When do two arms meet?

The idea

A triangle with two given angles exists when their sum is less than 180°; if the sum is 180° or more the arms never meet and there is no triangle, whatever the length of the included side.

Will the arms always meet?

The base AB is drawn. You draw an arm at A and an arm at B, and hope they meet at the third corner C.

Will the two arms always meet?

• Yes, whatever the angles are
• No, for some angles they never meet
• Only if AB is long enough

Swing the arm at B

∠A is fixed at 40°. Swing the arm at B bigger, little by little. Watch the two arms and the bar.

Leaning towards each other

The arms meet only if they lean towards each other. That happens when ∠A + ∠B is less than 180°.

At exactly 180°, AB is a transversal and the two angles on the same side add up to 180°. So the arms are parallel lines.

The length of AB has no part in this. It only makes the picture bigger or smaller.

A triangle with two given angles exists when their sum is less than 180°. If the sum is 180° or more, the arms never meet and there is no triangle, whatever the length of the included side.

• The two angles together | The arms
• less than 180° | meet at C. A triangle!
• equal to 180° | are parallel. They never meet.
• more than 180° | lean away. They never meet.

Notes

A triangle with two given angles exists when their sum is less than 180°. If the sum is 180° or more, the arms never meet and there is no triangle, whatever the length of the included side.

Check yourself

Which pair of angles at the ends of the base AB can give a triangle?

∠A = 65°. What is the biggest whole number of degrees ∠B can be so that the two arms still meet?

Answer: 114

The sum must be less than 180°. 65° + 115° = 180° is too big, and 65° + 114° = 179° is fine. So the biggest ∠B is 114°.

Ravi’s angles add up to 200°. He says: “If I draw a very, very long base AB, the arms will finally meet.” Is he right?

Both base angles are right angles: ∠A = 90° and ∠B = 90°. Why is there no triangle?

• 80° and 100°. 80° + 100° = 180°. The arms are parallel and never meet.
• 70° and 100° — correct. Yes! 70° + 100° = 170°, which is less than 180°. The arms meet.
• 120° and 90°. 120° + 90° = 210°, which is more than 180°. The arms lean away from each other.
• Yes. A longer base makes the arms meet.. A longer base only makes the picture bigger. The arms still lean away from each other.
• No. The sum is more than 180°, so the arms lean away from each other, whatever the length of AB. — correct. Yes! The length of AB has no part in it. The angles alone decide.
• Yes, but only if AB is more than 100 cm.. There is no such length. Whatever AB is, the arms lean away from each other.
• A triangle cannot have a right angle.. A triangle can have one right angle. The trouble here is that there are two right angles at the base, which add up to 180°.
• AB is too short.. The length of AB does not decide it. The two angles add up to 180°, and the arms never meet.
• 90° + 90° = 180°, so the arms are parallel and never meet. — correct. Yes! Both arms are perpendicular to AB, so they are parallel.