Types of Angle Pairs

Two angles often come as a pair that has a name of its own. Some pairs are named for the way the angles are placed (adjacent, vertical), others for the sum of their measures (complementary, supplementary). When a line crosses two parallel lines, eight angles appear, and their pairs follow simple rules that are used throughout geometry.

On this page: Adjacent angles · Complementary, supplementary · Vertical angles · Parallel lines and a transversal · When are lines parallel? · Solved examples · Common mistakes · Practice

Adjacent Angles

Adjacent angles are two angles that have the same vertex and a common side and do not overlap: they lie on different sides of the common side.

In the picture $\angle 1$ and $\angle 2$ are adjacent: they share the vertex $O$ and the side $OB.$ Together they make the larger angle $AOC,$ so ${\angle AOC=\angle 1+\angle 2}.$

Complementary and Supplementary Angles

The angles $35^\circ$ and $55^\circ$ are complementary, because ${35^\circ+55^\circ=90^\circ}.$ They do not have to be adjacent: only the sum matters. Each of the two angles is called the complement of the other. In the same way the angles $60^\circ$ and $120^\circ$ are supplementary, and each of them is the supplement of the other.

For an angle $\alpha$ the complement is ${90^\circ-\alpha}$ and the supplement is ${180^\circ-\alpha}.$ Only an acute angle has a complement.

Linear pair

A linear pair is a pair of adjacent angles whose other two sides form a straight line. The angles of a linear pair are supplementary: ${\angle 1+\angle 2=180^\circ}.$

The reason is simple: together the two angles make a straight angle, and a straight angle is $180^\circ.$

Example 1. Find the complement and the supplement of $37^\circ.$

Solution. The complement is ${90^\circ-37^\circ=53^\circ}$ and the supplement is ${180^\circ-37^\circ=143^\circ}.$

Example 2. Two angles form a linear pair, and one of them is four times the other. Find the angles.

Solution. Let the smaller angle be $x.$ Then the larger one is $4x$ and ${x+4x=180^\circ},$ so ${5x=180^\circ}$ and ${x=36^\circ}.$ The angles are $36^\circ$ and ${4\cdot36^\circ=144^\circ}.$

Vertical Angles

Two lines that intersect form four angles. Two of them that lie opposite each other, such as $\angle 1$ and $\angle 3,$ are called vertical angles (or vertically opposite angles): they are not adjacent, and the sides of one angle are the extensions of the sides of the other beyond the vertex. The second pair is $\angle 2$ and $\angle 4.$

Vertical angles are equal (US textbooks say congruent): ${\angle 1=\angle 3}$ and ${\angle 2=\angle 4}.$

Proof

The angles $\angle 1$ and $\angle 2$ form a linear pair, so ${\angle 1=180^\circ-\angle 2}.$ The angles $\angle 3$ and $\angle 2$ also form a linear pair, so ${\angle 3=180^\circ-\angle 2}.$ The right-hand sides are the same, so ${\angle 1=\angle 3}.$ The same reasoning gives ${\angle 2=\angle 4}.$

Example 3. Two lines intersect and one of the four angles is $68^\circ.$ Find the other three.

Solution. The angle vertical to the given one is also $68^\circ.$ Each of the other two angles forms a linear pair with the given one, so it is ${180^\circ-68^\circ=112^\circ}.$ The four angles are $68^\circ,$ $112^\circ,$ $68^\circ$ and $112^\circ.$

Pair	The two angles
Adjacent	share the vertex and one side and do not overlap
Complementary	add up to $90^\circ$
Supplementary	add up to $180^\circ$
Linear pair	are adjacent and their other sides form a line; they add up to $180^\circ$
Vertical	lie opposite each other where two lines cross; they are equal

Angle Pairs Formed by Parallel Lines Cut by a Transversal

A line that crosses two other lines at two different points is called a transversal. In the picture the transversal $t$ crosses the lines $m$ and $n$ and forms eight angles, four at each line.

The angles between the two lines, $\angle 3,$ $\angle 4,$ $\angle 5$ and $\angle 6,$ are interior angles. The angles outside the two lines, $\angle 1,$ $\angle 2,$ $\angle 7$ and $\angle 8,$ are exterior angles. The pairs of one angle at $m$ and one angle at $n$ that are used in geometry have these names:

Name	Where the two angles are	Pairs in the picture
Corresponding angles	on the same side of the transversal and in the same position at their lines (both above their line or both below their line)	$\angle 1$ and $\angle 5,$ $\angle 2$ and $\angle 6,$ $\angle 3$ and $\angle 7,$ $\angle 4$ and $\angle 8$
Alternate interior angles	between the lines, on opposite sides of the transversal	$\angle 3$ and $\angle 6,$ $\angle 4$ and $\angle 5$
Alternate exterior angles	outside the lines, on opposite sides of the transversal	$\angle 1$ and $\angle 8,$ $\angle 2$ and $\angle 7$
Same-side interior angles (also called co-interior or consecutive interior angles)	between the lines, on the same side of the transversal	$\angle 3$ and $\angle 5,$ $\angle 4$ and $\angle 6$
Same-side exterior angles	outside the lines, on the same side of the transversal	$\angle 1$ and $\angle 7,$ $\angle 2$ and $\angle 8$

These names describe only where the angles are, so they can be used for any two lines and a transversal. The pairs become useful when the two lines are parallel. We write ${m\parallel n}$ for “the line $m$ is parallel to the line $n$”.

Why the rules are true

We accept one rule without proof: when two lines are parallel, corresponding angles are equal. (This rule is equivalent to the parallel postulate: through a point that is not on a line there is exactly one line parallel to that line.) The other rules follow from it. Let ${m\parallel n}.$

Alternate interior angles. ${\angle 3=\angle 2}$ as vertical angles and ${\angle 2=\angle 6}$ as corresponding angles, so ${\angle 3=\angle 6}.$

Alternate exterior angles. ${\angle 2=\angle 6}$ as corresponding angles and ${\angle 6=\angle 7}$ as vertical angles, so ${\angle 2=\angle 7}.$

Same-side interior angles. The angles $\angle 4$ and $\angle 2$ form a linear pair, so ${\angle 4+\angle 2=180^\circ}.$ Since ${\angle 2=\angle 6}$ as corresponding angles, we get ${\angle 4+\angle 6=180^\circ}.$

Same-side exterior angles. The angles $\angle 2$ and $\angle 4$ form a linear pair, so ${\angle 2+\angle 4=180^\circ}.$ Since ${\angle 4=\angle 8}$ as corresponding angles, we get ${\angle 2+\angle 8=180^\circ}.$

A quick rule: acute and obtuse angles

When a transversal crosses two parallel lines and is not perpendicular to them, it forms four equal acute angles and four equal obtuse angles. Any acute angle and any obtuse angle among the eight add up to $180^\circ.$

So one angle is enough to find all eight. If the transversal is perpendicular to the lines, all eight angles are right angles.

How to Show That Two Lines Are Parallel

The rules also work in the other direction, and this is the usual way to prove that two lines are parallel.

Watch the direction of the reasoning. If you know that the lines are parallel, you may conclude that corresponding angles are equal. If you know that two corresponding angles are equal, you may conclude that the lines are parallel. If the lines are not parallel, then corresponding angles are not equal, and alternate angles are not equal either.

Solved Examples

Example 4. In the picture ${m\parallel n},$ the angles are numbered as before, and $\angle 1$ (the angle marked $53^\circ$) is given. Find $\angle 8$ and $\angle 7.$

Solution. The angles $\angle 1$ and $\angle 8$ are alternate exterior angles: they are outside the parallel lines, on opposite sides of the transversal. So ${\angle 8=\angle 1=53^\circ}.$ The angles $\angle 7$ and $\angle 8$ form a linear pair, so ${\angle 7=180^\circ-53^\circ=127^\circ}.$

Example 5. Find all eight angles in the picture of Example 4.

Solution. The angle $\angle 1$ is acute, so every acute angle is $53^\circ$ and every obtuse angle is ${180^\circ-53^\circ=127^\circ}.$ In the picture the acute angles are $\angle 1,$ $\angle 4,$ $\angle 5$ and $\angle 8,$ so ${\angle 1=\angle 4=\angle 5=\angle 8=53^\circ}$ and ${\angle 2}={\angle 3}={\angle 6}={\angle 7}={127^\circ}.$

Example 6. Two parallel lines are cut by a transversal. Two alternate interior angles are ${(3x+10)^\circ}$ and ${(5x-30)^\circ}.$ Find $x$ and the angles.

Solution. The lines are parallel, so the alternate interior angles are equal: ${3x+10=5x-30}.$ Then ${2x=40}$ and ${x=20}.$ Each angle is ${3\cdot20+10=70}$ degrees.

Example 7. Two parallel lines are cut by a transversal. Two same-side interior angles are ${(2x+15)^\circ}$ and ${(3x-5)^\circ}.$ Find the angles.

Solution. The lines are parallel, so the same-side interior angles add up to $180^\circ\text{:}$ ${(2x+15)+(3x-5)}={180}.$ Then ${5x+10=180}$ and ${x=34}.$ The angles are ${2\cdot34+15=83}$ and ${3\cdot34-5=97}$ degrees. Check: ${83^\circ+97^\circ=180^\circ}.$

Example 8. A transversal crosses the lines $m$ and $n,$ and the angles are numbered as in the picture of the eight angles above. Are the lines parallel if a) ${\angle 4=115^\circ}$ and ${\angle 6=65^\circ};$ b) ${\angle 2=72^\circ}$ and ${\angle 6=78^\circ}?$

Solution.

a) The angles $\angle 4$ and $\angle 6$ are same-side interior angles and ${115^\circ+65^\circ=180^\circ},$ so the lines are parallel.

b) The angles $\angle 2$ and $\angle 6$ are corresponding angles, but they are not equal, so the lines are not parallel.

Example 9. The lines $a$ and $b$ are parallel. Find the angle $x$ in the picture.

Solution. Draw the line through $K$ parallel to $a$ and $b$ (the dashed line in the picture). It divides the angle $x$ into two parts. The upper part and the angle of $35^\circ$ at $A$ are alternate interior angles for the line $a$ and the new line (the transversal is $AK$), so the upper part is $35^\circ.$ In the same way the lower part and the angle of $50^\circ$ at $B$ are alternate interior angles for the line $b$ and the new line (the transversal is $BK$), so the lower part is $50^\circ.$ So ${x=35^\circ+50^\circ=85^\circ}.$

Example 10. Prove that the angles of a triangle add up to $180^\circ.$

Proof. Draw the line through $C$ parallel to $AB.$ At $C$ the angle to the left of the triangle and the angle $\alpha$ at $A$ are alternate interior angles (the transversal is $AC$), so they are equal. In the same way the angle to the right of the triangle at $C$ equals $\beta$ (the transversal is $BC$). The three angles at $C$ make a straight angle, so ${\alpha+\gamma+\beta=180^\circ}.$

Common Mistakes

Mistake	Wrong	Right
Using the rules without parallel lines	$\angle 2$ and $\angle 6$ are corresponding angles, so they are equal	They are equal only if the lines are parallel
Same-side interior angles	Same-side interior angles formed by parallel lines are equal	They add up to $180^\circ;$ they are equal only when both are $90^\circ$
Complementary or supplementary?	Complementary angles add up to $180^\circ$	Complementary angles add up to $90^\circ,$ supplementary angles to $180^\circ$
Adjacent does not mean supplementary	Adjacent angles always add up to $180^\circ$	Only a linear pair does: the other two sides must form a line
Which angles are vertical	Two neighboring angles at the crossing of two lines are vertical	Vertical angles lie opposite each other; neighboring angles form a linear pair
Complementary does not mean adjacent	Complementary angles must be adjacent	Only the sum matters: $35^\circ$ and $55^\circ$ are complementary wherever they are

Practice Problems

1. Find the complement and the supplement of $64^\circ.$

Answer

The complement is ${90^\circ-64^\circ=26^\circ},$ the supplement is ${180^\circ-64^\circ=116^\circ}.$

2. Two complementary angles differ by $18^\circ.$ Find them.

Answer

If the smaller angle is $x,$ then ${x+(x+18^\circ)=90^\circ},$ so ${x=36^\circ}.$ The angles are $36^\circ$ and $54^\circ.$

3. The supplement of an angle is three times its complement. Find the angle.

Answer

For the angle $x$ we have ${180^\circ-x=3(90^\circ-x)},$ so ${2x=90^\circ}$ and ${x=45^\circ}.$ Check: the supplement is $135^\circ$ and the complement is $45^\circ.$

4. Two lines intersect, and two vertical angles add up to $150^\circ.$ Find all four angles.

Answer

Vertical angles are equal, so each of them is $75^\circ.$ Each of the other two is ${180^\circ-75^\circ=105^\circ}.$

5. In the picture of the eight angles ${m\parallel n}$ and ${\angle 1=118^\circ}.$ Find the other seven angles.

Answer

${\angle 4=\angle 5=\angle 8=118^\circ}$ and ${\angle 2}={\angle 3}={\angle 6}={\angle 7}={180^\circ-118^\circ}={62^\circ}.$

6. Name each pair in the picture of the eight angles: a) $\angle 4$ and $\angle 5;$ b) $\angle 2$ and $\angle 6;$ c) $\angle 1$ and $\angle 8;$ d) $\angle 3$ and $\angle 5.$

Answer

a) Alternate interior angles; b) corresponding angles; c) alternate exterior angles; d) same-side interior angles.

7. Two parallel lines are cut by a transversal. Two corresponding angles are ${(4x-12)^\circ}$ and ${(2x+40)^\circ}.$ Find $x$ and the angles.

Answer

${4x-12=2x+40},$ so ${x=26}.$ Each angle is ${4\cdot26-12=92}$ degrees.

8. Two parallel lines are cut by a transversal. One of two same-side interior angles is $40^\circ$ larger than the other. Find the angles.

Answer

${x+(x+40^\circ)=180^\circ},$ so ${x=70^\circ}.$ The angles are $70^\circ$ and $110^\circ.$

9. Are two lines parallel if a transversal forms with them a) alternate interior angles of $57^\circ$ and $123^\circ;$ b) same-side interior angles of $57^\circ$ and $123^\circ?$

Answer

a) No: alternate interior angles would have to be equal.

b) Yes: ${57^\circ+123^\circ=180^\circ}.$

10. A picture is like the one in Example 9, but with other numbers: the lines $a$ and $b$ are parallel, the angle at $A$ is $28^\circ$ and the angle $x$ at $K$ is $75^\circ.$ Find the angle at $B.$

Answer

As in Example 9, the angle at $K$ is the sum of the angles at $A$ and at $B,$ so the angle at $B$ is ${75^\circ-28^\circ=47^\circ}.$

11. A transversal is perpendicular to one of two parallel lines. Show that it is perpendicular to the other line too.

Answer

At the first line the transversal forms an angle of $90^\circ.$ The corresponding angle at the second line is equal to it, so it is also $90^\circ.$ So the transversal is perpendicular to the second line too.

12. Two parallel lines are cut by a transversal, and one of the eight angles is three times another. Find all eight angles.

Answer

The two angles are not equal, so one is acute and the other is obtuse, and they add up to $180^\circ.$ From ${x+3x=180^\circ}$ we get ${x=45^\circ}.$ Four of the angles are $45^\circ$ and four are $135^\circ.$

See also: Classifying angles · Measuring angles · Triangles · Parallelogram