Lines and angles

Lines and Angles
Important Definitions
Line:
A line has no thickness and has only length. A line can be extended infinitely through its both
ends.
Line segment: A line segment has a definite length, which means it has a definite
beginning point and a definite end point.
Angle – A figure formed by two rays having a common end point is called an angle. Rays are
called sides of the angle. Common end point which is shared by both the sides (rays) is called
vertex of the angle. Angles are usually measure in degree. Angle may be equal to 10 to 3600.
The symbol of angle is ∠.
Acute Angle – Angles less than 900 are called Acute Angles. For example – 200, 300, 850,
650, etc.
Obtuse Angle – Angles more than 900 are called Obtuse Angles. For example – 950, 1000,
800, 650, etc.
Right Angle – Angle equal to 900 is called Right Angle.

Straight Angle – Angle equal to 1800 is called Straight Angle.
Reflex Angles – Angles greater than 1800 are called reflex angles. For example 1900 is a
reflex angle.
Complementary Angles – When the sum of two angles is equal to 900, they are called
complementary angles. Both angles are called complementary to each others. For example
300 and 600, 400 and 500, 200 and 700, etc. are complementary angle since their sum is equal
to 900.
Example: Find the complementary angles for following angles:
(a) 15°
Solution: Let us assume the required complementary angle is x
So, 15° + x = 90°
Or, x = 90° - 15° = 75°

Hence, the required complementary angle = 75°
(b) 35°
So, 35° + x = 90°
Or, x = 90° - 35° = 55°
(c) 45°
So, 45° + x = 90°
Or, x = 90° - 45° = 45°
(d) 60°
So, 60° + x = 90°
Or, x = 90° - 60° = 30°
(e)70°
So, 70° + x = 90°
Or, x = 90° - 70° = 20°

Supplementary Angles - When the sum of two angles is equal to 1800, then they are called
supplementary angles. Both angles are called supplementary to each others. For example
1000 and 800, 300 and 1500, etc. are complementary angle.
Example: Find if the following angles make a pair of supplementary angles.
Solution: 60° + 120° = 180°
As the sum of these angles is equal to two right angles, so they make a pair of
supplementary angles.
Example: Find the supplementary angles for following angles:
(a)45°
Solution: Let us assume, the required angle = x
45° + x = 180°
Or, x = 180° - 45° = 135°
Hence, the required supplementary angle = 135°
(b) 25°
25° + x = 180°
Or, x = 180° - 25° = 155°

(c) 112°
112° + x = 180°
Or, x = 180° - 112° = 68°
(d) 130°
130° + x = 180°
Or, x = 180° - 130° = 50°
(e)78°
78° + x = 180°
Or, x = 180° - 78° = 102°
Adjacent Angles – When two angles have a common arm and common vertex and their non-
common arm are the either side of common arm, then they are called adjacent angles.
Example: Find the pairs of adjacent angles in the following figure.
Solution:
∠ 1 and ∠ 2 are adjacent to each other
∠ 2 and ∠ 3 are adjacent to each other

Example: Find the pairs of adjacent angles in the following figure.
Solution:
∠ APD and ∠ DPC are adjacent to each other
∠ DPC and ∠ CPB are adjacent to each other
Note: ∠ APD and ∠ CPB are not adjacent to each other, because they don’t have a common
arm in spite of having a common vertex.
Linear Pair of Angles
Two angles make a linear pair if their non-common arms are two opposite rays. In other
words, if the non-common arms of a pair of adjacent angles are in a straight line, these angles
make a linear pair.
Note: Two acute angles cannot make a linear pair because their sum will always be less than
180°. On the other hand, two right angles will always make a linear pair as their sum is equal
to 180°. It can also be said that angles of the linear pair are always supplementary to each
other.
Example: Find if following angles can make a linear pair.

Solution: 130° + 50° = 180°
Since the sum of these angles is equal to two right angles, so they can make a linear pair.
Example: Find if following angles can make a linear pair.
Solution: 110° + 70° = 180°
Since the sum of these angles is equal to two right angles, so they can make a linear pair.
Example: If following angles make a linear pair, find the value of q.

Conditions of adjacent angles –
Theyhavecommonvertex Non-common arms are the either side of common arm. Common
arm should be in the middle of rest two sides.
In the given figure ‘O’ is the common vertex and OB is the common arm. Hence, ∠a and ∠b
are called the adjacent angles.
Linear Pair - A pair of adjacent angles is called linear pair, if their two non-common arms
are opposite rays.
Here adjacent ∠A and ∠B are linear pair The angle sum of a linear pair is equal to 1800.
Vertical opposite angles: When two straight lines intersects, there are four angle formed.
The opposite angles are called vertically opposite angles. The vertically opposite angles are
equal.
In the given figure, ∠1 and ∠2,and ∠3 and ∠4 are called opposite angle.
Example: Find the values of x and y in following figure.

Example: An angle is equal to its complementary angle. What is the value of this angle?
Solution: Let us assume, the angle = x
Example: An angle is equal to its supplementary angle. What is the value of this angle?
Solution: Let us assume, the angle = x
Example: An angle is double its supplementary angle. What is the value of this angle?
Solution: Let us assume that the smaller angle = x, then the larger angle = 2

Example: An angle is double its complementary angle. What is the value of this angle?
Solution: Let us assume that the smaller angle = x, then the larger angle = 2x
Example: Find the value of x in each of the following figures.
Solution: The angles in these figures are on the same side of a line, so their sum is equal to
180°.

TRANSVERSAL: A line is called transversal line when it cut two lines at distinct points.
In the given picture AB is the transversal line which cut CD and EF at two different (distinct
points).
.
Transacting Lines:
When two lines pass through a common point, they are called as intersecting lines. The
common point, in this case, is called the point of intersection.
Parallel Lines: When two lines are as such that they don’t intersect anywhere; no matter in
which direction they are extended; they are called parallel lines.
Note: Two lines can be either intersecting or parallel.
Angles made by a transversal:
Understanding the concept of different angles; made by a transversal; is important for
understanding advanced concepts of geometry. Following are the names of different angles
formed when a transversal intersects two different lines.
(a) Interior angles: These are between the two lines which are being intersected by the
transversal, e.g. ∠ 3, ∠ 4, ∠ 5 and ∠ 6 are interior angles.

(b) Exterior angles: These are beyond the two lines which are being intersected by the
transversal, e.g. ∠ 1, ∠ 2, ∠ 7 and ∠ 8 are exterior angles.
(c) Pairs of corresponding angles: In the given figure, ∠ 1 and ∠ 5 make a pair of
corresponding angles. Similarly, other pairs of corresponding angles are; ∠ 2 and ∠ 6, ∠ 3
and ∠ 7, and ∠ 4 and ∠ 8.
(d) Pairs of alternate interior angles: In the given figure, ∠ 3 and ∠ 6 make a pair of alternate
interior angles. Similarly, ∠ 4 and ∠ 5 make another pair of alternate interior angles.
(e) Pairs of alternate exterior angles: In the given figure, ∠ 1 and ∠ 8 make a pair of alternate
exterior angles. Similarly, ∠ 2 and ∠ 7 make another pair of alternate exterior angles.
(f) Pairs of interior angles on the same side of transversal: In the given figure, ∠ 3 and ∠ 5
make a pair of interior angles on the same side of transversal. Similarly, ∠ 4 and ∠ 6 make
another pair.
Parallel Lines and transversal:
When a transversal intersects two parallel lines, then:
(a) Each pair of corresponding angles is composed of equal angles.
(b) Each pair of alternate interior angles is composed of equal angles.
(c) Each pair of alternate exterior angles is composed of equal angles.
(d) Interior angles on the same side of transversal are supplementary.
Note: The converse of above statements is also true, i.e. if a transversal intersects two given
lines at different point, then:
(a) If corresponding angles are equal, then the lines are parallel.
(b) If alternate interior angles are equal, then the lines are parallel.
(c) If alternate exterior angles are equal, then the lines are parallel.
(d) If interior angles on the same side of transversal are supplementary, then the lines are
parallel.

Example: If t is transversal intersecting a pair of parallel lines, find the value of x.
Solution: The known angle and the unknown angle make a pair of corresponding angles. We
know that corresponding angles are equal.
Hence, x⁰ = 127⁰
Example: If AB || CD and EF is a transversal, find the value of x.
Solution: The known angle and the unknown angle make a pair of alternate interior angles.
We know that alternate interior angles are equal.

Hence,
Example: If t is a transversal which intersects a pair of parallel lines, find the value of x.
Solution: The known angle and the unknown angle make a pair of alternate interior angles.
We know that alternate interior angles are equal.
Hence, x⁰ = 130⁰
Example: If ‘r’ and ‘s’ are parallel lines, find the values of ∠ 1 and ∠ 2.
Solution: The known angle and ∠ 1 make a pair of corresponding angles. We know that
corresponding angles are equal.
Hence, ∠ 1 = 76°

The known angle and ∠ 2 are on the same side of a line and are adjacent to each other. Hence
they are supplementary.
∠ 2 + 76° = 180°
Or, ∠ 2 = 180° - 76° = 104°
Example: It is given that DE || GB and AC is a transversal. Find the values of x, y and z in
following figure.
Solution: The known angle and ∠ z make a pair of corresponding angles and hence they are
equal.
∠ z = 125° (Corresponding angles are equal)
Now, ∠ z and ∠ y are vertically opposite angles and hence they are equal.
∠ y = ∠ z = 125° (Vertically opposite angles are equal)
Now, ∠ x and ∠ z are on the same side of a line and make a linear pair of angles. Hence,
these angles are supplementary.
∠ x + ∠ z = 180°
Or, ∠ x + 125° = 180°
Or, ∠ x = 180° - 125° = 55°
∠ x = 55°, ∠ y = 125° and ∠ z = 125°
Example: In the given figure, p || q and l is a transversal. Find the values of x and y.

Solution:
Construction: A transversal intersects two parallel lines on two distinct points.
Given: ∠ 1 = ∠ 2
Proof: ∠ 1 = ∠ 2 (Given)
∠ 1 = ∠ 3 (Vertically opposite angles are equal)
From above equations, it is clear;
∠ 3 = ∠ 2
Let us name the angle which is vertically opposite to ∠ 3, as ∠ 4.
Proof:
∠ 1 = ∠ 2 (Given)
∠ 1 = ∠ 3 (corresponding angles are equal)
From above equations, it is clear:
∠ 2 = ∠ 3

∠ 3 = ∠ 4 (Vertically opposite angles are equal.
From above equations, it is clear:
∠ 2 = ∠ 4
Solution: Let us name the angle adjacent to 120° as z.
120° + z = 180° (Linear pair of angles is supplementary)
Or, z = 180° - 120° = 60°

∠ x = ∠ z = 60° (Corresponding angles are equal)
Now,
∠ x = ∠ (3y + 6) (Corresponding angles are equal)
Or, 3y + 6 = 60°
Or, 3y = 60° - 6 = 54°
Or, y = 54 ÷ 3 = 18°
Hence, x = 60° and y = 18°
Example: In the following figure, find the pair of parallel lines.
Solution: ∠ MOW ≠ ∠ MPY
So, OW and PY are not parallel
∠ MOX = 50° + 30° = 80°
∠ MOZ = 52° + 28° = 80°
So, ∠ MOX = ∠ MOZ
Since corresponding angles are equal, so OX||OZ
Example: In the following figure, a transversal is intersecting two lines at distinct
points.

Solution:
∠ 113° + 67° = 180°
Since internal angles on the same side of transversal are supplementary,
Example: In the given figure, a transversal is intersecting two parallel lines at distinct points.
Find the value of x.
Example: If u and v are parallel lines, find the value of x.

Solution: Since corresponding angles are equal
Hence, x = 53⁰
.
Exercise 1 (Based on Lines and Angles)
Question - 1. Find the complement of each of the following angles.
Answer: (a) 57° (b) 38° (c) 69° (d) 15° (e) 45° (f) 36°

Question - 2. Find the supplement of each of the following angles.
Answer: (a) 126.3° (b) 64.8° (c) 68.9° (d) 101.8° (e) 39.3° (f) 136.5°
Question – 3. Find the pairs of complementary and supplementary angles from these pairs of
angles.
(a) 25°, 65°
(b) 32°, 58°
(c) 109°, 71°
(d) 78°, 12°
(e) 42°, 48°
(f) 112°, 68°
(g) 128°, 52°
(h) 98°, 82°
Answer: (a) complementary (b) Complementary (c) Supplementary (d) complementary (e)
complementary (f) supplementary (g) supplementary (h) supplementary
Question - 4. Find the values of x and y in following figures:

Answer: (a) x = y = 60° (b) x = 81°, y = 21° (c) x = 86°, y = 69° (d) x = 33°, y = 41°
Question -5. Find the value of x in following figure:
Answer: 30°
Question – 6. Find the value of x in the following figure.
Answer: 40°

Question – 7 . Find the value of y in the following figure.

Lines and angles

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