Revise notes Class 7 lines and angles

Revise notes
Class 7 Maths
Lines and Angles

Topics

1. Introduction
2. Angles
3. Parallel and intersecting lines
4. Transversal
5. Transversal of Parallel lines

Transversal of Parallel lines

The relationships between angles when a pair of parallel line is intersected by a transversal are:

Pairs of corresponding angles are equal.

∠2 = ∠6
∠1= ∠5
∠3 = ∠7
∠4 = ∠8

Pairs of alternate interior angles are equal.

∠3 =∠6
∠4=∠5

Pairs of interior angles on the same side of the transversal are supplementary.

∠4 + ∠6 = 180
∠3 + ∠5 = 180

Important points:

When a transversal cuts two lines, such that pairs of corresponding angles are equal, then the lines have to be parallel.
When a transversal cuts two lines, such that pairs of alternate interior angles are equal, the lines have to be parallel.
When a transversal cuts two lines, such that pairs of interior angles on the same side of the transversal are supplementary, the lines have to be parallel.

Problem: State the property that is used in each of the following statements?

(i) If a || b, then ∠1 = ∠5.

(ii) If ∠4 = ∠6, then a || b.

(iii) If ∠4 + ∠5 = 180°, then a || b.

Solution:

(i)The property that is being used is that when a pair of parallel lines is intersected by a transversal the pairs of corresponding angles are equal. Lines a and b are parallel.

Angle 1 and 5 are corresponding angles and thus are equal.

(ii) The property that is being used is that when a transversal cuts two lines, such that pairs of alternate interior angles are equal, the lines have to be parallel.

Given, Angle 4 and 6 are alternate interior angles and are equal. Lines a and b are parallel to each other.

(iii) The property that is being used is when a transversal cuts two lines, such that pairs of interior angles on the same side of the transversal are supplementary, the lines have to be parallel.

Angle 4 and 5 are interior angles on the same side of the transversal and are supplementary. Therefore, lines a and b are parallel.

Problem :In the adjoining figure, p || q. Find the unknown angles.

Solution:

Using the property that when a pair of parallel lines is intersected by a transversal the pairs of corresponding angles are equal. Lines p and q are parallel.

Angle d and 125 degrees are corresponding angles. Therefore, d= 125 degrees.

Now, angle d and angle b are vertically opposite angles as they are formed by the intersection of two lines and are opposite to each other.

Therefore, b = d= 125 degrees.

Angle a and b form a linear pair. So, a and b will be supplementary angles.

a + b = 180

a + 125 = 180

a = 55 degrees.

Angle a and c are vertically opposite angles as they are formed by the intersection of two lines and are opposite to each other.

c = a= 55 degrees.

Angle f and 125 degrees form a linear pair. So, f and 125 will be supplementary angles.

f + 125 =180

f =55 degrees.

Angle e and f are vertically opposite angles as they are formed by the intersection of two lines and are opposite to each other.

e = f= 55 degrees.

Problem :In the given figure, the arms of two angles are parallel.

If∠ABC = 70º, then find (i) ∠DGC (ii) ∠DEF

Solution:

The arms of two angles are parallel. So, we get

AB is parallel to DE

BC is parallel to EF

AB is parallel to DE. BC is a transversal. Using the property that when a pair of parallel lines is intersected by a transversal the pairs of corresponding angles are equal.

∠ABC = ∠DGC

∠DGC = 70 degrees.

BC is parallel to EF. DE is a transversal. Using the property that when a pair of parallel lines is intersected by a transversal the pairs of corresponding angles are equal.

∠DGC = ∠DEF

∠DEF = 70 degrees.

Therefore, ∠DEF = ∠DGC = 70 degrees

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