NCERT MCQ Solutions for Class 9 Maths Chapter 6 Lines and Angles

A straight angle is defined as an angle that measures exactly 180°. It represents a straight line and occurs when the arms of the angle form a continuous straight line without bending in either direction according to the standard types of angles.

A linear pair is formed when two adjacent angles have their non-common arms forming a straight line. Their sum is 180° by definition. This pair arises when a ray stands on a line creating two adjacent angles.

Supplementary angles are defined as two angles whose sum equals 180°. They do not necessarily have to be adjacent. Any two angles that add up to 180° meet the criteria of supplementary angles as per the geometric definition.

When two lines intersect each other, the opposite pairs of angles formed are called vertically opposite angles. According to Theorem, vertically opposite angles are always equal regardless of the measure of the intersecting angles.

Complementary angles are defined as two angles whose measures add up to 90°. In the given options, only 60° + 30° = 90°. Therefore, this pair is complementary. Complementary angles are often seen in right angle triangles and perpendicular lines.

In a triangle, the interior opposite angle is always less than the corresponding exterior angle. This is because the exterior angle is equal to the sum of the two remote interior angles, and hence it is greater than either of them.

If a ray stands on a line, it forms two adjacent angles. According to the Linear Pair Axiom, the sum of these adjacent angles is always 180°.

An obtuse angle as one that measures more than 90° but less than 180°. It lies between the range of a right angle and a straight angle. Obtuse angles typically appear in triangles and intersecting lines.

The total measure of angles formed around a point by rays or lines meeting at that point is 360°. This concept is used in multiple examples and is a fundamental property of circular angle measurement.

When two parallel lines are intersected by a transversal, alternate interior angles are equal. This is one of the key angle relationships discussed under properties of angles formed by parallel lines and a transversal.

Linear pair angles sum up to 180°. If ∠A is 120°, then
∠B = 180° – 120°
= 60°.
This follows directly from the definition and axiom that a ray standing on a line forms two adjacent angles adding to 180°.

If two angles are equal and complementary, their measures must be equal and sum to 90°.
Dividing 90° by 2 gives 45°.
Therefore, each angle measures 45° as per the definition and simple calculation.

Let the angle be x.
x = 5(180° – x).
Solve: x = 150°.

When two parallel lines are intersected by a transversal, the bisectors of the two pairs of interior angles form a rectangle. This is because the angle bisectors create right angles at their intersections due to the properties of parallel lines and transversal.

It defines parallel lines as lines in the same plane that never intersect, no matter how far they are extended. This is different from skew lines, which are in different planes. Parallelism is fundamental in geometric constructions.

It classifies a reflex angle as one that is greater than 180° but less than 360°.
270° fits in this range and is therefore a reflex angle.
Reflex angles represent the larger rotation between two arms of an angle.

Parallel is not a type of angle pair but a relation between lines. The other three options—complementary, supplementary and linear pair—are defined angle relationships discussed based on how angles are formed and measured.

A linear pair consists of two adjacent angles whose non-common arms form a straight line. Their sum is always 180° and they lie on a straight line. This makes them a special case of adjacent angles.

∠AOC + ∠BOC = 180°
3x + 2x = 180°
5x = 180°
x = 36°
So ∠AOC = 108° and ∠BOC = 72°
Thus x = 36°
Using the linear pair axiom from where adjacent angles on a straight line sum to 180°.

The sum of all the angles around a point is always 360°. This is a fundamental property of angles around a point in geometry.

Let the angle be x.
x = 180° – x.
Solve: x = 90°.

A full angle measures 360°. A quarter of 360° is 90°, which defines a right angle. It clearly categorizes a right angle as one that is exactly 90°, making it one-fourth of a complete angle.

Since they form a linear pair, their sum is 180°.
x + 2x – 30 = 180
3x – 30 = 180
3x = 210
x = 70°
But the correct form is: x + 2x – 30 = 180
3x = 210
x = 70°

When two lines intersect, they form two pairs of vertically opposite angles and two pairs of linear pairs. It provides diagrams and theorems that prove that vertically opposite angles are equal and linear pairs add up to 180°.

Supplementary angles sum to 180°.
If one angle is 130°, the other must be 180° – 130° = 50°.

Corresponding angles are equal when the lines being intersected by a transversal are parallel. This is supported by the corresponding angles axiom.

The sum of all the angles formed on the same side of a line at a given point on the line is always 180°. This is because the angles form a linear pair and add up to the measure of a straight angle.

Let the angle be x.
x = 90° – x.
Solve: x = 45°.

2x + 3x = 180°
5x = 180°
x = 36°
Then ∠ABC = 72° and ∠CBD = 108°
This uses the concept of a linear pair where the sum of adjacent angles on a line is 180°.

When a ray stands on a line, it forms a straight angle of 180°. The two adjacent angles formed in this situation always sum to 180°, confirming it to be a straight angle. This is a direct application of the Linear Pair Axiom.

Alternate interior angles lie between two lines but on opposite sides of the transversal. These angles are equal when the two lines are parallel. This property helps determine line relationships in geometric constructions.

Vertically opposite angles are always equal. If ∠AOC = 75°, then its vertically opposite angle ∠BOD must also be 75°. This is derived from Theorem 6.1 which proves equality of vertically opposite angles when two lines intersect.

If two adjacent angles sum to 180°, then their non-common arms form a straight line. This is the converse of the Linear Pair Axiom (Axiom 6.2) discussed and demonstrated through visual examples and constructions.

In a triangle, the sum of two interior opposite angles is always equal to the exterior angle formed by extending one side of the triangle. However, it is not necessarily equal to any of the given options (a, b or c).

Let the angle be x.
x = 4(90° – x).
Solve: x = 72°.