Angles Made by Parallel Lines and a Transversal

Two lines in a plane that stay the same distance apart forever, never meeting, are parallel. Draw a third line straight across both of them and you get eight angles, four at each crossing. Those eight angles take at most two different sizes, so once you know one of them you can work out all the rest.

Problem

In the diagram above, a transversal cuts two parallel lines (the matching arrowheads mark them parallel). The

6 5^{\circ}

angle and the marked angle sit in the same position at their two crossings, so they are corresponding angles. Find the marked angle. Give the number of degrees.

Show a hint

Corresponding angles are the pair that sit in the same spot at each crossing, like top-right at both. When the lines are parallel, that pair is always equal.
So the marked angle has the same measure as the $6 5^{\circ}$ angle. No adding or subtracting needed.

Show the full solution

The

6 5^{\circ}

angle and the marked angle are corresponding angles, matching positions at the two crossings. Across parallel lines, corresponding angles are equal, so the marked angle is

65

Problem

In the diagram above, the

7 2^{\circ}

angle and the marked angle are alternate interior angles, sitting between the parallel lines on opposite sides of the transversal. Find the marked angle. Give the number of degrees.

Show a hint

The two angles are alternate interior angles, and alternate interior angles formed by a transversal across parallel lines are always equal.
So the marked angle matches the $7 2^{\circ}$ angle. Just read off that value.

Show the full solution

Alternate interior angles across parallel lines are equal, so the marked angle matches the given one. That is

72

degrees.

Problem

In the diagram above, two parallel guide rails are crossed by a straight brace. The brace makes a

6 2^{\circ}

angle with the top rail, and the marked angle sits in the same position where the brace meets the bottom rail. Find the marked angle. Give the number of degrees.

Show a hint

The brace is a transversal cutting the two parallel rails. The $6 2^{\circ}$ angle and the marked angle are in matching positions, one at each rail, so they are corresponding angles.
Corresponding angles at parallel lines are equal, so the marked angle has the same measure as the given one.

Show the full solution

The two rails are parallel and the brace is the transversal. The

6 2^{\circ}

angle and the marked angle sit in the same position at each rail, so they are corresponding angles, which are equal. That gives

∠ = 6 2^{\circ} = 62

Problem

In the diagram above, the

4 8^{\circ}

angle and the marked angle are alternate exterior angles. They lie outside the parallel lines, on opposite sides of the transversal. Find the marked angle. Give the number of degrees.

Show a hint

Alternate exterior angles sit outside the two parallel lines, on opposite sides of the transversal. When the lines are parallel, what is true about a pair of alternate exterior angles?
Alternate exterior angles are equal. So the marked angle matches the $4 8^{\circ}$ angle exactly.

Show the full solution

Alternate exterior angles across parallel lines are equal, so the marked angle matches the given one. That is

48

degrees.

Problem

In the diagram above, the two marked angles sit side by side at the same crossing on the top line. One is

6 5^{\circ}

and the other is marked with a question mark. Find the marked angle. Give the number of degrees.

Show a hint

The two angles sit next to each other on a straight line, so together they form a linear pair.
Angles in a linear pair are supplementary, meaning their measures add to $18 0^{\circ}$ .

Show the full solution

The

6 5^{\circ}

angle and the marked angle form a linear pair on a straight line, so they are supplementary.

180 - 65 = 115

Problem

In the diagram above, the

11 0^{\circ}

angle and the marked angle are co-interior angles, sitting between the parallel lines on the same side of the transversal. Find the marked angle. Give the number of degrees.

Show a hint

Co-interior angles (the ones trapped between the parallel lines on the same side of the transversal) are supplementary, so the two angles add to $18 0^{\circ}$ .
Subtract the given angle from $18 0^{\circ}$ to get the marked one.

Show the full solution

The

11 0^{\circ}

angle and the marked angle are co-interior, so they add to

18 0^{\circ}

180 - 110 = 70

Co-interior is the interior pair that is supplementary instead of equal, so subtract rather than copy the angle across.

Problem

In the parallelogram

A B C D

above,

∠ A = 7 3^{\circ}

. Find

∠ C

, the angle opposite it. Give the number of degrees.

Show a hint

$∠ A$ and $∠ C$ sit across from each other, so they are opposite angles of the parallelogram.
Opposite angles of a parallelogram are equal, so $∠ C$ matches $∠ A$ .

Show the full solution

∠ A

and

∠ C

are opposite angles of the parallelogram, and opposite angles of a parallelogram are equal. So

∠ C = ∠ A = 73

Now put these pairs to work as equations. When the two angles are given as expressions in $x$ , the same two moves are all you need. Set the expressions equal for corresponding or alternate angles, since those pairs are equal. Set their sum to $180$ for co-interior angles, since those add to a straight angle. Either way you get one equation, and solving it for $x$ unlocks every angle in the figure.

Problem

In the diagram above, a transversal cuts two parallel lines. The

13 0^{\circ}

angle is at the top crossing and the marked angle is at the bottom crossing. Find the marked angle. Give the number of degrees.

Show a hint

The $13 0^{\circ}$ angle sits in the same position at the top crossing as one angle at the bottom crossing. Corresponding angles are equal, so that bottom angle is also $13 0^{\circ}$ .
That $13 0^{\circ}$ angle and the marked angle sit on a straight line together, so they form a linear pair and add to $18 0^{\circ}$ .

Show the full solution

The

13 0^{\circ}

angle and the angle in the matching spot at the bottom crossing are corresponding angles, so that angle is also

13 0^{\circ}

. The marked angle forms a linear pair with it, so the two add to

18 0^{\circ}

180 - 130 = 50

Problem

In the diagram above, a transversal crosses two parallel lines and meets the top line at a right angle, shown by the small square. Find the marked angle where the transversal crosses the bottom line. Give the number of degrees.

Show a hint

The transversal is perpendicular to the top line. Think about what that forces at the bottom line, since the two lines are parallel.
A line perpendicular to one of two parallel lines is perpendicular to the other one as well, so the marked angle is a right angle.

Show the full solution

The transversal meets the top line at a right angle. A line perpendicular to one of two parallel lines is perpendicular to the other one too, so the marked angle is

90

Problem

In the diagram above, a transversal cuts two parallel lines, and the two marked angles are corresponding angles labeled

2 x^{\circ}

and

(x + 40)^{\circ}

. Find

x

. Give the number of degrees (the value of

x

Show a hint

The two marked angles are corresponding angles, so they have equal measure. Set the two expressions equal to each other.
You get $2 x = x + 40$ . Subtract $x$ from both sides to get $x$ by itself.

Show the full solution

The two marked angles are corresponding angles, so they are equal. That gives

2 x = x + 40.

Subtracting

x

from both sides leaves

x = 40

, so

40

Problem

In the diagram above, a transversal cuts two parallel lines. The two co-interior angles are labeled

3 x^{\circ}

and

2 x^{\circ}

. Find

x

. Give the number of degrees (the value of

x

Show a hint

The two marked angles are co-interior (same-side interior), so they are supplementary and add to $18 0^{\circ}$ . Set $3 x + 2 x = 180$ .
Combine the like terms into $5 x = 180$ , then divide both sides by 5.

Show the full solution

The two marked angles are co-interior, so they are supplementary and add to

18 0^{\circ}

. That gives

3 x + 2 x = 180

5 x = 180

, and dividing by 5 gives

x = 36

Problem

In the diagram above, a transversal crosses two lines. The angles

3 x^{\circ}

and

7 5^{\circ}

sit in matching positions, so they are corresponding angles. For the two lines to be parallel, these corresponding angles must be equal. Find

x

. Give the number of degrees (the value of

x

Show a hint

Corresponding angles are only equal when the lines are parallel. So for these lines to be parallel, $3 x^{\circ}$ has to equal $7 5^{\circ}$ .
Set the two expressions equal and solve. What times 3 gives 75?

Show the full solution

The two marked angles are corresponding angles, and parallel lines force corresponding angles to be equal. So

3 x = 75.

Divide both sides by 3 to get

x = 25

Problem

In the diagram above, a transversal cuts two parallel lines. One angle at the top crossing measures

11 6^{\circ}

, and the acute angle marked at the bottom crossing is unknown. Find the marked angle. Give the number of degrees.

Show a hint

The $11 6^{\circ}$ angle at the top corresponds to the angle in the same position at the bottom crossing, so that angle is also $11 6^{\circ}$ .
That $11 6^{\circ}$ angle and the marked acute angle sit on a straight line together, so they form a linear pair and add to $18 0^{\circ}$ .

Show the full solution

The

11 6^{\circ}

angle at the top crossing corresponds to the angle in the same position at the bottom crossing, so that bottom angle is also

11 6^{\circ}

. The marked angle forms a linear pair with it.

180 - 116 = 64

Practice these ideas

Practice

In the diagram above, a transversal cuts two parallel lines. The

5 8^{\circ}

angle and the marked angle are corresponding angles. Find the marked angle. Give the number of degrees.

Show the solution

The

5 8^{\circ}

angle and the marked angle are corresponding angles, and corresponding angles are equal. So the marked angle is

58

degrees.

Practice

In the diagram above, a transversal cuts two parallel lines. The

5 1^{\circ}

angle and the marked angle are alternate interior angles. Find the marked angle. Give the number of degrees.

Show the solution

The marked angle and the

5 1^{\circ}

angle are alternate interior angles, and alternate interior angles are equal when the lines are parallel. So the marked angle is

51

degrees.

Practice

In the diagram above, a transversal cuts two parallel lines. The

11 2^{\circ}

angle and the marked angle are co-interior angles (same-side interior). Find the marked angle. Give the number of degrees.

Show the solution

Co-interior angles are supplementary, so they add to

18 0^{\circ}

180 - 112 = 68

Practice

In the diagram above, the

4 3^{\circ}

angle and the marked angle sit side by side at the same crossing and form a linear pair. Find the marked angle. Give the number of degrees.

Show the solution

The two angles form a linear pair, so they are supplementary.

180 - 43 = 137

Practice

In the diagram above, a transversal cuts two parallel lines. The

3 9^{\circ}

angle and the marked angle are alternate exterior angles. Find the marked angle. Give the number of degrees.

Show the solution

Alternate exterior angles are equal when the lines are parallel, so the marked angle matches the given one.

∠ = 3 9^{\circ} = 39

Practice

In the diagram above, a transversal cuts two parallel lines. The

12 5^{\circ}

angle and the marked angle are corresponding angles. Find the marked angle. Give the number of degrees.

Show the solution

Corresponding angles are equal, so the marked angle matches the given one.

∠ = 12 5^{\circ} = 125

Practice

In the diagram above, a transversal cuts two parallel lines. The two marked angles,

2 x^{\circ}

and

(x + 30)^{\circ}

, are corresponding angles. Find the value of

x

. Give the value of

x

Show the solution

Corresponding angles are equal, so the two expressions match.

2 x = x + 30

Subtract

x

from both sides.

x = 30

Practice

In the diagram above, a transversal cuts two parallel lines, and the two marked angles are

5 x^{\circ}

and

4 x^{\circ}

. They are co-interior angles, so they sit on the same side of the transversal between the two parallel lines. Find

x

. Give the value of

x

Show the solution

The two angles are co-interior, so they add to

18 0^{\circ}

. That gives

5 x + 4 x = 180,

9 x = 180

and

x = 20

Practice

In the parallelogram above,

∠ A = 10 8^{\circ}

. The angle you want,

∠ B

, is consecutive to

∠ A

. Find

∠ B

. Give the number of degrees.

Show the solution

Consecutive angles of a parallelogram are supplementary, so

∠ A

and

∠ B

add to

18 0^{\circ}

∠ B = 180 - 108 = 72

Practice

In the diagram above, a transversal cuts two lines, and the

4 x^{\circ}

angle and the

10 0^{\circ}

angle are corresponding angles. For the two lines to be parallel, what must

x

be? Give the value of

x

Show the solution

The lines are parallel exactly when the corresponding angles are equal, so

4 x^{\circ}

must equal

10 0^{\circ}

. Set them equal and solve.

4 x = 100 \Rightarrow x = 25

Practice

In the diagram above, the transversal meets the top parallel line at a right angle. Since the transversal is perpendicular to one of the parallel lines, it is perpendicular to the other as well. Find the marked angle where the transversal crosses the bottom line. Give the number of degrees.

Show the solution

The transversal hits the top line at

9 0^{\circ}

. Because the two lines are parallel, a transversal perpendicular to one is perpendicular to the other, so it meets the bottom line at a right angle as well.

∠ = 90^{\circ}

Practice

In the diagram above, a transversal cuts two parallel lines. The

5 3^{\circ}

angle and the marked angle are corresponding angles. Find the marked angle. Give the number of degrees.

Show the solution

Corresponding angles are equal, so the marked angle matches the

5 3^{\circ}

angle.

53

Practice

In the diagram above, a transversal cuts two parallel lines. The

11 4^{\circ}

angle at the top crossing and the marked angle at the bottom crossing are on the same side, and the marked angle forms a linear pair with the angle that corresponds to the

11 4^{\circ}

. Find the marked angle. Give the number of degrees.

Show the solution

Corresponding angles are equal, so the top

11 4^{\circ}

matches a

11 4^{\circ}

angle at the bottom crossing. The marked angle forms a linear pair with it.

180 - 114 = 66