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Geometry - Geometrical Terms

Grade 9IGCSE

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

🔑Concepts

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Vertical angles are the opposite angles formed by the intersection of two straight lines. They are always equal.

Diagram showing vertically opposite angles a and b are equal.
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When a transversal intersects two parallel lines, alternate angles (forming a 'Z' shape) are equal.

Diagram showing alternate interior angles x and y between parallel lines.
•

Corresponding angles (forming an 'F' shape) are equal when a transversal cuts parallel lines.

Diagram showing corresponding angles c1 and c2.
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Co-interior angles (forming a 'C' or 'U' shape) between parallel lines are supplementary, meaning they sum to 180∘180^\circ.

📐Formulae

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

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

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

Sum of exterior angles of any polygon=360∘Sum\ of\ exterior\ angles\ of\ any\ polygon = 360^\circ

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

Each exterior angle of a regular n−sided polygon=360∘nEach\ exterior\ angle\ of\ a\ regular\ n-sided\ polygon = \frac{360^\circ}{n}

💡Examples

Problem 1:

Find the value of xx if three angles on a straight line are 2x2x, 3x3x, and 40∘40^\circ.

Solution:

2x+3x+40∘=180∘2x + 3x + 40^\circ = 180^\circ 5x+40∘=180∘5x + 40^\circ = 180^\circ 5x=180∘−40∘5x = 180^\circ - 40^\circ 5x=140∘5x = 140^\circ x=140∘5x = \frac{140^\circ}{5} x=28∘x = 28^\circ

Explanation:

Since the angles lie on a straight line, their sum must be equal to 180∘180^\circ. We set up an algebraic equation and solve for xx.

Problem 2:

Calculate the size of each interior angle of a regular hexagon.

Solution:

A hexagon has n=6n = 6 sides. Sum of interior angles=(6−2)×180∘Sum\ of\ interior\ angles = (6 - 2) \times 180^\circ Sum=4×180∘=720∘Sum = 4 \times 180^\circ = 720^\circ Each interior angle=720∘6=120∘Each\ interior\ angle = \frac{720^\circ}{6} = 120^\circ

Explanation:

First, find the total sum of interior angles using the formula (n−2)×180∘(n-2) \times 180^\circ. Then, divide by the number of sides (nn) because it is a regular polygon.

Problem 3:

In a pair of parallel lines cut by a transversal, one of the co-interior angles is 75∘75^\circ. Find the size of the other co-interior angle yy.

Solution:

y+75∘=180∘y + 75^\circ = 180^\circ y=180∘−75∘y = 180^\circ - 75^\circ y=105∘y = 105^\circ

Explanation:

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

Problem 4:

In the following diagram, lines L1L1 and L2L2 are parallel. Find the value of angle aa.

Diagram with two parallel lines and a transversal showing co-interior angles 115 and a.

Solution:

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

Explanation:

The angles shown are co-interior angles between parallel lines. In geometry, co-interior angles sum to 180∘180^\circ. Therefore, we subtract the known angle from 180∘180^\circ to find the missing angle.

Problem 5:

Calculate the value of yy in the diagram where four angles meet at a central point.

Diagram showing four angles meeting at a point with values y, 90, 120, and 85 degrees.

Solution:

y+120∘+90∘+85∘=360∘y + 120^\circ + 90^\circ + 85^\circ = 360^\circ y+295∘=360∘y + 295^\circ = 360^\circ y=360∘−295∘y = 360^\circ - 295^\circ y=65∘y = 65^\circ

Explanation:

Angles at a point must sum to exactly 360∘360^\circ. By summing the three known angles and subtracting from 360∘360^\circ, we determine the unknown angle yy.