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Shapes and Angles - Angles in Shapes and Names

Grade 5CBSE

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

🔑Concepts

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An angle is formed when two rays meet at a common endpoint called a vertex. The space between these rays is measured in degrees (∘^{\circ}). Basic types include acute (<90∘< 90^{\circ}), right (90∘90^{\circ}), obtuse (>90∘> 90^{\circ} and <180∘< 180^{\circ}), and straight (180∘180^{\circ}).

Diagram showing an angle with a vertex and two rays.
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Closed shapes with straight sides are called polygons. The number of sides determines the name of the shape: 3 sides is a Triangle, 4 sides is a Quadrilateral (like squares or rectangles), 5 sides is a Pentagon, and 6 sides is a Hexagon.

A regular pentagon showing 5 sides and 5 angles.
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Angles in letters and numbers: Many capital letters and digits contain hidden angles. For example, the letter 'L' forms a 90∘90^{\circ} (right) angle, while the letter 'V' forms an acute angle.

Letter L showing a 90 degree right angle.
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Changing shapes: Using flexible joints (like matchsticks and rubber tubes), you can see that changing the angles of a shape can change the shape itself without changing the side lengths. This is why triangles are used in bridges; they are the only polygon that does not change shape when pressed.

📐Formulae

Sum of interior angles of a triangle = 180∘180^{\circ}

Sum of interior angles of a quadrilateral = 360∘360^{\circ}

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

💡Examples

Problem 1:

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

Solution:

Step 1: Add the known angles: 45∘+75∘=120∘45^{\circ} + 75^{\circ} = 120^{\circ}.\nStep 2: We know the sum of all angles in a triangle is 180∘180^{\circ}.\nStep 3: Subtract the sum of the known angles from 180∘180^{\circ}: 180∘−120∘=60∘180^{\circ} - 120^{\circ} = 60^{\circ}.

Explanation:

Since the total sum of angles in any triangle must be 180∘180^{\circ}, we find the missing value by subtracting the known parts from the total.

Problem 2:

A shape has four internal angles. Three of the angles are 90∘90^{\circ} each. What is the value of the fourth angle, and what type of angle is it?

Solution:

Step 1: Identify the shape as a quadrilateral since it has 4 angles.\nStep 2: Calculate the sum of the three known angles: 90∘+90∘+90∘=270∘90^{\circ} + 90^{\circ} + 90^{\circ} = 270^{\circ}.\nStep 3: The total sum for a quadrilateral is 360∘360^{\circ}. Subtract the known sum: 360∘−270∘=90∘360^{\circ} - 270^{\circ} = 90^{\circ}.\nStep 4: An angle of 90∘90^{\circ} is called a Right Angle.

Explanation:

We use the rule that a 4-sided shape's angles add up to 360∘360^{\circ} to find the missing corner. Since the result is exactly 90∘90^{\circ}, it is a right angle.

Problem 3:

Identify the number of right angles in the given shape of a staircase step (a simple L-shaped polygon).

An L-shaped polygon with two interior right angles marked.

Solution:

The shape has 2 right angles.

Explanation:

Looking at the interior corners of the L-shape, there is one 90∘90^{\circ} angle at the inner corner and another 90∘90^{\circ} angle at the base. In this specific polygon, we count the corners that form a square shape.

Problem 4:

In the following figure, identify the type of angle marked as xx. Is it acute, obtuse, or a right angle?

An obtuse angle marked x.

Solution:

xx is an obtuse angle.

Explanation:

By observing the opening of the rays, the angle xx is clearly wider than a 90∘90^{\circ} 'L' shape but smaller than a straight line. Therefore, it is an obtuse angle.