In mathematics, a line is used to represent a straight path that has length but no width or depth. Lines help us describe the shape and direction of objects around us.
An angle is formed when two lines, line segments, or rays meet at a common point. When two rays intersect in the same plane, the opening between them is called an angle. Angles help us understand turns, corners, and directions in geometry.
Lines
A line is a one-dimensional figure that extends infinitely in both directions and has no width. It is made up of infinitely many points placed close together. Euclid described a line as a breadthless length. In a Cartesian plane, a line is represented by the equation ax + by = c.
Types of Lines
Lines can also be categorized as,
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Angles
When two rays intersect at a point, they form an angle. Angles are measured in degrees and denoted by the symbol ∘, which shows the measure of rotation. An angle can have a value from 0∘ to 360∘ and is represented by the symbol ∠.
Types of Angles
There are various types of lines and angles in geometry based on the measurements and different scenarios. Let us learn here all those lines and angles along with their definitions.
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Theorems Related to Lines and Angles
There are various theorems related to lines and angles, some of these are:
Vertical Angles are Equal
The vertical opposite angles are always equal to each other. The image added below shows the pair of equal, vertically opposite angles
- ∠MON = ∠POQ
- ∠MOP = ∠NOQ
We can prove this as,
To Prove:
- ∠MON = ∠POQ
- ∠MOP =∠NOQ
Proof:
∠MOP+ ∠MON = 180°
∠MOP+ ∠POQ = 180°
Therefore:
∠MOP+ ∠MON = ∠MOP+ ∠POQ
Now subtracting ∠MOP at both sides
∠MOP + ∠MON - ∠MOP = ∠MOP + ∠POQ - ∠MOP
∠MON = ∠POQ
Similarly,
∠MON + ∠NOQ = 180°
∠POQ + ∠NOQ = 180°
Therefore,
∠MOP+ ∠NOQ = ∠POQ+ ∠NOQ
Subtracting ∠NOQ at both sides
∠MOP + ∠NOQ - ∠NOQ = ∠POQ + ∠NOQ - ∠NOQ
∠MOP = ∠POQ
Thus,
- ∠MON = ∠POQ
- ∠MOP =∠NOQ
Angles in a Triangle Sum Up to 180°
Sum of all the angles in any triangle is 180°. This is proved below, Suppose we have a triangle ABC as shown in the image below:
To Prove: ∠A + ∠B + ∠C = 180°
Proof:
Draw a line parallel to BC and pass through vertices A of the triangle.
Now as both lines PQ and BC are parallel, AB and AC are transversal.
So:
- ∠ PAB = ∠ ABC (Alternate Interior Angles)...(i)
- ∠ QAC = ∠ ACB (Alternate Interior Angles)...(ii)
Now, ∠ PAB + ∠ BAC + ∠ QAC = 180° (Linear Pair)...(iii)
From eq (i), (ii) and (iii)
∠ ABC + ∠ BAC + ∠ ACB = 180°
∠A + ∠B + ∠C = 180°
Proved.
Properties of Lines and Angles
In this section, we will learn about some general properties of lines and angles:
Properties of Lines
- Line has only one dimension i.e. length. It does not have breadth and height.
- A line has infinite points on it.
- Three points lying on a line are called collinear points
Properties of Angles
- Angles tell about how much a line has rotated from his position.
- Angles are formed when two lines meet and they are called arms of the angle.
Solved Examples on Lines and Angles
Example 1: Find the reflex angle of ∠x, if the value of ∠x is 75 degrees.
Solution:
Let the reflex angle of ∠x be ∠y.
Now, according to the properties of lines and angles, the sum of an angle and its reflex angle is 360°.
Thus,
∠x + ∠y = 360°
75° + ∠y = 360°
∠y = 360° − 75°
∠y = 285°
Thus, the reflex angle of 75° is 285°.
Example 2: Find the complementary angle of ∠x, if the value of ∠x is 75 degrees.
Solution:
Let the complementary angle of ∠x be ∠y.
Now, according to the properties of lines and angles, the sum of an angle and its complementary angle is 90°.
Thus,
∠x + ∠y = 90°
75° + ∠y = 90°
∠y = 90° − 75°
∠y = 15°
Thus, the complementary angle of 75° is 15°.
Example 3: Find the supplementary angle of ∠x, if the value of ∠x is 75 degrees.
Solution:
Let the supplementary angle of ∠x be ∠y.
Now, according to the properties of lines and angles, the sum of an angle and its supplementary angle is 180°.
Thus,
∠x + ∠y = 180°
75° + ∠y = 180°
∠y = 180° − 75°
∠y = 105°
Thus, the supplementary angle of 75° is 105°.
Example 4: Find the value of ∠A and ∠B if ∠A = 4x and ∠B = 6x are adjacent angles and they form a straight line.
Solution:
According to the properties of lines and angles, the sum of the adjacent linear angles formed by a line is 180°.
Thus,
∠A + ∠B = 180°
4x + 6x = 180°
10x = 180°
x = 180°/10 = 18°
Thus,
- ∠A = 4x = 4×18 = 72°
- ∠B = 6x = 6×18 = 108°
Practice Questions on Lines and Angles
Question 2: Find the complementary angle of ∠P, if the value of ∠P is 35 degrees.
Question 3: Find the supplementary angle of ∠Z, if the value of ∠Z is 120 degrees.
Question 1: Find the reflex angle of ∠A, if the value of ∠A is 110 degrees.
Question 5: Find the reflex angle of ∠B, if the value of ∠B is 45 degrees.
Question 4: Find the value of ∠X and ∠Y if ∠X = 5y and ∠Y = 7y are adjacent angles and they form a straight line.
Question 6: Find the value of ∠A and ∠B if ∠A = 3x and ∠B = 7x are adjacent angles and they form a straight line.