Lines and Angles | Properties and Examples (2024)

Introduction

What is a line?

A line is a single-dimensional or one-dimensional figure, with no thickness, and extends in both directions infinitely. Usually, it is called a straight line just to emphasize that there are no curves anywhere in it.

In the above picture, AB and BC are two lines meeting at point B. The point B is called the Vertex. The space between the two lines marked as “b” is called the Angle.

Other Definitions included as part of the lines are listed below for immediate reference.

Line segment: A part of a line with two endpoints is called a line segment. Line segment AB is denoted by AB.
Ray: A part of a line with one endpoint is called a ray.
Collinear points and non-collinear points: If three or more than three points are on the same line, then they are called collinear points, otherwise, they are non-collinear points.
Intersecting Lines and Non-intersecting Lines:Lines PQ and RS are intersecting lines because they are intersecting with each other at O.Lines AB and CD are non-intersecting (parallel) lines.

Note: The lengths of the common perpendicular at different points on these parallel lines are the same. This equal length is called the distance between two parallel lines.

Introduction to Lines and Angles-PDF

A line is a single-dimensional or one-dimensional figure, with no thickness, and extends in both directions infinitely. Let us read more about Lines and Angles.Here is a downloadable PDF to explore more.

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Introduction to Lines and Angles-PDF

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What are Angles?

Anangle is a space or the gap formed between two lines that meet at a point and the meeting point is called the Vertex. The two lines originating from the same point are called the arms of the angle.

The Angles are classified into different types such as Acute angle, Right angle, Obtuse angle, Straight angle, Reflex angle, and Full angle. These angles are explained in the table given below along with images showing different angles.

When a pair of angles are formed, they are named differently based on a few criteria. The names of the pair of angles and their criteria are given below.

Complementary Angles: Two angles whose sum is 90° are called complementary angles.
Supplementary Angles: Two angles whose sum is 180° are called supplementary angles.
Adjacent Angles: Two angles are adjacent if they have a common vertex, a common arm and their non-common arms are on different sides of the common arm.∠ABD and ∠DBC are the adjacent angles. Ray BD is their common arm and point B is their common vertex. Ray BA and Ray BC are non-common arms.

Note: ∠ABC = ∠ABD + ∠DBC

Vertically Opposite Angles: The vertically opposite angles formed when two lines intersect each other at a point. Two lines AB and CD intersect each other at point O, then, there are two pairs of vertically opposite angles. One pair is ∠AOD and ∠BOC and another pair is ∠AOC and ∠BOD.

If two lines intersect each other, then the vertically opposite angles are equal. ∠AOD = ∠BOC; ∠COA = ∠DOB

Linear Pair of Angles: When the sum of two adjacent angles is 180°, then they are called a linear pair of angles.

(i) If a ray stands on a line, then the sum of two adjacent angles so formed is 180°.
(ii) If the sum of two adjacent angles is 180°, then a ray stands on a line (that is the non-common arms form a line). In the above image, Ray OC stands on line AB. ∠AOC + ∠BOC = 180°

Parallel Lines and a Transversal: A line that intersects two or more lines at distinct points is called a transversal.

Here, line l is a transversal of the lines m and n, respectively.
Line l intersects m and n at P and Q respectively, then four angles are formed at each of the points P and Q namely ∠1, ∠2, ∠3,…, ∠8
∠1, ∠2, ∠7, and ∠8 are called exterior angles.
∠3, ∠4, ∠5, and ∠6 are called interior angles.
We classify these eight angles into the following groups:
1. Corresponding angles
∠1 and ∠5
∠2 and ∠6
∠4 and ∠8
∠3 and ∠7

2. Alternate interior angles
∠4 and ∠6
∠3 and ∠5

Angles Sum Property of a Triangle

1. The sum of the angles of the triangle is 180°.
∠1 + ∠2 + ∠3 = 180°

2. If a side of a triangle is produced, then the exterior angle, formed is equal to the sum of the two interior opposite angles.

∠4 = ∠1 + ∠2

Note: An exterior angle of a triangle is greater than either of its interior opposite angles.

Applications of Lines & Angles

Angles are used in daily life. Civil Engineers and Architects use angles for building infrastructures such as roads, construction work, animations, bearing tools, crossroads, etc. Angles are also used by carpenters to make chairs, tables, cots, and sofas. Angles are also used by Artists in, their creations such as portraits and paintings.

Some examples of Angles in daily life.

Example Problems

Question 1: In the angles of a triangle are in the ratio 2:3:4, then the triangle formed will be
(A) Right-angled triangle (B) Isosceles triangle
(C) Scalene triangle (D) Obtuse angled triangle

Advanced Level Problems

Question 1: In the given figure, if OCD is an isosceles triangle in which OD and OC are equal, then what will be the value of ∠OCD?

(A) 70° (B) 50°
(C) 65° (D) 45°

Answer:
\(x + 10° + x + x +20°= 180°\)
\(3x +30° = 180°\)
\(3x = 150°\)
\(x = 50°\)
∠OCD = ∠ODC [∵OD = OC]
Now,
\(∠OCD + ∠ODC + x = 180°\)
\(∠OCD + ∠OCD + 50° = 180°\)
\(2∠OCD = 130°\)
\(∠OCD = 65°\)
Therefore the correct answer is (C)65°.

Question 2: In the given figure, AB∥DC. Then the value of x is _______.

(A) 25° (B) 30°
(C) 45° (D) 50°

Answer: Given AB II CD, extending the line CD to meet the line AE at F, as shown in the below figure.

∠EAB and ∠EFC are corresponding angles. Therefore ∠EAB = ∠EFC = 50°
Now, in △AFC, the exterior angle at C is 100°
We know that the exterior angle is equal to the sum of two opposite angles in the triangle.
Therefore,
∠EFC+∠FEC = ∠ECD
\(50°+x=100°\)
\(x=100 - 50 = 50°\)
The correct answer is option D.

(D) 50°

Learning Outcomes

Identify lines, line segments, rays, and angles.
Classify angles as acute, right, obtuse, or straight.
Identify complementary and supplementary angles.
Find measures of angles.

For more details you can refer to the blogs from the below links:

https://www.cuemath.com/geometry//angles/
https://www.cuemath.com/geometry//lines/
https://www.cuemath.com/learn/types-of-lines/