This article on lines and angles is part of the elementary math that is asked in various competitive exams like SSC, banking, CLAT, etc. So it helps the students to get a basic understanding of the topic and makes them capable of solving the objective question based on this topic.
Table of Contents
Fundamental terms of Lines and Angles
Line segment and Ray
A line segment is the part of a straight line whose both ends are fixed. If one point of a line is fixed then it is called as Ray.
Collinear and Non-Collinear points
If three or more points line on a straight line then they are called collinear points. If three or more points do not lie on a straight line then they are called non-collinear points.
Types of angles
The types of angle are as follows:
Acute Angle: If an angle lies between 0° and 90°, it is called an acute angle.
Right Angle: An angle whose value is 90° is called a right angle.
Obtuse Angle: If an angle lies between 90° and 180° .it is called obtuse angle
Straight Angle: An angle whose value is 180 ° is called a straight angle.
Reflex Angle: An angle lies between 180 ° and 360 °, it is called a reflex angle.
If the sum of two angles is equal to 90° then they mutually formed a set of complementary angles. The complementary angle of 40° is 50° and the complementary angle of 50° is 40°.
θ1 + θ2 = 90° ⇒ Set of Complementary Angles
If the addition of the two angles is 180° then they are said to be Supplementary to each other. The supplementary angle of 50° is 130° and the supplementary angle of 130° is 50°.
θ1 + θ2 = 180° ⇒ Set of Supplementary Angles
Two angles are said to be adjacent angles if they have a common vertex, one common side, and their uncommon sides must be situated at 2 different sides of the common side.
In the given fig 3
∠AOB and ∠COB are Adjacent angles because point O is common to both of them and their uncommon side OA and OC are opposite to the common side OB.
∠BOC and ∠DOC ; ∠AOD and ∠COD are Adjacent angles
But ∠AOB and ∠DOC; ∠BOC and ∠AOD are not adjacent angles as they do not have a common side.
Linear Pair of angles
In the given figure 4. ∠AOC and ∠COB are adjacent angles and AOB is a straight line i.e, uncommon sides of adjacent angles form a straight line. Such angles are called linear pairs of angles.
Vertically opposite Angles
If Two straight lines AB and CD intersect each other at point O then the angle facing each other is called Vertically opposite angles.
In the given figure 5. ∠AOC and ∠DOB are one pair of Vertically opposite angles, while ∠AOD and ∠COB are another pair of Vertically opposite angles.
A straight line that intersects two or more than two lines at distinct points is termed as a transversal line.
In the given fig 6. straight line a intersect two or more lines at b and c at point p and q, so a is a transversal line .
Exterior angle and interior angle
In the given fig 7, a transversal line intersects two straight lines b and c respectively at p and q. Around each point p and q, four angles are formed , among these angles ∠1 , ∠4, ∠7, ∠6 are called exterior angles while ∠2 , ∠3 , ∠5 , ∠8 are called interior angles.
Corresponding angles and alternate angles
In the fig 7 , the name of angles are as follows:
- ∠1 and ∠5 ; ∠4 and ∠8 ; ∠2 and ∠6 ; ∠3 and ∠7 are called pair of corresponding angles.
- ∠2 and ∠8 ; ∠3 and ∠5 are called pair of alternate interior angle.
- ∠1 and ∠7 ; ∠4 and ∠6 are called alternate exterior angle.
- ∠2 and ∠5 ; ∠3 and ∠8 are called consecutive interior angle.
The Exterior angle and Interior opposite angle of a triangle
In the above fig 8,
∠A, ∠B, and ∠C are the interior angles of a triangle.
∠ACD, ∠CBF, and ∠BAE are the Exterior angles of a triangle.
Interior angles ∠A and ∠B are called interior opposite angles to the exterior angle ∠ACD
Interior angles ∠A and ∠C are called interior opposite angles to the exterior angle ∠CBF
Interior angles ∠B and ∠C are called interior opposite angles to the exterior angle ∠BAE
Types of Triangle according to size of sides:
Equilateral triangle: When the magnitude of all the three sides of a triangle is equal then the triangle is said to be an Equilateral triangle.
Isosceles triangle: When any two sides of a triangle are of equal magnitude then the triangle is said to be an isosceles triangle.
Scalene triangle: When all the three sides of a triangle are of different magnitude then the triangle is said to be a scalene triangle.
Types of triangles according to their angles
Acute angle triangle: If all the 3 angles of a triangle are acute then the triangle is said to be an acute angle triangle.
Right angle triangle: If one of the angles of a triangle is right angle i.e 90° then the triangle is said to be right angle triangle.
Obtuse angle triangle: If one of the angles of a triangle is obtuse i.e greater than 90° and less than 180 ° then the triangle is said to be an obtuse angle triangle. Any triangle can have only 1 obtuse angle.