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Geometry - Calculating missing angles on a straight line and at a point

Grade 5Cambridge (IGCSE)

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

🔑Concepts

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Angles on a straight line always add up to 180∘180^\circ. This is known as the sum of angles on a straight line or supplementary angles.

Diagram showing two angles a and b on a straight line summing to 180 degrees.
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Angles at a point (around a full circle) always add up to 360∘360^\circ.

Three angles meeting at a central point summing to 360 degrees.
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A right angle is exactly 90∘90^\circ and is usually marked with a small square symbol.

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To find a missing angle, subtract the sum of the known angles from the total (180∘180^\circ for a line, 360∘360^\circ for a point).

📐Formulae

Angle a+Angle b=180∘\text{Angle } a + \text{Angle } b = 180^\circ (for two angles on a straight line)

∑Angles on a straight line=180∘\sum \text{Angles on a straight line} = 180^\circ

∑Angles at a point=360∘\sum \text{Angles at a point} = 360^\circ

💡Examples

Problem 1:

A straight line is divided into two angles. One angle is 125∘125^\circ. Find the size 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 85∘85^\circ. Calculate the fourth angle yy.

Solution:

y=75∘y = 75^\circ

Explanation:

Angles around a point sum to 360∘360^\circ. First, add the known angles: 90+110+85=285∘90 + 110 + 85 = 285^\circ. Then, subtract from the total: 360∘−285∘=75∘360^\circ - 285^\circ = 75^\circ.

Problem 3:

On a straight line, there are three angles: 40∘40^\circ, a right angle, and an unknown angle zz. Find zz.

Solution:

z=50∘z = 50^\circ

Explanation:

A right angle is 90∘90^\circ. The total for a straight line is 180∘180^\circ. So, z=180∘−(40∘+90∘)=180∘−130∘=50∘z = 180^\circ - (40^\circ + 90^\circ) = 180^\circ - 130^\circ = 50^\circ.

Problem 4:

Three angles lie on a straight line. Two of the angles are 55∘55^\circ and 72∘72^\circ. Calculate the value of the third angle aa.

A straight line split into three angles: 55 degrees, 72 degrees, and angle a.

Solution:

  1. Sum the known angles: 55∘+72∘=127∘55^\circ + 72^\circ = 127^\circ
  2. Subtract from 180∘180^\circ: 180∘−127∘=53∘180^\circ - 127^\circ = 53^\circ
  3. Therefore, a=53∘a = 53^\circ.

Explanation:

Since the angles are on a straight line, their total must be 180∘180^\circ. By subtracting the known values from 180∘180^\circ, we find the remaining angle.

Problem 5:

Find the missing angle xx in the diagram where four angles meet at a point. The known angles are 120∘120^\circ, 80∘80^\circ, and 45∘45^\circ.

Four angles meeting at a point labeled 120, 80, 45, and x.

Solution:

  1. Sum the known angles: 12080+45245\begin{array}{r} 120 \\ 80 \\ + 45 \\ \hline 245 \end{array}
  2. Subtract the sum from 360∘360^\circ: 360−245115\begin{array}{r} 360 \\ - 245 \\ \hline 115 \end{array}
  3. So, x=115∘x = 115^\circ.

Explanation:

The sum of all angles around a point is 360∘360^\circ. Adding the three given angles gives 245∘245^\circ. Subtracting this from 360∘360^\circ leaves 115∘115^\circ for the missing angle.