A Tale of Three Intersecting Lines
Finding the third angle
Construct triangles from two angles and see what you notice about the third angle.
Take two angles that add to less than 180°, say 60° and 70°, with a base of 5 cm. Construct the triangle and measure the third angle. Now use a base of 7 cm and construct again. Does the third angle change?
No! Once two angles are fixed, the third doesn't depend on the side. So there must be a rule linking the three angles, and we should be able to find the third without drawing.
In ∆ABC, ∠B = 50° and ∠C = 70°. Find ∠A. The trick: through the top corner A, draw a line XY parallel to the base BC.
Two new angles appear at A:
- ∠XAB = ∠B = 50°: they are alternate angles (AB is a transversal of the parallels XY and BC).
- ∠YAC = ∠C = 70°: alternate angles again (AC is a transversal).
The three angles at A (∠XAB, ∠BAC, ∠YAC) sit side by side along the straight line XY, so they add to 180°: 50° + ∠BAC + 70° = 180°, so ∠BAC = 60°. Check by drawing it with any base. Try the same method for 36° and 72°, 150° and 15°, 90° and 30°.
Practise this lesson
Questions that get harder as you improve, a Tutor that helps without giving the answer away, and progress you can see. Free for every learner.
No account, no email address. Opens the real product.