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Angles on lines and at a point

Choose the correct angle relationship and state why it applies.

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Enlarged diagram

A straight turn is 180°

Two adjacent angles that together form a straight angle add to 180°. The outer arms must form a straight line: any two angles touching at a vertex do not automatically add to 180°.

Two parts of a straight angle

A, O and C are collinear. Ray OB separates the two marked sectors.

Angle a is 68°. Use the given value; the drawing identifies the openings.
View figure data
Given geometry and angles
GivenInformation
GeometryA, O and C lie on one straight line.
Angle a68°

Here a + b = 180°, so b = 180°-68° = 112°. State the total before subtracting: it tells you which geometric fact your calculation uses. Check 68° + 112° = 180°.

A complete turn is 360°

Angles around a point, with no gap and no overlap, total 360°. Include every marked sector exactly once, even if some of it lies below the point or outside the smaller opening between two rays.

Three sectors fill the turn

The marked sectors a, b and c surround O without a gap or overlap.

Angle a is 95° and angle b is 140°. Find c from the complete turn.
View figure data
Given geometry and angles
GivenInformation
GeometryThe marked sectors cover a complete turn without gaps or overlap.
Angle a95°
Angle b140°

The total is 360°, so c = 360°-95°-140° = 125°. Check 95° + 140° + 125° = 360°. Subtracting from 180° would account for only half a turn.

Check your understanding 1 Follow the marked arc

The smaller opening a is 60°. The arc b follows the rest of the turn, not the same smaller opening again. State the total of a and b, then find b.

Two rays make two sectors

The short arc a and the long arc b together go all the way around O.

Angle a is 60°. The long arc b marks a reflex angle.
View figure data
Given geometry and angles
GivenInformation
GeometryThe marked sectors cover a complete turn without gaps or overlap.
Angle a60°
Show answer

The total is 360°. Angle b = 360°-60° = 300°. Its arc goes the long way around O. Using 180° would give 120°, which is neither the given smaller opening nor the marked reflex sector.

Opposite angles at intersecting lines

When two straight lines intersect, vertically opposite angles are equal. To explain why, call one angle a and its neighbour b. Both a + b and the opposite angle plus b equal 180°, so the opposite angle equals a. This is a property of the two intersecting lines, not of opposite corners in any shape.

Check your understanding 2 Use a reason for each step

Lines AOC and BOD are straight and intersect at O. Angle a is 73°. Find b, c and d, then check the full turn.

Read opposite and adjacent sectors

The marked sectors are a, b, c and d in order around O. AOC and BOD are straight lines.

Angle a is 73°. Angles a and c are across the vertex; b and d are adjacent to a.
View figure data
Given geometry and angles
GivenInformation
GeometryB, O and D lie on a straight line; A, O and C lie on a straight line.
Angle a73°
Show answer

The vertically opposite angle c is 73°. Each adjacent angle is 180°-73° = 107°, so b and d are both 107°. The four sum to 73° + 107° + 73° + 107° = 360°. A total check is useful, but it does not replace identifying which angles are opposite and which are adjacent.

Change the diagram condition

Check your understanding 3 Choose the total from the geometry

OA is perpendicular to OC. Ray OB splits this right angle into a and b. Angle a is 34°. State their total and find b. Explain why subtracting from 180° would be wrong.

A right angle, not a straight turn

The given perpendicular outer rays form a right angle. OB lies inside it.

Angle a is 34°. Use the stated perpendicular relationship.
View figure data
Given geometry and angles
GivenInformation
GeometryOA is perpendicular to OC.
Angle a34°
Show answer

The total is 90°, so b = 90°-34° = 56°. The outer rays form a right angle. Two touching angles add to 180° only when they fill a straight opening.

Check your understanding 4 Notice when information is missing

Two adjacent angles share one ray, and one angle is 68°. Is the other necessarily 112°? State the extra condition that would make that calculation valid.

Show answer

No. Touching angles do not have a fixed total by themselves. The outer rays must form a straight line and the two angles must fill that straight opening without overlap. Only then does 180°-68° = 112° follow.

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