Theorems of Lines & Angles - Class 9 PDF Download

Lines And Angles

(a) Segment: A part of line with two end points is called a line-segment.

A line segment is denoted by AB and its length is is denoted by AB.

(b) Ray: A part of a line with one end-point is called a ray.

We can denote a line-segment AB, a ray AB and length AB and line AB by the same symbol AB.

(c) Collinear points: If three or more points lie on the same line, then they are called collinear points, otherwise they are called non-collinear points.

(d) Angle: An angle is formed by two rays originating from the same end point.

The rays making an angle are called the arms of the angle and the end-points are called the vertex of the angle.

(d) Types of Angles:

(i) Acute angle: An angle whose measure lies between 0° and 90°, is called an acute angle.

(ii) Right angle: An angle, whose measure is equal to 90°, is called a right angle.

(iii) Obtuse angle: An angle, whose measure lies between 90° and 180°, is called an obtuse angle.

(iv) Straight angle: The measure of a straight angle is 180°.

(v) Reflex angle: An angle which is greater than 180° and less than 360°, is called the reflex angle.

(vi) Complimentary angle: Two angles, whose sum is 90°, are called complimentary angle.

(vii) Supplementary angle: Two angles whose sum is 180º, are called supplementary angle.

(viii) Adjacent angle: Two angles are adjacent, if they have a common vertex, common arm and their non-common arms are on different sides of the common arm.

(ix) Linear pair of angles: If the sum of two adjacent angles is 180º, then their non-common lines are in the same straight line and two adjacent angles form a linear pair of angles.

(x) Vertically opposite angles: - When two lines AB and CD intersect at a point O, the vertically opposite angles are formed.

(e) Intersecting lines and non-intersecting lines: - Two lines are intersecting if they have one point in common. We have observed in the above figure that lines AB and CD are intersecting lines, intersecting at O, their point of intersection.

Parallel lines: - If two lines do not meet at a point if extended to both directions, such lines are called parallel lines.

Lines PQ and RS are parallel lines.

The length of the common perpendiculars at different points on these parallel lines is same. This equal length is called the distance between two parallel lines.

Axiom 1. If a ray stands on a line, then the sum of two adjacent angles so formed is 180º.

Conversely if the sum of two adjacent angles is 180º, then a ray stands on a line (i.e., the non-common arms form a line).

Axiom 2. If the sum of two adjacent angles is 180º, then the non-common arms of the angles form a line. It is called Linear Pair Axiom.

(f) Theorem 1. If two lines intersect each other, then the vertically opposite angles are equal.

Solution: Given: Two lines AB and CD intersect each other at O.

To Prove:

Parallel Lines And A Transversal

Axiom 4. If a transversal intersects two lines such that a pair of corresponding angles is equal, then the two lines are parallel to each other.

Theorem 2. If a transversal intersects two parallel lines, then each pair of alternate interior angles is equal.

Solution: Given: Let PQ and RS are two parallel lines and AB be the transversal which intersects them on L and M respectively.

Theorem 3. If a transversal intersects two lines such that a pair of alternate interior angles is equal, then the two lines are parallel.

Solution: Given: - A transversal AB intersects two lines PQ and RS such that

But these are corresponding angles.

We know that if a transversal intersects two lines such that a pair of corresponding angles is equal, then the two lines ate parallel to each other.

Hence, PQ¦RS Proved.

Theorem 4. If a transversal intersects two parallel lines, then each pair of interior angles on the same side of the transversal is supplementary.

Solution:

Given: - Transversal EF intersects two parallel lines AB and CD at G and H respectively.

Theorem 5. If a transversal intersects two lines such that a pair of interior angles on the same side of the transversal is supplementary, then the two lines are parallel.

Solution:

Given: - A transversal EF intersects two lines AB and CD at P and Q respectively.

But these are alternate interior angles. We know that if a transversal intersects two lines such that the pair of alternate interior angles are equal, then the lines are parallel.

Hence, AB¦CD Proved.

Theorem 6. Lines which are parallel to the same line are parallel to each other.

Solution:

Given: - Three lines AB, CD and EF are such that AB¦CD, CD¦EF.

To Prove: - AB¦EF.

Construction: - Let us draw a transversal GH which intersects the lines AB, CD and EF at P, Q and R respectively.

Proof: - Since, AB¦CD and GH is the transversal. Therefore,

But these are corresponding angles.

We know that if a transversal intersects two lines such that a pair of corresponding angles is equal, then the two lines ate parallel to each other.

Hence, AB¦ EF Proved.

Angle Sum Property of Triangle: -

Theorem 7. The sum of the angles of a triangle is 180º.

Solution:

Given: - Δ ABC.

Hence Proved.

Theorem 8. If a side of a triangle is produced, then the exterior angle so formed is equal to the sum of the two interior opposite angles.

Solution: