AC9M7M05 · Year 7 · Measurement

Interior angle sum of a triangle

ACARA v9 CONTENT DESCRIPTION “demonstrate that the interior angle sum of a triangle in the plane is 180 degrees and apply this to determine the interior angle sum of other shapes and the size of unknown angles”

Every triangle, no matter how it is shaped, hides the same secret: its three angles always add up to exactly 180 degrees. A long thin triangle, a wide squat one, a right-angled one, all obey this single rule. This year you learn to demonstrate why the angle sum is 180 degrees and to use it to find unknown angles in triangles and other shapes.

This fact is one of the most useful in geometry, because it means you never need to measure all three angles of a triangle. Knowing any two tells you the third, and the rule extends to working out angles in more complicated figures built from triangles.

Why the angles total 180

There is a wonderfully simple way to see the rule. Draw any triangle on paper, then tear off its three corners. Lay the three torn angles side by side, with their points meeting, and they fit together perfectly along a straight line, leaving no gap and no overlap. A straight line measures 180 degrees, so the three angles of the triangle must total 180 degrees too.

Three corners make a straight line
Tear off the three angles of a triangle and they fit along a straight line.
Tear the three corners off any triangle and lay them side by side. They always fit exactly along a straight line, and a straight line is 180 degrees. This is why the three angles of every triangle add up to 180.

This can be shown more formally using the angle relationships on parallel lines from earlier. Drawing a line through the top corner parallel to the base creates alternate angles equal to the two base angles, and those, together with the top angle, lie along a straight line. Whether you tear paper or reason with parallel lines, the conclusion is the same: the interior angles of a triangle always sum to 180 degrees.

Using the angle sum

The rule becomes a tool the moment two angles are known. If a triangle has angles of 50 and 60 degrees, the third must be 180 minus 50 minus 60, which is 70 degrees. There is no need to measure it; the angle sum delivers it by subtraction. This works for any triangle, and it is the single most common way unknown angles are found.

Finding the third angle
If two angles are known, subtract their sum from 180 to find the third.
Because the three angles total 180 degrees, knowing two lets you find the third. With angles of 50 and 60 degrees, the third is 180 minus 50 minus 60, which is 70 degrees. The angle sum turns two known angles into a third.

The rule also explains the shapes of special triangles. An equilateral triangle has three equal angles, so each must be 180 divided by 3, which is 60 degrees. A right-angled triangle uses 90 of its 180 degrees on the right angle, leaving just 90 to share between the other two, which is why a triangle can never contain two right angles. From finding a single missing angle to reasoning about whole families of triangles, the angle sum of 180 degrees is a rule you will use constantly, and it extends naturally to finding angles in quadrilaterals and other polygons by splitting them into triangles.

Teaching tip: the torn-corners demonstration is worth doing with real paper. Have the student draw several very different triangles, tear the corners from each, and line them up. Seeing completely different triangles all produce the same straight line is convincing in a way that simply stating the rule can never be.

Encourage checking that found angles are sensible. After calculating a missing angle, a quick mental check that all three add to 180 confirms the work. If they do not, an error has crept in, and the check catches it immediately.

Builds on: Angles on a transversal (AC9M7M04). That unit used angle relationships on parallel lines; this unit applies them to prove a triangle has 180 degrees.
Quick self-check
1. What do the three interior angles of any triangle add up to?
2. Two angles of a triangle are 40 degrees and 75 degrees. What is the third angle?
3. In an equilateral triangle, all three angles are equal. What is each angle?
4. A right-angled triangle has one angle of 90 degrees and another of 30 degrees. What is the third?
5. Why can a triangle never have two right angles?