A point has a position but no size. A moving point traces out a line, and a moving line sweeps out a surface.
Through the same two points A and B, a line goes on forever both ways, a ray starts at A and goes on forever towards B, and a line segment keeps only the part between A and B.
An angle is how far two rays from one point open apart; with the right angle (90°) and the straight angle (180°) as markers, angles divide into acute and obtuse.
On this page you change the size of an angle to sort it into its type, and see why vertically opposite angles, formed where two lines cross, are always equal.
Points, lines, surfaces and three kinds of line
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A line, a ray and a line segment through points A and B, with their symbols
👀 Points, lines and surfaces
①A point has a position but no size. A moving point traces a line, and a moving line sweeps a surface
②Countless lines pass through one point, but exactly one line passes through two different points
③Line AB and line BA are the same line, and segment AB and segment BA are the same segment
④Two rays are the same only if they start at the same point and go the same way. Ray AB goes from A towards B and ray BA from B towards A, so they are different
⑤The length of segment AB is called the distance between A and B
Angle size and type
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Acute, right, obtuse and straight angles as ∠AOB changes
60
🔍 Kinds of angle
①∠AOB is how far the two rays OA and OB, drawn from the vertex O, open apart
②An angle greater than 0° and less than 90° is acute, and an angle of exactly 90° is a right angle. Set the picture to 90° and a right-angle mark appears at the vertex
③An angle greater than 90° and less than 180° is obtuse
④When the two rays form a straight line through the vertex, the angle is 180°, a straight angle
Vertically opposite, complementary and supplementary angles
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The four angles a, b, c, d formed where two lines cross
50°
Vertically opposite angles
∠a = ∠c, ∠b = ∠d
Where two lines cross at a point, the angles facing each other are equal
Complementary angles
∠x + ∠y = 90°
Two angles that add up to 90° are complementary; each is the complement of the other
Supplementary angles
∠x + ∠y = 180°
Two angles that add up to 180° are supplementary. ∠a and ∠b in the picture are an example
💡 Why vertically opposite angles are equal
①In the picture, ∠a and ∠b sit side by side on one straight line, so they add up to 180°
②∠c and ∠b sit side by side on the other line, so they also add up to 180°
③So ∠a = 180° − ∠b = ∠c, and in the same way ∠b = ∠d
④Move the slider to change the angle between the lines: the two gold angles always stay equal to each other, and so do the two blue ones
Check with examples
Example 1
Two lines cross at a point, and one of the angles they make is 65°. Find the angle vertically opposite it.
1
The two angles facing each other where the lines cross are vertically opposite angles.
(opposite angle) = (given angle)
2
Vertically opposite angles are equal.
∠x = 65°
▸ 65°
The neighbouring angle is 180° − 65° = 115°. Set ∠a to 65° in the picture and it shows ∠c = 65° and ∠b = ∠d = 115°.
Example 2
At a point on a straight line, two angles of 3x and 2x together make a straight angle. Find x.
1
A straight angle is 180°, so the two angles add up to 180°.
3x + 2x = 180°
2
Simplify and solve for x.
5x = 180° ⇒ x = 36°
▸ x = 36°
The angles are 3x = 108° and 2x = 72°, which add up to 180° as they should. Turning straight and right angles into equations lets you find an unknown angle.
Exam checklist
Key facts
one line through two points · vertically opposite angles are equal
line AB = line BA, segment AB = segment BA, ray AB ≠ ray BA
Grade 7 school test style
At a point on a straight line, the straight angle is split into three angles of 90°, x and 30°. What is x?
①50°
②60°
③70°
④80°
⑤90°
▸ ② 60°
1
It is a straight angle, so the three angles add up to 180°.
90° + x + 30° = 180°
2
Solve for x.
x = 180° − 120° = 60°
🎯 Exam checklist
①Exactly one line passes through two different points
②Line AB = line BA and segment AB = segment BA, but ray AB ≠ ray BA
③Acute angles lie between 0° and 90°, obtuse angles between 90° and 180°. 90° and 180° themselves belong to neither
④Vertically opposite angles are equal
⑤For angles that together make a straight angle (180°) or a right angle (90°), write their sum as an equation