Vertically Opposite Angles

When two straight lines intersect, they form four angles around the point of intersection. The two pairs of vertically opposite angles , the ones diagonally across from each other , are always equal. This compact fact, proved in three lines from the linear pair axiom, is one of the most-used results in geometry.

Definitions

When two lines intersect at a point $O$ , four angles are formed. The pair of angles that are diagonally opposite at the intersection (sharing only the vertex, not an arm) are called vertically opposite angles. There are two such pairs.

The theorem and its proof

Theorem (Vertically opposite angles). If two straight lines intersect, then the vertically opposite angles are equal.

Proof. Let lines $\overleftrightarrow{AB}$ and $\overleftrightarrow{CD}$ intersect at $O$ , forming four angles labelled $\angle 1, \angle 2, \angle 3, \angle 4$ in order around $O$ , so that $\angle 1$ and $\angle 3$ are vertically opposite (similarly $\angle 2$ and $\angle 4$ ).

$\angle 1$ and $\angle 2$ form a linear pair on line $\overleftrightarrow{AB}$ : $\angle 1 + \angle 2 = 180^\circ$ .

$\angle 2$ and $\angle 3$ form a linear pair on line $\overleftrightarrow{CD}$ : $\angle 2 + \angle 3 = 180^\circ$ .

Subtract: $\angle 1 - \angle 3 = 0$ , so $\angle 1 = \angle 3$ . Similarly $\angle 2 = \angle 4$ . Q.E.D.

The proof uses only the linear pair axiom and a Common Notion (if equals are subtracted from equals, the remainders are equal). That is the entire content.

Using the theorem

Once you know vertically opposite angles are equal, you can read pairs off any crossing-lines figure at a glance.

Example. Two lines cross at $O$ making an angle of $35^\circ$ at one of the four positions. Then:

The vertically opposite angle is also $35^\circ$ .
The two adjacent angles (linear pairs with $35^\circ$ ) are each $180 - 35 = 145^\circ$ .
The remaining angle is $145^\circ$ (vertically opposite to $145^\circ$ ).

So all four angles are determined by any one of them.

Combining with the linear pair

The two ideas team up beautifully. If you know one of the four angles formed by crossing lines, the other three are forced. If two of the angles are given as algebraic expressions, you usually set up one equation (either equality of vertically opposite or sum of linear pair = $180^\circ$ ) and solve.

Example. Two lines cross. One angle is $(3x + 10)^\circ$ and its vertically opposite angle is $(2x + 30)^\circ$ . Find $x$ .

Vertically opposite are equal: $3x + 10 = 2x + 30 \Rightarrow x = 20$ .

Example. Two lines cross. One angle is $(4x)^\circ$ and an adjacent angle is $(2x + 30)^\circ$ . Find $x$ .

Adjacent (linear pair) sum to $180^\circ$ : $4x + 2x + 30 = 180 \Rightarrow 6x = 150 \Rightarrow x = 25$ .

A useful corollary

Corollary. If two lines intersect at $O$ , then the four rays $\overrightarrow{OA}, \overrightarrow{OB}, \overrightarrow{OC}, \overrightarrow{OD}$ make four angles whose sum is $360^\circ$ .

Proof. Each pair of adjacent angles is a linear pair summing to $180^\circ$ . There are two such pairs, but together they cover all four angles. So total is $360^\circ$ . (Equivalently, the four angles together form a complete turn around $O$ .)

Worked examples

Example 1. Two lines cross at $P$ . One angle is $50^\circ$ . State all four angles.

$50^\circ, 130^\circ, 50^\circ, 130^\circ$ in order. (Vertically opposite pair: $50^\circ$ each; linear-pair partner: $130^\circ$ each.)

Example 2. Two lines cross at $O$ . One angle is $4x^\circ$ and the vertically opposite angle is $(x + 60)^\circ$ . Find $x$ and the four angles.

$4x = x + 60 \Rightarrow x = 20$ . Angle: $80^\circ$ . All four: $80^\circ, 100^\circ, 80^\circ, 100^\circ$ .

Example 3. Two intersecting lines make four angles whose ratio is $1 : 4 : 1 : 4$ . Find them.

Adjacent angles are linear pairs, so $1k + 4k = 180^\circ \Rightarrow 5k = 180 \Rightarrow k = 36$ . Angles: $36^\circ, 144^\circ, 36^\circ, 144^\circ$ .

Example 4. Three lines meet at $O$ forming six angles. If three of them are $30^\circ, 50^\circ, 100^\circ$ in order, find the other three.

The three given angles plus the three opposite ones cover the full $360^\circ$ . By the vertically opposite property, the opposite angles match: the remaining three are also $30^\circ, 50^\circ, 100^\circ$ . Sum: $30 + 50 + 100 = 180$ on each side; total $360$ . Checks out.

Example 5. Two lines $\ell$ and $m$ cross at $P$ . An angle at $P$ is $5x^\circ$ and an adjacent angle is $(2x + 5)^\circ$ . Find $x$ and the four angles.

$5x + 2x + 5 = 180 \Rightarrow 7x = 175 \Rightarrow x = 25$ . Angles: $125^\circ, 55^\circ, 125^\circ, 55^\circ$ .

Try it yourself

State the vertically-opposite-angles theorem.
Sketch two crossing lines making an angle of $40^\circ$ and label all four angles.
Two lines cross. One angle is $(3x + 20)^\circ$ and its vertically opposite is $(2x + 35)^\circ$ . Find $x$ .
Two lines cross. One angle is $70^\circ$ . Find the other three.
Two lines cross at $O$ making four angles in the ratio $2 : 3 : 2 : 3$ . Find them.
Two intersecting lines form a right angle at the intersection. What are the other three angles?
Two lines cross at $A$ . If one angle is $9x^\circ$ and an adjacent angle is $(x + 10)^\circ$ , find $x$ .
Prove the vertically-opposite-angles theorem in your own words.
Two lines $\ell$ and $m$ cross at $O$ . If $\ell$ is reflected onto $m$ by a half-turn about $O$ , what happens to vertically opposite angles?
State whether true or false: "two adjacent angles can be vertically opposite". Justify.

Pitfalls / Insight

Vertically opposite means opposite at the vertex , sharing the vertex but no arm. Beginners sometimes call adjacent angles "vertical" by mistake.
The theorem requires two straight lines. If the figure has a bent line, vertically opposite reasoning may not apply.
Always check linear pair separately. Vertically opposite gives equality; linear pair gives the sum $180^\circ$ . Both relations live at the same intersection.

Insight. Vertically opposite angles are equal because of the linear pair axiom. Once you internalise this, every crossing-lines figure becomes a four-angle pattern $a, 180-a, a, 180-a$ , and finding any one angle determines all four.