Neighbours on a straight line
Two angles side by side on a line always make 180°.
A puzzle without a protractor
Two lines cross. The angle marked 120° is ∠a. The angle next to it is ∠b. We do not draw it and we do not measure it.
What can ∠b be?
What this lesson covers
The idea
Adjacent angles formed by two intersecting lines are called linear pairs; together they form a straight angle, so a linear pair always adds up to 180°.
A puzzle without a protractor
Two lines cross. The angle marked 120° is ∠a. The angle next to it is ∠b. We do not draw it and we do not measure it.
What can ∠b be?
- 60°
- 120°
- We cannot know without measuring
Fit the angles on a bar
Each pair of neighbouring angles is laid end to end on a bar. The bar is one straight angle.
Linear pairs
Neighbouring angles like ∠a and ∠b share one arm, and together they lie along one straight line. Together they make a straight angle, which is 180°.
So if ∠a = 120°, then ∠b = 180° − 120° = 60°. Then ∠b + ∠c = 180° gives ∠c = 120°, and ∠c + ∠d = 180° gives ∠d = 60°.
Adjacent angles formed by two intersecting lines are called linear pairs. Together they form a straight angle, so a linear pair always adds up to 180°.
- Linear pair | Adds up to
- ∠a and ∠b | 120° + 60° = 180°
- ∠b and ∠c | 60° + 120° = 180°
- ∠c and ∠d | 120° + 60° = 180°
- ∠d and ∠a | 60° + 120° = 180°
Notes
Adjacent angles formed by two intersecting lines are called linear pairs. Together they form a straight angle, so a linear pair always adds up to 180°.
Check yourself
∠a and ∠b are a linear pair. ∠a = 115°. Find ∠b.
Answer: 65 °
∠a + ∠b = 180°, so ∠b = 180° − 115° = 65°.
Which angles make a linear pair with ∠a?
∠PQR = 60° and ∠RQS = 50° are side by side and share the arm QR. Do they form a linear pair?
Two lines cross. ∠b = 70°. Find ∠c, the angle next to ∠b.
Answer: 110 °
∠b + ∠c = 180°, so ∠c = 180° − 70° = 110°.
- ∠c. ∠c is across from ∠a, not next to it. Angles that are only across from each other are not a linear pair.
- ∠b and ∠c. ∠c is across from ∠a. Only the angles next to ∠a form a linear pair with it.
- ∠b and ∠d — correct. Yes! ∠b is next to ∠a, and so is ∠d. Each one makes a straight line with ∠a.
- Yes. They are next to each other.. Being next to each other is not enough. The two outer arms must also make one straight line.
- No. The arms QP and QS do not make one straight line. — correct. Yes, that is it. Together they make 60° + 50° = 110°, not a straight angle.
- Yes. 110° is close enough to 180°.. A linear pair adds up to exactly 180°. Here the outer arms are not on one line, so the sum is 110°.