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Geometry - Lines and Angles: Pairs of angles (Complementary, Supplementary, Adjacent, Vertically Opposite)

Grade 7ICSE

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

πŸ”‘Concepts

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Complementary Angles: Two angles are said to be complementary if the sum of their measures is exactly 90∘90^{\circ}. Visually, if a right angle (often indicated by a small square symbol at the vertex) is divided by a ray into two parts, those two angles are complements of each other.

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Supplementary Angles: Two angles are supplementary if the sum of their measures is 180∘180^{\circ}. When two supplementary angles are placed side-by-side, their exterior arms form a straight line. For example, if a ray stands on a straight line, the two angles formed are supplementary.

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Adjacent Angles: These are two angles that share a common vertex and a common arm, but do not overlap (they have no common interior points). The non-common arms lie on opposite sides of the common arm, appearing like two side-by-side slices of a pizza.

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Linear Pair: A linear pair consists of two adjacent angles whose non-common arms are opposite rays, forming a straight line. By the Linear Pair Axiom, the sum of these two angles is always 180∘180^{\circ}. This looks like a 'T' or a slanted ray sitting on a flat horizontal line.

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Vertically Opposite Angles: When two straight lines intersect at a single point, they form four angles. The pairs of angles that are opposite to each other at the vertex (not sharing an arm) are called vertically opposite angles. They resemble an 'X' shape, and the angles facing each other are always equal in measure.

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Angles at a Point: The sum of all angles around a single point is 360∘360^{\circ}, which represents a complete rotation or a full circle. If multiple rays originate from the same vertex, the sum of all resulting adjacent angles is 360∘360^{\circ}.

πŸ“Formulae

Complementary Condition: ∠A+∠B=90∘\angle A + \angle B = 90^{\circ}

Supplementary Condition: ∠A+∠B=180∘\angle A + \angle B = 180^{\circ}

Measure of Complement: ComplementΒ ofΒ x=(90βˆ˜βˆ’x)\text{Complement of } x = (90^{\circ} - x)

Measure of Supplement: SupplementΒ ofΒ x=(180βˆ˜βˆ’x)\text{Supplement of } x = (180^{\circ} - x)

Vertically Opposite Angles: If lines ABAB and CDCD intersect at OO, then ∠AOC=∠BOD\angle AOC = \angle BOD and ∠AOD=∠BOC\angle AOD = \angle BOC

Sum of Angles in a Linear Pair: θ1+θ2=180∘\theta_1 + \theta_2 = 180^{\circ}

πŸ’‘Examples

Problem 1:

Find the measure of an angle which is 24∘24^{\circ} less than its supplement.

Solution:

  1. Let the angle be xx. \n2. Its supplement is (180βˆ˜βˆ’x)(180^{\circ} - x). \n3. According to the question: x=(180βˆ˜βˆ’x)βˆ’24∘x = (180^{\circ} - x) - 24^{\circ}. \n4. Simplify: x=156βˆ˜βˆ’xx = 156^{\circ} - x. \n5. Move xx to one side: x+x=156βˆ˜β€…β€ŠβŸΉβ€…β€Š2x=156∘x + x = 156^{\circ} \implies 2x = 156^{\circ}. \n6. Solve for xx: x=156∘2=78∘x = \frac{156^{\circ}}{2} = 78^{\circ}.

Explanation:

We use the definition of supplementary angles (sum = 180∘180^{\circ}) and set up a linear equation based on the given relationship.

Problem 2:

In an 'X' shape formed by two intersecting lines, one of the angles is 2x+10∘2x + 10^{\circ} and its vertically opposite angle is 3xβˆ’20∘3x - 20^{\circ}. Find the value of xx and the measure of these angles.

Solution:

  1. Since vertically opposite angles are equal, we set the expressions equal to each other: 2x+10∘=3xβˆ’20∘2x + 10^{\circ} = 3x - 20^{\circ}. \n2. Subtract 2x2x from both sides: 10∘=xβˆ’20∘10^{\circ} = x - 20^{\circ}. \n3. Add 20∘20^{\circ} to both sides: x=30∘x = 30^{\circ}. \n4. Substitute xx back into either expression: 2(30∘)+10∘=60∘+10∘=70∘2(30^{\circ}) + 10^{\circ} = 60^{\circ} + 10^{\circ} = 70^{\circ}. \n5. Therefore, both vertically opposite angles are 70∘70^{\circ}.

Explanation:

We apply the property that vertically opposite angles are always equal to solve for the unknown variable xx and then calculate the specific angle measure.