Unit 3 · Topic 02 · Geometry and Triangles
Kabir drew a triangle with angles 70°, 80°, and 40°, and Zara said one angle had to be wrong before she even measured.
Kabir added them up himself: 70+80+40 = 190.
"Every triangle's angles add up to exactly 180," Zara said. "Yours don't. Redraw it."
Kabir sketched a triangle freehand and estimated its three angles as 70°, 80°, and 40°. Zara didn't even pick up a protractor. "Add them," she said. 70+80+40 = 190, not 180. "The angle sum property says the three angles of ANY triangle always add up to exactly 180°. Your triangle can't have those three angles — one of your estimates is off."
She proved it with a quick paper trick: tear off the three corners of a triangle and place them together, side by side, along a straight line. The three angles always fit perfectly along that straight 180° line, no matter what shape the triangle was. "That's the angle sum property in a single picture," she said.
Kabir then asked about a triangle where he only knew two angles: 55° and 65°. "Third angle = 180 − 55 − 65 = 60°," Zara said. "Knowing any two angles always gives you the third, instantly."
She then extended one side of the triangle outward to form an EXTERIOR angle at that vertex. "Here's the exterior angle theorem: an exterior angle of a triangle equals the SUM of the two interior angles that are not adjacent to it — the two 'remote' interior angles." She demonstrated: if the triangle's angles were 55°, 65°, and 60°, the exterior angle at the 60°-vertex equals 55+65=120°. She checked it against the linear pair rule too: 180−60=120, matching exactly.
"So the exterior angle theorem is really just a shortcut," Kabir said, "instead of finding the third angle then subtracting from 180, you just add the two OTHER interior angles directly." Zara agreed, and showed one more use: if you know an exterior angle (say 100°) and one remote interior angle (say 40°), the other remote interior angle is 100−40=60°, again without ever finding the actual interior angle at that exterior vertex.
By the end, Kabir had two tools that worked together: the angle sum property (all three interior angles sum to 180°) and the exterior angle theorem (an exterior angle equals the sum of the two remote interior angles) — either one solving problems the other made faster.
Angle Sum Property: the three interior angles of any triangle always sum to exactly 180°.
Given any two angles of a triangle, the third is found by subtracting their sum from 180°.
Exterior Angle Theorem: an exterior angle of a triangle equals the sum of the two remote (non-adjacent) interior angles.
The exterior angle and its adjacent interior angle always form a linear pair (sum to 180°), which gives a second way to check or compute the exterior angle.
Draw a triangle with two known angles of 48° and 77°. Find the third angle, then extend one side to form an exterior angle and verify the exterior angle theorem.
Given a triangle's exterior angle of 130° and one remote interior angle of 55°, find the other remote interior angle and the full interior angle at the exterior vertex.
Loading questions…