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Geometry - Angle properties on a line and at a point

Grade 6IGCSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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Angles on a straight line: The sum of all angles that meet at a point on a straight line is exactly 180∘180^\circ. These are also called supplementary angles.

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Angles at a point: The sum of all angles around a single point (a full turn) is exactly 360∘360^\circ.

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Vertically opposite angles: When two straight lines intersect, the angles opposite each other at the vertex are equal.

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Right angles: A right angle is exactly 90∘90^\circ, often indicated by a small square symbol in the corner.

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Perpendicular lines: Lines that meet at a 90∘90^\circ angle.

📐Formulae

Sum of angles on a line: a+b+c=180∘a + b + c = 180^\circ

Sum of angles at a point: a+b+c+d=360∘a + b + c + d = 360^\circ

Vertically opposite angles: Angle A=Angle B\text{Angle } A = \text{Angle } B

💡Examples

Problem 1:

Two angles lie on a straight line. One angle is 125∘125^\circ. Find the value of the missing angle xx.

Solution:

x=55∘x = 55^\circ

Explanation:

Since angles on a straight line add up to 180∘180^\circ, we calculate 180∘−125∘=55∘180^\circ - 125^\circ = 55^\circ.

Problem 2:

Four angles meet at a point. Three of the angles are 90∘90^\circ, 110∘110^\circ, and 75∘75^\circ. Find the fourth angle yy.

Solution:

y=85∘y = 85^\circ

Explanation:

Angles around a point sum to 360∘360^\circ. First, add the known angles: 90+110+75=275∘90 + 110 + 75 = 275^\circ. Then, subtract from the total: 360∘−275∘=85∘360^\circ - 275^\circ = 85^\circ.

Problem 3:

Two straight lines intersect to form an X-shape. If one of the angles is 42∘42^\circ, what is the size of the angle vertically opposite to it?

Solution:

42∘42^\circ

Explanation:

Vertically opposite angles are always equal. Therefore, the angle directly across the intersection is also 42∘42^\circ.