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Geometry and Measurement - Angles on a Line, at a Point, and in Triangles

Grade 7IB

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

🔑Concepts

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Angles on a Straight Line: When two or more angles meet at a point on a straight line, their measures always add up to 180∘180^{\circ}. Visually, these angles together form a perfect semi-circle resting on the flat edge of the line.

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Angles at a Point: The sum of all angles that meet at a single vertex to complete a full turn is 360∘360^{\circ}. In a diagram, these angles radiate out from the center point and together form a complete circle.

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Vertically Opposite Angles: When two straight lines intersect, the angles directly across from each other at the vertex are equal. Visually, these form an 'X' shape, where the top angle is equal to the bottom angle and the left angle is equal to the right angle.

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Interior Angles of a Triangle: The three angles inside any triangle always sum to 180∘180^{\circ}. If you visualize cutting the three corners of a paper triangle and placing them side-by-side, their edges would perfectly align to form a straight line.

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Exterior Angle of a Triangle: An exterior angle is formed when one side of a triangle is extended outwards. The measure of this exterior angle is equal to the sum of the two interior angles that are opposite to it.

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Isosceles Triangles: This type of triangle has at least two equal sides and two equal angles, known as base angles. Visually, the triangle appears symmetrical, with the two equal angles located at the base of the two equal sides.

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Equilateral Triangles: In an equilateral triangle, all three sides are of equal length and all three interior angles are equal. Since the total sum is 180∘180^{\circ}, each angle in an equilateral triangle must be exactly 60∘60^{\circ}.

📐Formulae

Sum of angles on a line=180∘\text{Sum of angles on a line} = 180^{\circ}

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

∠A+∠B+∠C=180∘ (Triangle interior sum)\angle A + \angle B + \angle C = 180^{\circ} \text{ (Triangle interior sum)}

Exterior angle (d)=∠a+∠b (where a,b are opposite interior angles)\text{Exterior angle } (d) = \angle a + \angle b \text{ (where } a, b \text{ are opposite interior angles)}

Angle in an equilateral triangle=180∘3=60∘\text{Angle in an equilateral triangle} = \frac{180^{\circ}}{3} = 60^{\circ}

💡Examples

Problem 1:

Calculate the value of xx if three angles on a straight line are given as xx, 42∘42^{\circ}, and 88∘88^{\circ}.

Solution:

  1. Use the property that angles on a straight line sum to 180∘180^{\circ}: x+42∘+88∘=180∘x + 42^{\circ} + 88^{\circ} = 180^{\circ} 2. Add the known values together: x+130∘=180∘x + 130^{\circ} = 180^{\circ} 3. Subtract 130∘130^{\circ} from both sides to solve for xx: x=180∘−130∘x = 180^{\circ} - 130^{\circ} 4. The result is: x=50∘x = 50^{\circ}

Explanation:

This problem is solved by identifying that the angles are supplementary because they sit on a straight line, meaning their total must be 180∘180^{\circ}.

Problem 2:

An isosceles triangle has a vertex angle (the angle between the two equal sides) of 40∘40^{\circ}. What is the size of each base angle?

Solution:

  1. Let each base angle be represented by bb. Since it is an isosceles triangle, both base angles are equal. 2. Set up the triangle sum equation: 40∘+b+b=180∘40^{\circ} + b + b = 180^{\circ} 3. Simplify the equation: 40∘+2b=180∘40^{\circ} + 2b = 180^{\circ} 4. Subtract 40∘40^{\circ} from both sides: 2b=140∘2b = 140^{\circ} 5. Divide by 22 to find the value of one base angle: b=140∘2=70∘b = \frac{140^{\circ}}{2} = 70^{\circ}

Explanation:

In an isosceles triangle, we subtract the vertex angle from 180∘180^{\circ} and divide the remainder by 22 because the two remaining angles are equal.