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Geometry - Angle Properties (Polygons and Parallel Lines)

Grade 12IGCSE

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

🔑Concepts

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Parallel Lines: When a transversal intersects two parallel lines, alternate angles are equal (Z-shape), corresponding angles are equal (F-shape), and co-interior angles sum to 180° (C-shape).

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Vertically Opposite Angles: Angles formed opposite each other at the intersection of two straight lines are always equal.

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Sum of Interior Angles: For any n-sided polygon, the sum of all interior angles depends on the number of triangles it can be split into (n-2).

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Exterior Angles: The exterior angle of a polygon is formed by extending one of its sides. The sum of exterior angles for any convex polygon is always 360°.

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Regular Polygons: A polygon where all sides are of equal length and all interior angles are of equal measure.

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Interior-Exterior Relationship: At any vertex of a polygon, the interior angle and the exterior angle are supplementary (sum to 180°).

📐Formulae

Sum of interior angles = (n−2)×180∘(n - 2) \times 180^\circ

Each interior angle (regular polygon) = (n−2)×180∘n\frac{(n - 2) \times 180^\circ}{n}

Sum of exterior angles = 360∘360^\circ

Each exterior angle (regular polygon) = 360∘n\frac{360^\circ}{n}

Number of sides (n) = 360∘Exterior Angle\frac{360^\circ}{\text{Exterior Angle}}

Interior Angle + Exterior Angle = 180∘180^\circ

💡Examples

Problem 1:

A regular polygon has an interior angle of 150∘150^\circ. Calculate the number of sides (n) of this polygon.

Solution:

180∘−150∘=30∘180^\circ - 150^\circ = 30^\circ. n=360∘/30∘=12n = 360^\circ / 30^\circ = 12.

Explanation:

First, find the exterior angle using the supplementary rule (Interior + Exterior = 180°). Then, use the property that the sum of exterior angles is 360° divided by the measure of one exterior angle to find the number of sides.

Problem 2:

In a pentagon, four of the interior angles are 110∘,90∘,120∘,110^\circ, 90^\circ, 120^\circ, and 100∘100^\circ. Find the size of the fifth angle.

Solution:

Sum = (5−2)×180∘=540∘(5-2) \times 180^\circ = 540^\circ. Fifth angle = 540∘−(110∘+90∘+120∘+100∘)=540∘−420∘=120∘540^\circ - (110^\circ + 90^\circ + 120^\circ + 100^\circ) = 540^\circ - 420^\circ = 120^\circ.

Explanation:

Calculate the total sum of interior angles for a pentagon (n=5). Subtract the sum of the known four angles from the total sum to find the remaining angle.

Problem 3:

Line L1L_1 and L2L_2 are parallel. A transversal cuts them. If a pair of co-interior angles are represented by (2x+10)∘(2x + 10)^\circ and (3x+20)∘(3x + 20)^\circ, find the value of xx.

Solution:

(2x+10)+(3x+20)=180⇒5x+30=180⇒5x=150⇒x=30(2x + 10) + (3x + 20) = 180 \Rightarrow 5x + 30 = 180 \Rightarrow 5x = 150 \Rightarrow x = 30.

Explanation:

Co-interior angles between parallel lines are supplementary, meaning they add up to 180°. Set up an algebraic equation summing the two expressions to 180 and solve for x.