Triangle Angles, Exterior Angles and Solving for x | Grade 8 Math | LevelKite

A triangle's three angles are tied together. Once you know two of them, the third is already decided, and so is the angle you get by extending a side. In this lesson you will use that tie to find missing angles inside a triangle, to find angles just outside it, and to write and solve equations for an unknown number $x$ .

Every angle measure here is in degrees. $\angle A$ names the angle at corner $A$ ; a corner is called a vertex (plural: vertices). We write $\angle A = 52^\circ$ to mean that this angle measures $52$ degrees. A three-letter name such as $\angle ACD$ names the angle at the middle letter, $C$ , with one side running toward $A$ and the other toward $D$ .

The interior angles of a triangle are the three angles inside it, one at each vertex. A straight angle is the angle along a straight line, and it measures $180^\circ$ .

Here is the fact that ties a triangle together: in every triangle drawn on a flat surface, the three interior angle measures add to exactly $180^\circ$ , the same as one straight angle. The size and shape of the triangle do not change this total.

A familiar case: a square has four $90^\circ$ corners, and a diagonal (a segment joining opposite corners) cuts two of them exactly in half. Each half of the square is a triangle with angles of $90^\circ$ , $45^\circ$ and $45^\circ$ , and $90 + 45 + 45 = 180$ .

Because the total is fixed, the angles share it. If two interior angles together take up $150^\circ$ , only $30^\circ$ is left for the third. So when two interior angles of a triangle are stated, you can find the third: add the two you know and subtract that sum from $180^\circ$ .

Worked example

Finding a missing interior angle

In triangle $ABC$ , you are told that $\angle A = 52^\circ$ and $\angle B = 71^\circ$ . Find $\angle C$ .

Write the triangle fact as an equation. The three interior angles of every triangle add to 180°, and two of the measures are stated.
$52^\circ + 71^\circ + \angle C = 180^\circ$
Add the two known measures, so the left side has just one known number.
$52^\circ + 71^\circ = 123^\circ, \quad \text{so} \quad 123^\circ + \angle C = 180^\circ$
Subtract 123° from both sides. Taking the same amount from both sides keeps them equal, and it leaves angle C by itself.
$\angle C = 180^\circ - 123^\circ = 57^\circ$
Check. The measure is positive, and all three measures add to 180°, as the triangle fact requires.
$52^\circ + 71^\circ + 57^\circ = 180^\circ$

Answer

Angle $C$ measures $57^\circ$ .

Now step just outside the triangle. Take triangle $ABC$ and extend side $BC$ in a straight line past $C$ to a new point $D$ . Then $B$ , $C$ and $D$ lie on one straight line, with $C$ between $B$ and $D$ .

$\angle ACD$ , the angle at $C$ between side $CA$ and the extension $CD$ , is an exterior angle of the triangle.
The exterior angle sits beside the interior angle $\angle ACB$ . Together they fill the straight angle along line $BD$ , so their measures add to $180^\circ$ . Two angles whose measures add to $180^\circ$ are called supplementary.
The interior angles at the other two vertices, $\angle A$ and $\angle B$ , are the remote interior angles for this exterior angle. Remote means far away: they are the two interior angles that are not at $C$ .

Now compare two true statements about this triangle:

\angle A + \angle B + \angle ACB = 180^\circ \qquad \text{and} \qquad \angle ACD + \angle ACB = 180^\circ

Both statements add something to $\angle ACB$ to reach $180^\circ$ . In the first, that something is $\angle A + \angle B$ . In the second, it is $\angle ACD$ . Each of them equals $180^\circ - \angle ACB$ , so they equal each other: $\angle ACD = \angle A + \angle B$ .

This result is the exterior angle theorem (a theorem is a fact that has been shown to be true). In words: an exterior angle of a triangle, made by extending one side in a straight line, equals the sum of its two remote interior angles.

Key concept

Two equations that hold in every triangle

In every triangle on a flat surface, the three interior angles add to $180^\circ$ . When one side is extended in a straight line past a vertex, the exterior angle formed there equals the sum of its two remote interior angles, the interior angles at the other two vertices. In the formulas, triangle $ABC$ has side $BC$ extended past $C$ to $D$ . Each stated angle relationship gives you an equation, and an equation with one unknown lets you find the missing measure.

\angle A + \angle B + \angle ACB = 180^\circ \qquad \angle ACD = \angle A + \angle B

Worked example

Finding an exterior angle

In triangle $ABC$ , you are told that $\angle A = 48^\circ$ and $\angle B = 67^\circ$ . Side $BC$ is extended in a straight line past $C$ to point $D$ . Find the exterior angle $\angle ACD$ .

Name the parts. The exterior angle ACD is at vertex C. Its remote interior angles are the two interior angles that are not at C, which are angle A and angle B.
Use the exterior angle theorem. Because side BC is extended in a straight line, the exterior angle equals the sum of its two remote interior angles.
$\angle ACD = 48^\circ + 67^\circ = 115^\circ$
Check by a second route. The triangle sum gives the interior angle at C: subtract the sum of the two stated angles from 180°.
$\angle ACB = 180^\circ - (48^\circ + 67^\circ) = 180^\circ - 115^\circ = 65^\circ$
Angle ACB and angle ACD sit side by side along the straight line through B, C and D, so they are supplementary and must add to 180°. They do, so the answer agrees with both facts.
$65^\circ + 115^\circ = 180^\circ$

Answer

The exterior angle $\angle ACD$ measures $115^\circ$ .

Watch out: $180^\circ$ minus the two remote angles is not the exterior angle. With $\angle A = 48^\circ$ and $\angle B = 67^\circ$ , it is tempting to work out $180 - 48 - 67 = 65$ and call $65^\circ$ the exterior angle at $C$ . This fails because the $180^\circ$ total counts only the three angles inside the triangle, so $65^\circ$ is the interior angle at $C$ . The exterior angle lies outside the triangle, beside that interior angle on a straight line, so it measures $180^\circ - 65^\circ = 115^\circ$ . That matches $48^\circ + 67^\circ = 115^\circ$ . Before you calculate, ask: is the angle I want inside the triangle or outside it?

Often an angle measure is given as an expression, such as $(2x + 10)^\circ$ . Here $x$ is an unknown number, and the whole expression is a measure in degrees. Only values of $x$ that make every angle measure positive make sense.

A stated angle relationship gives you an equation to solve for $x$ . Work in three moves:

Translate the stated relationship into an equation. Every measure is in degrees, so you can write the equation with numbers alone.
Solve for $x$ , doing the same thing to both sides at each step so that the two sides stay equal.
Substitute and check. Every angle measure must come out positive, and the stated relationship must be true.

For the first move, notice that every relationship in the table below says one of two things: two amounts are equal, or some measures add to a fixed total of $90^\circ$ or $180^\circ$ .

Two terms appear in the table. Parallel lines are lines on a flat surface that never meet, however far they are extended. A transversal is a line that crosses two other lines, so it makes one crossing on each. The equations in the last three rows hold only when the two crossed lines are parallel, so use them only when the problem states that the lines are parallel.

Angle relationships as equations. Each letter is an angle measure in degrees.

Stated relationship	What it means	Equation
Complementary angles	Two angles whose measures add to $90^\circ$ , the measure of a right angle	$a + b = 90^\circ$
Supplementary angles	Two angles whose measures add to $180^\circ$ , such as two angles side by side that fill a straight line	$a + b = 180^\circ$
Vertical angles	The two angles opposite each other where two straight lines cross	$a = b$
Interior angles of a triangle	The three angles inside one triangle, one at each vertex	$a + b + c = 180^\circ$
Exterior angle of a triangle	Angle $e$ is made by extending one side; $a$ and $b$ are its two remote interior angles	$e = a + b$
Corresponding angles (lines stated parallel)	One angle at each crossing, in the same corner position at both (for example, both upper left)	$a = b$
Alternate interior angles (lines stated parallel)	One angle at each crossing, both between the parallel lines, on opposite sides of the transversal	$a = b$
Same-side interior angles (lines stated parallel)	One angle at each crossing, both between the parallel lines, on the same side of the transversal	$a + b = 180^\circ$

The stated relationship decides the equation; the expressions alone do not. Take two angles that measure $(3x)^\circ$ and $(x + 20)^\circ$ .

If they are stated to be vertical angles, or alternate interior angles on lines stated to be parallel, the measures are equal: $3x = x + 20$ . Subtract $x$ from both sides to get $2x = 20$ , then divide both sides by $2$ to get $x = 10$ . Each angle measures $30^\circ$ , since $3 \times 10 = 30$ and $10 + 20 = 30$ .
If they are stated to be supplementary, or same-side interior angles on lines stated to be parallel, the measures add to $180^\circ$ : $3x + (x + 20) = 180$ , which simplifies to $4x + 20 = 180$ . Subtract $20$ from both sides to get $4x = 160$ , then divide both sides by $4$ to get $x = 40$ . The angles measure $120^\circ$ and $60^\circ$ (from $3 \times 40$ and $40 + 20$ ), and $120 + 60 = 180$ .

Same expressions, different stated relationships, different answers.

Worked example

Solving for x with an exterior angle

In triangle $PQR$ , side $QR$ is extended in a straight line past $R$ to point $S$ . The exterior angle is $\angle PRS = (4x - 15)^\circ$ . Its remote interior angles are $\angle P = (x + 20)^\circ$ and $\angle Q = (2x - 10)^\circ$ . Find $x$ and the measure of each of these three angles.

Translate. By the exterior angle theorem, the exterior angle equals the sum of its two remote interior angles. Every measure is in degrees, so write the equation with numbers alone.
$4x - 15 = (x + 20) + (2x - 10)$
Simplify the right side by adding the x terms and adding the numbers: x + 2x = 3x and 20 - 10 = 10. This rewrites the right side without changing its value.
$4x - 15 = 3x + 10$
Subtract 3x from both sides. Taking the same amount from both sides keeps them equal, and 4x - 3x = x.
$x - 15 = 10$
Add 15 to both sides, which also keeps them equal, so that x is by itself: 10 + 15 = 25.
$x = 25$
Substitute x = 25 into each expression, using 2 × 25 = 50 and 4 × 25 = 100.
$\angle P = (25 + 20)^\circ = 45^\circ, \quad \angle Q = (50 - 10)^\circ = 40^\circ, \quad \angle PRS = (100 - 15)^\circ = 85^\circ$
Check. All three measures are positive, and the two remote interior angles add to the exterior angle, as the theorem requires. The interior angle at R is 180° - 85° = 95°, and 45° + 40° + 95° = 180°, so the triangle sum agrees too.
$45^\circ + 40^\circ = 85^\circ$

Answer

$x = 25$ , so $\angle P = 45^\circ$ , $\angle Q = 40^\circ$ and the exterior angle $\angle PRS = 85^\circ$ .

Worked example

Solving for x with stated parallel lines

In triangle $ABC$ , point $D$ lies on side $AB$ between $A$ and $B$ , and point $E$ lies on side $AC$ between $A$ and $C$ . You are told that line $DE$ is parallel to line $BC$ and that, with line $AB$ as the transversal, $\angle ADE$ and $\angle ABC$ are corresponding angles. The angle at vertex $A$ measures $38^\circ$ , $\angle ABC = (3x + 10)^\circ$ and $\angle AED = (2x + 12)^\circ$ . Find $x$ and the three interior angles of triangle $ADE$ .

Corresponding angles sit in the same corner position at the two points where a transversal crosses two lines. Here line AB crosses DE at D and BC at B, and each angle has one side running toward A and the other running across the triangle toward side AC. Because lines DE and BC are stated to be parallel, corresponding angles are equal.
$\angle ADE = \angle ABC = (3x + 10)^\circ$
Triangle ADE has interior angles at A, D and E. Its angle at A is the stated 38° angle, because sides AD and AE lie along sides AB and AC. The interior angles of every triangle add to 180°. Every measure is in degrees, so write the equation with numbers alone.
$38 + (3x + 10) + (2x + 12) = 180$
Simplify the left side by adding the x terms and adding the numbers: 3x + 2x = 5x and 38 + 10 + 12 = 60.
$5x + 60 = 180$
Subtract 60 from both sides, which keeps the two sides equal: 180 - 60 = 120.
$5x = 120$
Divide both sides by 5. Dividing both sides by the same nonzero number keeps them equal: 120 ÷ 5 = 24.
$x = 24$
Substitute x = 24, using 3 × 24 = 72 and 2 × 24 = 48, then check. Every measure is positive, and the three angles of triangle ADE add to 180°.
$\angle ADE = (72 + 10)^\circ = 82^\circ, \quad \angle AED = (48 + 12)^\circ = 60^\circ, \quad 38^\circ + 82^\circ + 60^\circ = 180^\circ$

Answer

$x = 24$ , and triangle $ADE$ has interior angles of $38^\circ$ at $A$ , $82^\circ$ at $D$ and $60^\circ$ at $E$ .

Takeaway. A triangle's angles are tied together by equations. In every triangle on a flat surface, the three interior angles add to $180^\circ$ , and an exterior angle made by extending a side in a straight line equals the sum of its two remote interior angles. When measures are given as expressions in $x$ , translate the stated relationship into an equation, solve it, and substitute back to check that every angle is positive and the relationship is true. Use a parallel-line relationship only when the lines are stated to be parallel.

Next, you will look at arguments that explain why these triangle facts are true, and you will use the angle-angle criterion to decide when two triangles are similar.