Basic Concepts Of Geometry - Class 9 Free MCQ Test with solutions

Let the angle = x
Supplement of angle= 180-x

ATQ

180-x = x
180 = 2x

x = 90°

And its supplement will be 180-90=90°

Two lines in a plane are said to be parallel if they do not intersect, when extended infinitely in both the direction.

Also, the distance between the two lines is the same throughout.

Let the angle be x
complement of x is (90 − x)°
supplement of x is (180 − x)°
Given (90 − x)° = (1/3) × (180 − x)°
⇒ 270° − 3x = 180°− x
⇒ 270° − 180° = 3x − x
⇒ 2x = 90°
∴ x = 45°

• Linear Pair: A linear pair is a pair of adjacent angles whose non-common sides are opposite rays.


• Linear Pair Theorem: If two angles form a linear pair, then they are supplementary, which means their measures add up to 180°.


• Statement A: If angles forming a linear pair are equal, then each of these angles measures 90°.




• Since the angles are equal, let's say they are both x.


• According to the Linear Pair Theorem, x + x = 180°.


• Therefore, 2x = 180°, and x = 90°.


• So, each angle in the linear pair measures 90°.




• Statement B: Angles forming a linear pair cannot both be acute angles because their sum must be 180°.


• Statement C: If two adjacent angles are equal, they can each measure 90° only if they form a right angle.


• Statement D: Two angles whose sum is 90° are called complementary angles, not supplementary angles which have a sum of 180°.

Given, four points such that no three of them are collunear.
we kow that line can be formed using two points. therfore, 

Two angles form a linear pair if they have; 

A common arm

A common vertex

Their interiors do not overlap

The sum of two angles is 180°.

Therefore, linear pair of angles are adjacent angles whose non-common arms are opposite rays. 

Note: 

All adjacent angles do not form a linear pair.

From the above figure we can observe; OX and OY are two opposite rays and ∠XOZ and ∠YOZ are the adjacent angles. Therefore, ∠XOZ and ∠YOZ form a linear pair. 
If you measure ∠XOZ and ∠YOZ with the help of the protractor, you will find the sum of their measures equal to 180°.

Thus, the sum of the angles in a linear pair is 180°.

Let the angle be x. Its complement will be 90 - x. Therefore,

x = 90 - x 

=> 2x = 90 

=> x = 45 degree

The answer is 45 degree.

If two straight lines intersect each other at any point then the angle which are vetically opposite are equal.

Given two lines AB and CD intersect each other at the point O.
To prove: ∠1 = ∠3 and ∠2 = ∠4
Proof:
From the figure, ∠1 + ∠2 = 180°  [Linear pair]  → (1)
∠2 + ∠3 = 180°  [Linear pair]  → (2)
From (1) and (2), we get
∠1 + ∠2 = ∠2 + ∠3
∴ ∠1 = ∠3
Similarly, we can prove ∠2 = ∠4 also.

Let the angle be = x  the supplement = 180^o – x

According to the question,
180^o – x
180^o – 122^o
58^o

Let the angles be x
As they are in ratio of 4:5, so we can write it in the form of 4x ,5x
We know supplementary angle =180°
so,According to the question
4x+5x = 180°
9x = 180°
x = 180° ÷ 9
x = 20°
so, 1 st angle = 4x = 4×20° = 80°
2 nd angle = 5x = 5×20° = 100°
PROOF
4x + 5x = 180°
4×20° +5×20° = 180°
80°+ 100°= 180°
180° = 180°