left of the transversal
- Look for the co-interior angle. Co-interior angles are between the parallel lines on the same side of the transversal.The angles c and f make a C or U shape.
- Write the answer. angle f
Check. Both angles are marked on the diagram, and c and f are co-interior. ✓
Watch out. Alternate and corresponding angles are equal; co-interior angles add up to 180°.
Key words. Parallel lines always stay the same distance apart and never meet (they are marked with arrows). A transversal is a line that crosses them. It makes alternate angles (a Z shape), corresponding angles (an F shape) and co-interior angles (a C or U shape).
Method. Find which angle fact links the two angles, then use it.
- Look at the diagram. The arrows on the two lines show that they are parallel, so the parallel-line angle facts can be used. We know one angle and want another.Known: the 62° angle = 62°
Wanted: b - Find which angle fact links the 62° angle and b. The two angles sit side by side on one straight line, so together they make a half turn.They make a straight line, so the fact is: Angles on a straight line add up to 180°.
- Use the fact to find b. The two angles add up to 180°, so subtract the known angle from 180°.the 62° angle = 62°
180 − 62 = 118°
Check. Does the answer look right on the diagram? The angle marked b looks obtuse (between 90° and 180°), and 118° is obtuse (between 90° and 180°). ✓
Watch out. Alternate (Z), corresponding (F) and co-interior (C) angles only work between parallel lines. Check the arrows on the lines first.
Key words. Parallel lines always stay the same distance apart and never meet (they are marked with arrows). A transversal is a line that crosses them. It makes alternate angles (a Z shape), corresponding angles (an F shape) and co-interior angles (a C or U shape).
Method. Find which angle fact links the two angles, then use it.
- Look at the diagram. The arrows on the two lines show that they are parallel, so the parallel-line angle facts can be used. We know one angle and want another.Known: the 62° angle = 62°
Wanted: c - Find which angle fact links the 62° angle and c. The two angles are opposite each other where two straight lines cross (an X shape).They make an X shape, so the fact is: Vertically opposite angles are equal.
- Use the fact to find c. The two angles are equal, so the new angle is the same size as the known one.the 62° angle = 62°
c = 62°
Check. Does the answer look right on the diagram? The angle marked c looks acute (less than 90°), and 62° is acute (less than 90°). ✓
Watch out. Alternate (Z), corresponding (F) and co-interior (C) angles only work between parallel lines. Check the arrows on the lines first.
Key words. Parallel lines always stay the same distance apart and never meet (they are marked with arrows). A transversal is a line that crosses them. It makes alternate angles (a Z shape), corresponding angles (an F shape) and co-interior angles (a C or U shape).
Method. Find which angle fact links the two angles, then use it.
- Look at the diagram. The arrows on the two lines show that they are parallel, so the parallel-line angle facts can be used. We know one angle and want another.Known: the 62° angle = 62°
Wanted: d - Find which angle fact links the 62° angle and d. The two angles sit side by side on one straight line, so together they make a half turn.They make a straight line, so the fact is: Angles on a straight line add up to 180°.
- Use the fact to find d. The two angles add up to 180°, so subtract the known angle from 180°.the 62° angle = 62°
180 − 62 = 118°
Check. Does the answer look right on the diagram? The angle marked d looks obtuse (between 90° and 180°), and 118° is obtuse (between 90° and 180°). ✓
Watch out. Alternate (Z), corresponding (F) and co-interior (C) angles only work between parallel lines. Check the arrows on the lines first.
Key words. Parallel lines always stay the same distance apart and never meet (they are marked with arrows). A transversal is a line that crosses them. It makes alternate angles (a Z shape), corresponding angles (an F shape) and co-interior angles (a C or U shape).
Method. Find which angle fact links the two angles, then use it.
- Look at the diagram. The arrows on the two lines show that they are parallel, so the parallel-line angle facts can be used. We know one angle and want another.Known: the 62° angle = 62°
Wanted: e - Find which angle fact links the 62° angle and e. The two angles are in the same position at the two crossing points: an F shape (parallel lines).They make an F shape, so the fact is: Corresponding angles are equal.
- Use the fact to find e. The two angles are equal, so the new angle is the same size as the known one.the 62° angle = 62°
e = 62°
Check. Does the answer look right on the diagram? The angle marked e looks acute (less than 90°), and 62° is acute (less than 90°). ✓
Watch out. Alternate (Z), corresponding (F) and co-interior (C) angles only work between parallel lines. Check the arrows on the lines first.
Key words. Parallel lines always stay the same distance apart and never meet (they are marked with arrows). A transversal is a line that crosses them. It makes alternate angles (a Z shape), corresponding angles (an F shape) and co-interior angles (a C or U shape).
Method. Use one angle fact at a time, starting from the angle we are given, and write down the reason for each step.
- Look at the diagram. The arrows on the two lines show that they are parallel, so the parallel-line angle facts can be used. We know one angle and want another.Known: the 62° angle = 62°
Wanted: f
(this needs 2 steps) - Find which angle fact links the 62° angle and the angle at the top crossing, above the line and to the left of the transversal. The two angles sit side by side on one straight line, so together they make a half turn.They make a straight line, so the fact is: Angles on a straight line add up to 180°.
- Use the fact to find the angle at the top crossing, above the line and to the left of the transversal. The two angles add up to 180°, so subtract the known angle from 180°.the 62° angle = 62°
180 − 62 = 118° - Find which angle fact links the angle at the top crossing, above the line and to the left of the transversal and f. The two angles are in the same position at the two crossing points: an F shape (parallel lines).They make an F shape, so the fact is: Corresponding angles are equal.
- Use the fact to find f. The two angles are equal, so the new angle is the same size as the known one.the angle at the top crossing, above the line and to the left of the transversal = 118°
f = 118°
Check. Does the answer look right on the diagram? The angle marked f looks obtuse (between 90° and 180°), and 118° is obtuse (between 90° and 180°). ✓
Watch out. Alternate (Z), corresponding (F) and co-interior (C) angles only work between parallel lines. Check the arrows on the lines first.
Key words. Parallel lines always stay the same distance apart and never meet (they are marked with arrows). A transversal is a line that crosses them. It makes alternate angles (a Z shape), corresponding angles (an F shape) and co-interior angles (a C or U shape).
Method. Use one angle fact at a time, starting from the angle we are given, and write down the reason for each step.
- Look at the diagram. The arrows on the two lines show that they are parallel, so the parallel-line angle facts can be used. We know one angle and want another.Known: the 62° angle = 62°
Wanted: g
(this needs 2 steps) - Find which angle fact links the 62° angle and the angle at the top crossing, below the line and to the left of the transversal. The two angles are opposite each other where two straight lines cross (an X shape).They make an X shape, so the fact is: Vertically opposite angles are equal.
- Use the fact to find the angle at the top crossing, below the line and to the left of the transversal. The two angles are equal, so the new angle is the same size as the known one.the 62° angle = 62°
the angle at the top crossing, below the line and to the left of the transversal = 62° - Find which angle fact links the angle at the top crossing, below the line and to the left of the transversal and g. The two angles are in the same position at the two crossing points: an F shape (parallel lines).They make an F shape, so the fact is: Corresponding angles are equal.
- Use the fact to find g. The two angles are equal, so the new angle is the same size as the known one.the angle at the top crossing, below the line and to the left of the transversal = 62°
g = 62°
Check. Does the answer look right on the diagram? The angle marked g looks acute (less than 90°), and 62° is acute (less than 90°). ✓
Watch out. Alternate (Z), corresponding (F) and co-interior (C) angles only work between parallel lines. Check the arrows on the lines first.
Key words. Parallel lines always stay the same distance apart and never meet (they are marked with arrows). A transversal is a line that crosses them. It makes alternate angles (a Z shape), corresponding angles (an F shape) and co-interior angles (a C or U shape).
Method. Use one angle fact at a time, starting from the angle we are given, and write down the reason for each step.
- Look at the diagram. The arrows on the two lines show that they are parallel, so the parallel-line angle facts can be used. We know one angle and want another.Known: the 62° angle = 62°
Wanted: h
(this needs 2 steps) - Find which angle fact links the 62° angle and the angle at the top crossing, below the line and to the right of the transversal. The two angles sit side by side on one straight line, so together they make a half turn.They make a straight line, so the fact is: Angles on a straight line add up to 180°.
- Use the fact to find the angle at the top crossing, below the line and to the right of the transversal. The two angles add up to 180°, so subtract the known angle from 180°.the 62° angle = 62°
180 − 62 = 118° - Find which angle fact links the angle at the top crossing, below the line and to the right of the transversal and h. The two angles are in the same position at the two crossing points: an F shape (parallel lines).They make an F shape, so the fact is: Corresponding angles are equal.
- Use the fact to find h. The two angles are equal, so the new angle is the same size as the known one.the angle at the top crossing, below the line and to the right of the transversal = 118°
h = 118°
Check. Does the answer look right on the diagram? The angle marked h looks obtuse (between 90° and 180°), and 118° is obtuse (between 90° and 180°). ✓
Watch out. Alternate (Z), corresponding (F) and co-interior (C) angles only work between parallel lines. Check the arrows on the lines first.
Method. Recall the three facts for parallel lines.
- Recall the fact. Alternate and corresponding angles are equal; co-interior angles add up to 180°.Co-interior angles add up to 180°.
- Compare. Co-interior angles are equal only when both are 90°.The statement is false in general.
Check. For 70° and 110°: they add to 180 but are not equal. ✓
Method. Recall when the facts work.
- Recall the condition. Alternate angles are equal only if the two lines are parallel.Alternate angles are equal. (parallel lines)
- Decide. If the lines are not parallel, the angles are different.False
Check. Draw two lines that meet far away: the alternate angles are not equal. ✓
Method. Identify the correct pair of angles by their shape.
- Look at the shape made by the two angles. The 70° angle and x are between the parallel lines on opposite sides of the transversal: a Z shape.They are alternate angles.
- Spot Ali’s mistake. Ali used the co-interior rule (180°) but these are not co-interior angles.wrong fact
- Use the correct fact. Alternate angles are equal.x = 70°
Check. On the diagram both angles look acute, so they are equal, not 110°. ✓
Watch out. Choose the fact by the shape (Z, F or C).
Key words. Parallel lines always stay the same distance apart and never meet (they are marked with arrows). A transversal is a line that crosses them. It makes alternate angles (a Z shape), corresponding angles (an F shape) and co-interior angles (a C or U shape).
Method. Decide which angle fact connects the two angles, form an equation, solve it, then substitute to find the angles.
- Say how the two angles are related. The two angles are between the parallel lines on opposite sides of the transversal: a Z shape (parallel lines).They are alternate angles, so they are equal. Reason: Alternate angles are equal.
- Write the equation. The two expressions are equal (or the angles add to the right total), so we can write an equation and solve it.2x + 30 = x + 40
- Collect the x terms on one side. Subtract x from both sides, so that there is only one x term left.x + 30 = 40
- Collect the numbers on the other side. Subtract 30 from both sides.x = 10
- Read off the value of x. x = 10
- Substitute x = 10 to find the first angle. Now we know the value of x, put it into the expression for the angle.2x + 30 = 2 × 10 + 30 = 50°
- Substitute x = 10 to find the second angle. Now we know the value of x, put it into the expression for the angle.x + 40 = 1 × 10 + 40 = 50°
Check. Check: the two angles are 50° and 50°; they are equal. ✓
Watch out. Substitute back into the expressions to check. Then make sure you answer what was asked: the value of x, or the size of an angle.
Key words. Parallel lines always stay the same distance apart and never meet (they are marked with arrows). A transversal is a line that crosses them. It makes alternate angles (a Z shape), corresponding angles (an F shape) and co-interior angles (a C or U shape).
Method. Decide which angle fact connects the two angles, form an equation, solve it, then substitute to find the angles.
- Say how the two angles are related. The two angles are in the same position at the two crossing points: an F shape (parallel lines).They are corresponding angles, so they are equal. Reason: Corresponding angles are equal.
- Write the equation. The two expressions are equal (or the angles add to the right total), so we can write an equation and solve it.2x + 60 = 3x + 40
- Collect the x terms on one side. Subtract 2x from both sides (the smaller x term), which keeps the number in front of x positive.60 = x + 40
- Collect the numbers on the other side. Subtract 40 from both sides.20 = x
- Read off the value of x. x = 20
- Substitute x = 20 to find the first angle. Now we know the value of x, put it into the expression for the angle.2x + 60 = 2 × 20 + 60 = 100°
- Substitute x = 20 to find the second angle. Now we know the value of x, put it into the expression for the angle.3x + 40 = 3 × 20 + 40 = 100°
Check. Check: the two angles are 100° and 100°; they are equal. ✓
Watch out. Substitute back into the expressions to check. Then make sure you answer what was asked: the value of x, or the size of an angle.
Method. Use the angle facts one at a time, starting from the 90° angle.
- Use angles on a straight line. Angles on a straight line add up to 180°.The angle next to 90° is 180 − 90 = 90°
- Use vertically opposite angles. Vertically opposite angles are equal.The angle opposite a 90° angle is 90°
- Use corresponding angles. Corresponding angles are equal.The angles at the other crossing are copies, so they are 90° as well
- State the conclusion. Every angle is reached by one of these facts.All eight angles are 90°: the transversal is perpendicular to both lines.
Check. If the transversal is perpendicular to one of the lines, it is perpendicular to the other too. ✓
Exercise 17E Parallel lines: multi-step angle problems and reasons
Non-calculatorKey words. Parallel lines always stay the same distance apart and never meet (they are marked with arrows). A transversal is a line that crosses them. It makes alternate angles (a Z shape), corresponding angles (an F shape) and co-interior angles (a C or U shape).
Method. Use one angle fact at a time, starting from the angle we are given, and write down the reason for each step.
- Look at the diagram. The arrows on the two lines show that they are parallel, so the parallel-line angle facts can be used. We know one angle and want another.Known: the 128° angle = 128°
Wanted: x
(this needs 2 steps) - Find which angle fact links the 128° angle and the angle at the top crossing, below the line and to the left of the transversal. The two angles sit side by side on one straight line, so together they make a half turn.They make a straight line, so the fact is: Angles on a straight line add up to 180°.
- Use the fact to find the angle at the top crossing, below the line and to the left of the transversal. The two angles add up to 180°, so subtract the known angle from 180°.the 128° angle = 128°
180 − 128 = 52° - Find which angle fact links the angle at the top crossing, below the line and to the left of the transversal and x. The two angles are in the same position at the two crossing points: an F shape (parallel lines).They make an F shape, so the fact is: Corresponding angles are equal.
- Use the fact to find x. The two angles are equal, so the new angle is the same size as the known one.the angle at the top crossing, below the line and to the left of the transversal = 52°
x = 52°
Check. Does the answer look right on the diagram? The angle marked x looks acute (less than 90°), and 52° is acute (less than 90°). ✓
Watch out. Alternate (Z), corresponding (F) and co-interior (C) angles only work between parallel lines. Check the arrows on the lines first.
Key words. Parallel lines always stay the same distance apart and never meet (they are marked with arrows). A transversal is a line that crosses them. It makes alternate angles (a Z shape), corresponding angles (an F shape) and co-interior angles (a C or U shape).
Method. Use one angle fact at a time, starting from the angle we are given, and write down the reason for each step.
- Look at the diagram. The arrows on the two lines show that they are parallel, so the parallel-line angle facts can be used. We know one angle and want another.Known: the 62° angle = 62°
Wanted: y
(this needs 2 steps) - Find which angle fact links the 62° angle and the angle at the bottom crossing, above the line and to the left of the transversal. The two angles are opposite each other where two straight lines cross (an X shape).They make an X shape, so the fact is: Vertically opposite angles are equal.
- Use the fact to find the angle at the bottom crossing, above the line and to the left of the transversal. The two angles are equal, so the new angle is the same size as the known one.the 62° angle = 62°
the angle at the bottom crossing, above the line and to the left of the transversal = 62° - Find which angle fact links the angle at the bottom crossing, above the line and to the left of the transversal and y. The two angles are in the same position at the two crossing points: an F shape (parallel lines).They make an F shape, so the fact is: Corresponding angles are equal.
- Use the fact to find y. The two angles are equal, so the new angle is the same size as the known one.the angle at the bottom crossing, above the line and to the left of the transversal = 62°
y = 62°
Check. Does the answer look right on the diagram? The angle marked y looks acute (less than 90°), and 62° is acute (less than 90°). ✓
Watch out. Alternate (Z), corresponding (F) and co-interior (C) angles only work between parallel lines. Check the arrows on the lines first.
Key words. Parallel lines always stay the same distance apart and never meet (they are marked with arrows). A transversal is a line that crosses them. It makes alternate angles (a Z shape), corresponding angles (an F shape) and co-interior angles (a C or U shape).
Method. PAQ is parallel to BC, so angles PAB and ABC are alternate angles (a Z shape).
- Spot the pair. The angle at B and the angle at A are between the parallel lines on opposite sides of AB.angle PAB and angle ABC form a Z shape
- Use the fact. Alternate angles are equal.angle PAB = angle ABC = 55°
Check. Both angles look acute and the same size on the diagram. ✓
Method. The angles PAB, BAC and QAC all lie on the straight line PAQ, so they add up to 180°.
- Write the sum. Angles on a straight line add up to 180°.55 + angle BAC + 65 = 180
- Add and subtract. 55 + 65 = 120.180 − 120 = 60°
Check. The angles of triangle ABC: 55 + 60 + 65 = 180°. ✓
Method. Draw an extra line through P that is parallel to the other two lines, then use alternate angles twice.
- Draw the extra line. This line through P is parallel to the other two, so we can use the parallel-line facts for it.a third parallel line through P
- Find the upper part. The upper part of angle x is alternate to the 35° angle at A.upper part = 35° (alternate angles are equal)
- Find the lower part. The lower part of angle x is alternate to the 48° angle at C.lower part = 48° (alternate angles are equal)
- Add the two parts. Together they make angle APC.35 + 48 = 83°
Check. Test the pattern: the angle at P should be bigger than each of the two outside angles, and 83 is bigger than 35 and bigger than 48. ✓
Watch out. Drawing a parallel line through the point in the middle is a very useful trick.
Key words. Parallel lines always stay the same distance apart and never meet (they are marked with arrows). A transversal is a line that crosses them. It makes alternate angles (a Z shape), corresponding angles (an F shape) and co-interior angles (a C or U shape).
Method. Use one angle fact at a time, starting from the angle we are given, and write down the reason for each step.
- Look at the diagram. The arrows on the two lines show that they are parallel, so the parallel-line angle facts can be used. We know one angle and want another.Known: the 130° angle = 130°
Wanted: x
(this needs 2 steps) - Find which angle fact links the 130° angle and the angle at the top crossing, below the line and to the right of the transversal. The two angles sit side by side on one straight line, so together they make a half turn.They make a straight line, so the fact is: Angles on a straight line add up to 180°.
- Use the fact to find the angle at the top crossing, below the line and to the right of the transversal. The two angles add up to 180°, so subtract the known angle from 180°.the 130° angle = 130°
180 − 130 = 50° - Find which angle fact links the angle at the top crossing, below the line and to the right of the transversal and x. The two angles are in the same position at the two crossing points: an F shape (parallel lines).They make an F shape, so the fact is: Corresponding angles are equal.
- Use the fact to find x. The two angles are equal, so the new angle is the same size as the known one.the angle at the top crossing, below the line and to the right of the transversal = 50°
x = 50°
Check. Does the answer look right on the diagram? The angle marked x looks acute (less than 90°), and 50° is acute (less than 90°). ✓
Watch out. Alternate (Z), corresponding (F) and co-interior (C) angles only work between parallel lines. Check the arrows on the lines first.
Method. Add the two alternate angles at P (use a third parallel line through P).
- Use alternate angles for the top part. Alternate angles are equal.angle between PA and the parallel line = 40°
- Use alternate angles for the bottom part. Alternate angles are equal.angle between PC and the parallel line = 55°
- Add. Angle APC is made of the two parts.40 + 55 = 95°
Check. This agrees with the pattern: the middle angle equals the sum of the two outside angles. ✓