Interior angle sum of a triangle
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.
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.
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.
From triangles to other shapes
The descriptor asks you to extend the rule to other shapes. Any polygon can be split into triangles from a single corner: a four-sided shape gives two, a five-sided shape gives three. Each triangle contributes 180 degrees, so the whole angle sum is the triangle count times 180.
Interior and exterior angles
At every corner the interior angle and the exterior angle together lie along a straight line. That makes them add to 180, so one is always 180 minus the other. This straight-line pairing is the same idea that gave the triangle its angle sum.
Unknown angles in special triangles
The angle sum shines when a triangle has equal sides. An isosceles triangle has two equal base angles, so once the apex is known, the two base angles share what is left of 180 equally. Each is 180 minus the apex, divided by two.