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Geometry and Measure - Angle properties of triangles and quadrilaterals

Grade 7IGCSE

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.

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Angles around a point always sum to 360∘360^\circ.

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Vertically opposite angles are equal.

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The sum of interior angles in any triangle is exactly 180∘180^\circ.

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In an equilateral triangle, all angles are 60∘60^\circ.

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In an isosceles triangle, the two angles opposite the equal sides are equal.

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The exterior angle of a triangle is equal to the sum of the two opposite interior angles.

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The sum of interior angles in any quadrilateral is exactly 360∘360^\circ.

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In a parallelogram and rhombus, opposite angles are equal.

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In a kite, one pair of opposite angles (between the unequal sides) are equal.

📐Formulae

Sum of angles in a triangle: a+b+c=180∘a + b + c = 180^\circ

Sum of angles in a quadrilateral: a+b+c+d=360∘a + b + c + d = 360^\circ

Exterior angle of a triangle: ext=int1+int2ext = int_1 + int_2

Sum of interior angles of a polygon with nn sides: (n−2)×180∘(n - 2) \times 180^\circ

💡Examples

Problem 1:

In an isosceles triangle ABCABC, the angle at the vertex AA is 40∘40^\circ. Find the size of the two base angles BB and CC.

Solution:

70∘70^\circ

Explanation:

The sum of angles in a triangle is 180∘180^\circ. Subtract the vertex angle: 180∘−40∘=140∘180^\circ - 40^\circ = 140^\circ. Since it is an isosceles triangle, the base angles are equal. Therefore, each base angle is 140∘÷2=70∘140^\circ \div 2 = 70^\circ.

Problem 2:

A quadrilateral has three angles measuring 110∘110^\circ, 80∘80^\circ, and 75∘75^\circ. Calculate the size of the fourth angle.

Solution:

95∘95^\circ

Explanation:

The sum of interior angles in a quadrilateral is 360∘360^\circ. Add the known angles: 110∘+80∘+75∘=265∘110^\circ + 80^\circ + 75^\circ = 265^\circ. Subtract this sum from 360∘360^\circ: 360∘−265∘=95∘360^\circ - 265^\circ = 95^\circ.

Problem 3:

In triangle PQRPQR, the side QRQR is extended to a point SS. If angle PQR=50∘PQR = 50^\circ and angle QPR=60∘QPR = 60^\circ, find the exterior angle PRSPRS.

Solution:

110∘110^\circ

Explanation:

According to the exterior angle property, the exterior angle of a triangle is equal to the sum of the two interior opposite angles. Thus, angle PRS=anglePQR+angleQPR=50∘+60∘=110∘PRS = angle PQR + angle QPR = 50^\circ + 60^\circ = 110^\circ.