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Lines and Angles - Special Types of Angles

Grade 6CBSE

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

🔑Concepts

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Complementary Angles: Two angles are called complementary if the sum of their measures is 90∘90^\circ. Each angle is called the complement of the other.

Two angles x and y forming a right angle.
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Supplementary Angles: Two angles are called supplementary if the sum of their measures is 180∘180^\circ. Each angle is called the supplement of the other.

Two supplementary angles on a straight line.
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Adjacent Angles: Two angles that have a common vertex and a common arm, but no common interior points, are called adjacent angles.

Diagram showing adjacent angles with a common vertex and common arm.
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Linear Pair: A pair of adjacent angles is called a linear pair if their non-common arms are opposite rays. The sum of angles in a linear pair is always 180∘180^\circ.

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Vertically Opposite Angles: When two lines intersect, the angles formed opposite to each other at the vertex are called vertically opposite angles. They are always equal.

Two intersecting lines showing vertically opposite angles 1, 2, 3, and 4.

📐Formulae

Complement of ∠A=90∘−∠A\text{Complement of } \angle A = 90^\circ - \angle A

Supplement of ∠A=180∘−∠A\text{Supplement of } \angle A = 180^\circ - \angle A

∠1+∠2=180∘ (Linear Pair Property)\angle 1 + \angle 2 = 180^\circ \text{ (Linear Pair Property)}

If lines L1 and L2 intersect, then ∠V.O.A1=∠V.O.A2\text{If lines } L_1 \text{ and } L_2 \text{ intersect, then } \angle \text{V.O.A}_1 = \angle \text{V.O.A}_2

💡Examples

Problem 1:

Find the measure of an angle which is equal to its complement.

Solution:

Let the angle be xx. Since it is equal to its complement: x=90∘−xx = 90^\circ - x x+x=90∘x + x = 90^\circ 2x=90∘2x = 90^\circ x=90∘2=45∘x = \frac{90^\circ}{2} = 45^\circ

Explanation:

If an angle is equal to its complement, their sum (x+xx + x) must equal 90∘90^\circ.

Problem 2:

Find the supplement of 125∘125^\circ.

Solution:

To find the supplement, we subtract the given angle from 180∘180^\circ: 180−12555\begin{array}{r} 180 \\ - 125 \\ \hline 55 \end{array} The supplement is 55∘55^\circ.

Explanation:

Supplementary angles add up to 180∘180^\circ. Subtracting 125∘125^\circ from 180∘180^\circ gives the remaining angle.

Problem 3:

In the given figure, two lines intersect. If one of the vertically opposite angles is 70∘70^\circ, find the other three angles.

Solution:

Let the four angles be ∠1,∠2,∠3,\angle 1, \angle 2, \angle 3, and ∠4\angle 4. Let ∠1=70∘\angle 1 = 70^\circ.

  1. ∠3=∠1=70∘\angle 3 = \angle 1 = 70^\circ (Vertically Opposite Angles are equal).
  2. ∠1+∠2=180∘\angle 1 + \angle 2 = 180^\circ (Linear Pair).
  3. ∠2=180∘−70∘=110∘\angle 2 = 180^\circ - 70^\circ = 110^\circ.
  4. ∠4=∠2=110∘\angle 4 = \angle 2 = 110^\circ (Vertically Opposite Angles).

Explanation:

We use the property that vertically opposite angles are equal and adjacent angles on a straight line form a linear pair (sum to 180∘180^\circ).

Problem 4:

In the following figure, if ∠AOC\angle AOC and ∠BOC\angle BOC form a linear pair and the measure of ∠AOC=2x−10∘\angle AOC = 2x - 10^\circ and ∠BOC=3x+20∘\angle BOC = 3x + 20^\circ, find the value of xx.

A straight line AB with a ray OC forming two angles AOC and BOC.

Solution:

Since ∠AOC\angle AOC and ∠BOC\angle BOC form a linear pair, their sum must be 180∘180^\circ. (2x−10∘)+(3x+20∘)=180∘(2x - 10^\circ) + (3x + 20^\circ) = 180^\circ 5x+10∘=180∘5x + 10^\circ = 180^\circ 5x=180∘−10∘5x = 180^\circ - 10^\circ 5x=170∘5x = 170^\circ x=170∘5x = \frac{170^\circ}{5} x=34∘x = 34^\circ

Explanation:

We use the Linear Pair Property which states that the sum of adjacent angles on a straight line is 180∘180^\circ. We set up an algebraic equation with the given expressions and solve for xx.

Problem 5:

Find the angle which is double its supplement.

Diagram representing an angle and its supplement on a straight line.

Solution:

Let the angle be xx. Then its supplement is (180∘−x)(180^\circ - x). According to the problem: x=2(180∘−x)x = 2(180^\circ - x) x=360∘−2xx = 360^\circ - 2x x+2x=360∘x + 2x = 360^\circ 3x=360∘3x = 360^\circ x=360∘3x = \frac{360^\circ}{3} x=120∘x = 120^\circ The angle is 120∘120^\circ.

Explanation:

Supplementary angles add up to 180∘180^\circ. By representing the angle as xx, we express the supplement in terms of xx and solve the resulting linear equation based on the given ratio.