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Geometry - Angle properties (parallel lines and polygons)

Grade 11A Level

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

🔑Concepts

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When a transversal line crosses two parallel lines, several angle relationships are formed: Corresponding angles are equal (forming an 'F' shape), Alternate angles are equal (forming a 'Z' shape), and Co-interior angles sum to 180∘180^\circ (forming a 'C' or 'U' shape).

Parallel lines with a transversal showing corresponding and alternate angles.
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For any nn-sided polygon, the sum of the exterior angles is always 360∘360^\circ. For a regular polygon, each exterior angle is calculated as 360∘n\frac{360^\circ}{n}.

Pentagon showing an extended side to illustrate an exterior angle.
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The sum of interior angles of a polygon with nn sides is (n−2)×180∘(n-2) \times 180^\circ. A regular polygon has all interior angles equal.

A quadrilateral divided into two triangles to show the sum of interior angles.
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Vertically opposite angles are equal whenever two straight lines intersect.

📐Formulae

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

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

Sum of exterior angles=360∘\text{Sum of exterior angles} = 360^\circ

Individual exterior angle of a regular polygon=360∘n\text{Individual exterior angle of a regular polygon} = \frac{360^\circ}{n}

Interior angle+Exterior angle=180∘\text{Interior angle} + \text{Exterior angle} = 180^\circ

💡Examples

Problem 1:

Two parallel lines are intersected by a transversal. If one of the co-interior angles is 115∘115^\circ, find the value of its pair.

Solution:

180∘−115∘=65∘180^\circ - 115^\circ = 65^\circ

Explanation:

Co-interior angles (also known as allied angles) between parallel lines are supplementary, meaning they add up to 180°.

Problem 2:

Calculate the number of sides of a regular polygon if each interior angle is 144∘144^\circ.

Solution:

n=10n = 10

Explanation:

First, find the exterior angle: 180∘−144∘=36∘180^\circ - 144^\circ = 36^\circ. Since the sum of exterior angles is 360∘360^\circ, use the formula n=360/exterior anglen = 360 / \text{exterior angle}. Therefore, n=360/36=10n = 360 / 36 = 10.

Problem 3:

A pentagon has four interior angles of 100∘,110∘,120∘,100^\circ, 110^\circ, 120^\circ, and 90∘90^\circ. Find the fifth interior angle.

Solution:

120∘120^\circ

Explanation:

First, find the total sum of interior angles for a pentagon (n=5n=5): (5−2)×180∘=540∘(5-2) \times 180^\circ = 540^\circ. Subtract the known angles from the total: 540−(100+110+120+90)=540−420=120∘540 - (100 + 110 + 120 + 90) = 540 - 420 = 120^\circ.

Problem 4:

In the diagram, ABAB is parallel to CDCD. Find the value of xx.

Parallel lines AB and CD with co-interior angles 130 and 2x+10.

Solution:

130+(2x+10)=180130 + (2x + 10) = 180 140+2x=180140 + 2x = 180 2x=402x = 40 x=20x = 20

Explanation:

The two angles marked are co-interior angles because they lie between the parallel lines on the same side of the transversal. Co-interior angles sum to 180∘180^\circ. We set up an equation adding the two expressions to 180180 and solve for xx.

Problem 5:

Calculate the size of one interior angle of a regular octagon.

A regular octagon showing one interior angle labeled x.

Solution:

n=8n = 8 Sum=(8−2)×180=1080∘\text{Sum} = (8 - 2) \times 180 = 1080^\circ Interior angle=10808=135∘\text{Interior angle} = \frac{1080}{8} = 135^\circ

Explanation:

An octagon has 8 sides. First, calculate the total sum of all interior angles using the formula (n−2)×180∘(n-2) \times 180^\circ. Since it is a regular octagon, all interior angles are equal, so divide the total sum by the number of sides (8).