Three angles, one straight line

The idea

The sum of the three angles of any triangle is 180°: a line through a vertex parallel to the opposite side makes alternate angles equal to the other two angles, and these with the third angle form a straight angle.

Is the third angle decided?

A triangle has corners B and C. You know ∠B and ∠C, but you do not measure ∠A.

Can you tell ∠A without measuring it?

• Yes, ∠B and ∠C decide it
• No, ∠A could be anything
• Only if the triangle is a right triangle

Turn the corners onto A

A line through A is drawn parallel to BC. Turn the corners at B and C onto A. Then try other triangles.

Alternate angles fit on the line

Each corner turned half way round the middle of its side, and landed at A next to the parallel line XY. That works because alternate angles are equal.

The three angles at A sit side by side on the straight line XY. A straight angle is 180°, so ∠B + ∠A + ∠C = 180°.

The three angles of any triangle add up to 180°. Draw the line through a vertex parallel to the opposite side: the other two angles become alternate angles at that vertex, and with the third angle they make a straight angle.

• The angle at A | Equals
• ∠XAB | ∠B (alternate angles)
• ∠BAC | ∠A itself
• ∠YAC | ∠C (alternate angles)

Notes

The three angles of any triangle add up to 180°. Draw the line through a vertex parallel to the opposite side: the other two angles become alternate angles there, and with the third angle they make a straight angle.

Check yourself

Two angles of a triangle are 40° and 75°. How big is the third angle?

Answer: 65

The three angles add up to 180°. 40° + 75° = 115°, and 180° − 115° = 65°.

A triangle is said to have angles 90°, 60° and 50°. Can that be?

In triangle ABC, ∠A = ∠B and ∠C = 80°. How big is ∠A?

Answer: 50

∠A + ∠B = 180° − 80° = 100°. They are equal, so each is 100° ÷ 2 = 50°.

Why do the corners at B and C fit exactly on the line through A?

• Yes. A triangle can have a right angle.. A triangle can have a right angle, but the three angles must still add up to 180°. Here 90° + 60° + 50° = 200°.
• Yes. Every angle is less than 180°.. That is not enough. The three angles must add up to exactly 180°.
• No. The three angles add up to 200°, not 180°. — correct. Yes! 90° + 60° + 50° = 200°. The three angles of a triangle always make exactly 180°.
• Because the sides AB and AC are equal.. AB and AC do not have to be equal. It works for every triangle, because the line through A is parallel to BC.
• The line is parallel to BC, so the angles at A are alternate angles of ∠B and ∠C. — correct. Yes! Alternate angles are equal, so the corners fit exactly against the parallel line.
• Because the triangle is small.. The size does not matter. It works for every triangle, because the line through A is parallel to BC.