Finding Missing Angles in Triangles and Quadrilaterals | Grade 7 Math | LevelKite

You can already sort triangles and quadrilaterals into classes. Now you will find the size of an angle that no one has told you.

An interior angle is the angle inside a figure at one corner, between the two sides that meet there. You measure angles in degrees and mark them with a small raised circle. A right angle measures $90^\circ$ . A straight angle, whose two sides form one straight line, measures $180^\circ$ .

A triangle has three interior angles, and a quadrilateral has four. When you know all of them but one, that last one is the missing angle. You will not measure it or guess it from a drawing. You will work it out from the angles you know and one fact about the figure's total. Every figure in this lesson lies on a flat surface and has straight sides.

Start with what the total means for a triangle. Try this with any triangle cut from paper, or just imagine it. Tear off the three corners and place them point to point, side by side. They fit with no gap and no overlap, and their outer edges form one straight line. Together the three corners make a straight angle, so the three interior angles add to $180^\circ$ .

This is not luck. Mathematicians have proved it for every triangle on a flat surface, whatever its shape, size or class. A larger triangle has longer sides, but its three interior angles still add to exactly $180^\circ$ .

Now the quadrilateral. Start with a case you know. A rectangle has four right angles by definition, so its interior angles add to $4 \times 90 = 360$ degrees. A square is a rectangle, so its total is $360^\circ$ as well.

The same total holds for every quadrilateral on a flat surface whose four sides do not cross, not only for rectangles. Here is why. A diagonal is a straight segment that joins two corners that do not share a side. Every such quadrilateral has a diagonal that stays inside the figure. That diagonal cuts the quadrilateral into two triangles. The angles of those two triangles together make up exactly the four interior angles of the quadrilateral, and each triangle brings $180^\circ$ :

180^\circ + 180^\circ = 360^\circ

Key concept

Angle sums for triangles and quadrilaterals

The interior angles of every triangle on a flat surface add to exactly $180^\circ$ . The interior angles of every quadrilateral on a flat surface, with four straight sides that do not cross, add to exactly $360^\circ$ .

In the formula, $a$ , $b$ and $c$ are the measures, in degrees, of a triangle's three interior angles. The letters $p$ , $q$ , $r$ and $s$ are the measures, in degrees, of a quadrilateral's four interior angles.

These totals do not depend on the figure's size or side lengths. So when you know every interior angle but one, the total fixes the last one.

a + b + c = 180 \quad \text{and} \quad p + q + r + s = 360

The two angle sums, side by side

Figure	Interior angles	Sum of the interior angles	Angles you must know to find one missing angle	How to find it
Triangle	$3$	$180^\circ$	$2$	Subtract the sum of the $2$ known angles from $180^\circ$
Quadrilateral	$4$	$360^\circ$	$3$	Subtract the sum of the $3$ known angles from $360^\circ$

Here is a routine for a triangle or a quadrilateral with exactly one missing angle. Turn the angle sum into an equation.

Name the unknown. Let $x$ be the measure, in degrees, of the missing angle. Because $x$ counts degrees, the equation uses plain numbers, and the degree symbol returns in your answer.
Write the sum. Add $x$ to the known measures and set the result equal to the figure's total: $180$ for a triangle, $360$ for a quadrilateral.
Add, then subtract. Add the known measures. Then subtract that sum from both sides. Taking the same number from both sides keeps them equal, so the equation stays true and $x$ stands alone.
Check. All the angles must add to the total, and $x$ must be greater than $0$ . If the known angles already reach the total or pass it, no such figure exists.

Worked example

A triangle with two known angles

A triangle has interior angles that measure $47^\circ$ and $68^\circ$ . What is the measure of its third interior angle?

Name the unknown. Let x be the measure, in degrees, of the third interior angle. Because x counts degrees, the equation will use plain numbers.
Write the angle sum. The three interior angles of a triangle on a flat surface add to exactly 180 degrees, so x and the two known measures must total 180.
$x + 47 + 68 = 180$
Add the two known measures and write their sum in their place. This does not change the value of the left side, so the equation stays true.
$47 + 68 = 115 \quad \text{so} \quad x + 115 = 180$
Subtract 115 from both sides. Taking the same number from both sides keeps them equal, and it leaves x alone on the left.
$x + 115 - 115 = 180 - 115 \quad \text{so} \quad x = 65$
Check. The three measures must add to 180, and x must be greater than 0. The sum below is 180, and 65 is greater than 0, so both checks pass.
$47 + 68 + 65 = 115 + 65 = 180$

Answer

The third interior angle of the triangle measures $65^\circ$ .

Worked example

A quadrilateral with three known angles

A quadrilateral has interior angles that measure $85^\circ$ , $110^\circ$ and $72^\circ$ . What is the measure of its fourth interior angle?

Let x be the measure, in degrees, of the fourth interior angle. The four interior angles of a quadrilateral on a flat surface add to exactly 360 degrees, so x and the three known measures must total 360.
$x + 85 + 110 + 72 = 360$
Add the three known measures, two at a time, and write their sum in their place. The left side keeps the same value.
$85 + 110 = 195, \quad 195 + 72 = 267 \quad \text{so} \quad x + 267 = 360$
Subtract 267 from both sides. The two sides stay equal, and x stands alone.
$x = 360 - 267 = 93$
Check that the four measures add to 360 and that x is greater than 0.
$85 + 110 + 72 + 93 = 267 + 93 = 360$

Answer

The fourth interior angle of the quadrilateral measures $93^\circ$ .

Mixing up the two totals. A common mistake is to subtract from $180$ for every figure. Try it on the quadrilateral from the last example: its known angles sum to $267$ , and $180 - 267 = -87$ . An interior angle is always greater than $0^\circ$ , so $-87^\circ$ is impossible.

This fails because $180^\circ$ is the total for the three angles of a triangle. A quadrilateral has four angles and splits into two triangles, so its total is $360^\circ$ . That gives $360 - 267 = 93$ .

The reverse mix-up gives no negative sign to warn you. Using $360$ for the triangle with angles of $47^\circ$ and $68^\circ$ gives $360 - 115 = 245$ , but each angle of a triangle is less than $180^\circ$ . So count the sides first, then choose the total.

Worked example

Two missing angles that are equal

A quadrilateral has interior angles that measure $95^\circ$ and $125^\circ$ . Its other two interior angles are equal to each other. What is the measure of each of those two angles?

Only two angles are known, but the problem states a fact that links the other two: they are equal. So one letter can stand for both. Let x be the measure, in degrees, of each missing angle. The four interior angles of a quadrilateral on a flat surface add to exactly 360 degrees.
$x + x + 95 + 125 = 360$
Simplify the left side. Two angles of x degrees each make 2 times x degrees, and the known measures add to 220. Neither change alters the value of the left side.
$95 + 125 = 220 \quad \text{so} \quad 2x + 220 = 360$
Subtract 220 from both sides so that the sides stay equal. The two missing angles share the 140 degrees that remain.
$2x = 360 - 220 = 140$
Divide both sides by 2. Dividing both sides by the same nonzero number keeps them equal, and it splits 140 into two equal parts.
$x = 140 \div 2 = 70$
Check that the four measures add to 360 and that x is greater than 0.
$95 + 125 + 70 + 70 = 220 + 140 = 360$

Answer

Each of the two missing interior angles measures $70^\circ$ .

Takeaway. The interior angles of every triangle on a flat surface add to exactly $180^\circ$ . The interior angles of every quadrilateral on a flat surface, with sides that do not cross, add to exactly $360^\circ$ , because a diagonal splits it into two triangles. To find a missing angle, count the sides, choose the matching total, write an equation for the unknown measure in degrees, and subtract the sum of the known angles from both sides. A stated fact or a definition, such as two equal angles or a right angle, can supply what a measure does not. Always check that the angles add back to the total and that each one is greater than $0^\circ$ .