When all four angles are equal
Equal angles at a crossing are right angles.
Four equal angles?
Two lines cross and make four angles. Usually two are bigger and two are smaller.
Can two crossing lines make four equal angles? How big would each angle be?
What this lesson covers
The idea
Perpendicular lines are a pair of lines that intersect each other at right angles; if two intersecting lines form four equal angles, each angle must be 90°.
Four equal angles?
Two lines cross and make four angles. Usually two are bigger and two are smaller.
Can two crossing lines make four equal angles? How big would each angle be?
- No: two of the angles are always bigger
- Yes: each angle is 45°
- Yes: each angle is 90°
Make all four equal
Turn line m and watch the four angles. The bar shows ∠a and ∠b together on a straight angle.
Right angles
Two neighbours ∠a and ∠b fill a straight angle, 180°. If they are equal, each is half of it: 180° ÷ 2 = 90°. ∠c and ∠d are equal to them, so all four angles are 90°.
If two lines intersect and all four angles are equal, each angle is a right angle (90°). Two lines that intersect at right angles are perpendicular to each other.
- m at 60° | m at 90°
- ∠a = 120°, ∠b = 60° | ∠a = ∠b = 90°
- ∠c = 120°, ∠d = 60° | ∠c = ∠d = 90°
- Not equal | All four equal
Notes
If two lines intersect and all four angles are equal, each angle is a right angle (90°). Two lines that intersect at right angles are perpendicular to each other.
Check yourself
Two lines cross and all four angles are equal. How big is each angle?
Answer: 90 °
Two equal neighbours fill 180°, so each is 180° ÷ 2 = 90°.
Two lines cross and ∠a = ∠b. What can we say about the lines?
Lines l and m are perpendicular and ∠a = 90°. What do ∠b + ∠c + ∠d add up to?
Answer: 270 °
∠b = 180° − 90° = 90°. ∠c = ∠a = 90° and ∠d = ∠b = 90°. So ∠b + ∠c + ∠d = 90° + 90° + 90° = 270°.
Which sentence describes two perpendicular lines?
- They are parallel.. Parallel lines never meet. These two lines cross each other.
- Nothing, because only two of the angles are equal.. Two neighbours that are equal must be 90° each, and ∠c and ∠d match them. So the lines are perpendicular.
- They are perpendicular: ∠a + ∠b = 180° and they are equal, so each is 90°. — correct. Yes! Equal neighbours on a straight angle are 90° each. Then ∠c and ∠d are 90° too.
- They cross each other at right angles (90°). — correct. Yes! Perpendicular lines meet at right angles, so all four angles are 90°.
- They never meet, however far they are extended.. That describes parallel lines. Perpendicular lines do cross each other.
- They cross and make angles of 45°.. Perpendicular lines make right angles, 90°, not 45°.