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Shape and Space - Measuring and Calculating Angles

Grade 6IB

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}. These are also known as supplementary angles when two angles add up to this total.

Diagram showing two angles a and b on a straight line adding up to 180 degrees.
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The sum of interior angles in any triangle is exactly 180∘180^{\circ}. This property allows us to find a missing angle if two are known.

Triangle ABC illustrating that the sum of interior angles is 180 degrees.
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Angles around a point sum to 360∘360^{\circ}, forming a full rotation.

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Vertically opposite angles are formed when two straight lines intersect. These angles are always equal to each other.

Two intersecting lines showing vertically opposite angles are equal.

📐Formulae

Sum of angles on a straight line=180∘\text{Sum of angles on a straight line} = 180^{\circ}

Sum of angles at a point=360∘\text{Sum of angles at a point} = 360^{\circ}

∠A+∠B+∠C=180∘ (for a triangle)\angle A + \angle B + \angle C = 180^{\circ} \text{ (for a triangle)}

∠A+∠B+∠C+∠D=360∘ (for a quadrilateral)\angle A + \angle B + \angle C + \angle D = 360^{\circ} \text{ (for a quadrilateral)}

Complementary Angles sum to 90∘\text{Complementary Angles sum to } 90^{\circ}

Supplementary Angles sum to 180∘\text{Supplementary Angles sum to } 180^{\circ}

💡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=180∘−125∘=55∘x = 180^{\circ} - 125^{\circ} = 55^{\circ}

Explanation:

Since angles on a straight line must sum to 180∘180^{\circ}, we subtract the known angle from 180∘180^{\circ} using vertical subtraction: 180−12555\begin{array}{r} 180 \\ - 125 \\ \hline 55 \end{array}

Problem 2:

In a triangle, two angles are 45∘45^{\circ} and 75∘75^{\circ}. Calculate the third angle yy.

Solution:

y=180∘−(45∘+75∘)=180∘−120∘=60∘y = 180^{\circ} - (45^{\circ} + 75^{\circ}) = 180^{\circ} - 120^{\circ} = 60^{\circ}

Explanation:

The sum of interior angles in a triangle is 180∘180^{\circ}. First, find the sum of the known angles (45+75=12045 + 75 = 120), then subtract from 180180.

Problem 3:

Four angles meet at a point. Three of the angles are 90∘90^{\circ}, 110∘110^{\circ}, and 40∘40^{\circ}. Find the fourth angle zz.

Solution:

z=360∘−(90∘+110∘+40∘)=360∘−240∘=120∘z = 360^{\circ} - (90^{\circ} + 110^{\circ} + 40^{\circ}) = 360^{\circ} - 240^{\circ} = 120^{\circ}

Explanation:

Angles around a point sum to 360∘360^{\circ}. We add the known angles together to get 240∘240^{\circ} and subtract this from 360∘360^{\circ}.

Problem 4:

Calculate the value of the missing angle aa in the given quadrilateral where three angles are 110∘110^{\circ}, 80∘80^{\circ}, and 70∘70^{\circ}.

A quadrilateral with three known angles and one unknown angle labeled a.

Solution:

  1. Recall that the sum of angles in a quadrilateral is 360∘360^{\circ}.
  2. Add the known angles: 110∘+80∘+70∘=260∘110^{\circ} + 80^{\circ} + 70^{\circ} = 260^{\circ}.
  3. Subtract the sum from 360∘360^{\circ}: a=360∘−260∘a = 360^{\circ} - 260^{\circ}
  4. a=100∘a = 100^{\circ}

Explanation:

Since any quadrilateral can be split into two triangles, its total internal angle sum is always 2×180∘=360∘2 \times 180^{\circ} = 360^{\circ}. By subtracting the known values from 360∘360^{\circ}, we find the remaining angle.

Problem 5:

Find the value of angle bb which is complementary to an angle of 35∘35^{\circ}.

A right angle divided into two parts, one labeled 35 degrees and the other labeled b.

Solution:

  1. Complementary angles sum to 90∘90^{\circ}.
  2. Set up the equation: b+35∘=90∘b + 35^{\circ} = 90^{\circ}
  3. Subtract 35∘35^{\circ} from both sides: b=90∘−35∘b = 90^{\circ} - 35^{\circ}
  4. b=55∘b = 55^{\circ}

Explanation:

Complementary angles form a right angle (90∘90^{\circ}). If one part is known, the other is the difference between 90∘90^{\circ} and the known angle.