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Geometry - Angle properties on a line and at a point

Grade 6Cambridge (IGCSE)

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

🔑Concepts

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Angles on a straight line always sum to 180∘180^\circ. If a line is split into multiple angles, their total must be exactly half a full turn.

A straight line split into two angles a and b summing to 180 degrees
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Angles at a point (around a center) always sum to 360∘360^\circ, representing a full rotation.

Three angles meeting at a central point summing to 360 degrees
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Vertically opposite angles are equal. When two straight lines intersect, the angles opposite each other at the vertex have the same measure.

Two intersecting lines showing equal vertically opposite angles A and B
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A right angle is indicated by a square symbol and represents exactly 90∘90^\circ.

📐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.

Problem 4:

Calculate the value of angle xx in the diagram where three angles on a straight line are 55∘55^\circ, xx, and 40∘40^\circ.

Straight line divided into three angles: 55, x, and 40 degrees

Solution:

55∘+x+40∘=180∘55^\circ + x + 40^\circ = 180^\circ 95∘+x=180∘95^\circ + x = 180^\circ x=180∘−95∘x = 180^\circ - 95^\circ x=85∘x = 85^\circ

Explanation:

Since the angles lie on a straight line, their sum must be 180∘180^\circ. We add the known angles and subtract the sum from 180∘180^\circ to find the unknown.

Problem 5:

Five angles meet at a point. Four of the angles are 70∘70^\circ, 80∘80^\circ, 60∘60^\circ, and 90∘90^\circ. Find the size of the fifth angle zz.

Five angles around a point labeled 90, 70, 80, 60, and z

Solution:

70∘+80∘+60∘+90∘+z=360∘70^\circ + 80^\circ + 60^\circ + 90^\circ + z = 360^\circ 300∘+z=360∘300^\circ + z = 360^\circ z=360∘−300∘z = 360^\circ - 300^\circ z=60∘z = 60^\circ

Explanation:

The sum of all angles around a point is 360∘360^\circ. Adding the four given angles gives 300∘300^\circ. Subtracting this from 360∘360^\circ gives the remaining angle zz.