CBSE Class 7 Mathematics Lines And Angles Worksheet Set 05

Question 1. Find the complement of the angles : (a) \(56^\circ\) (b) \(47^\circ\) (c) \(12^\circ\) (d) \(81^\circ\)
Answer: Two angles are complementary if their sum is \(90^\circ\).
(a) Complement of \(56^\circ = 90^\circ - 56^\circ = 34^\circ\)
(b) Complement of \(47^\circ = 90^\circ - 47^\circ = 43^\circ\)
(c) Complement of \(12^\circ = 90^\circ - 12^\circ = 78^\circ\)
(d) Complement of \(81^\circ = 90^\circ - 81^\circ = 9^\circ\)
In simple words: Subtract the given angle from \(90^\circ\) to find its complement.
Exam Tip: Always remember that complementary angles add up to a right angle (\(90^\circ\)).

Question 2. Find the supplement of the angles : (a) \(135^\circ\) (b) \(87^\circ\) (c) \(39^\circ\) (d) \(112^\circ\)
Answer: Two angles are supplementary if their sum is \(180^\circ\).
(a) Supplement of \(135^\circ = 180^\circ - 135^\circ = 45^\circ\)
(b) Supplement of \(87^\circ = 180^\circ - 87^\circ = 93^\circ\)
(c) Supplement of \(39^\circ = 180^\circ - 39^\circ = 141^\circ\)
(d) Supplement of \(112^\circ = 180^\circ - 112^\circ = 68^\circ\)
In simple words: Subtract the given angle from \(180^\circ\) to find its supplement.

Exam Tip: Supplementary angles form a straight line and always add up to \(180^\circ\).

Question 3. Find the supplement of the angles : (a)                                  (b)
Answer:
(a) From the intersecting lines diagram, the angle \(49^\circ\) and angle \(z\) are vertically opposite angles, so \(z = 49^\circ\).
The angles \(x\) and \(49^\circ\) form a linear pair on a straight line, so:
\(x + 49^\circ = 180^\circ\)
\(x = 180^\circ - 49^\circ = 131^\circ\)
Similarly, \(y\) and \(z\) form a linear pair, so:
\(y = 180^\circ - z = 180^\circ - 49^\circ = 131^\circ\)
(b) From the intersecting lines diagram, the angle \(25^\circ\) and angle \(z\) are vertically opposite angles, so \(z = 25^\circ\).
The angle \(x\) and \(60^\circ\) along with \(z\) form a straight line sum of \(180^\circ\), or using adjacent angles on a straight line:
\(x + 60^\circ + 25^\circ = 180^\circ\)
\(x + 85^\circ = 180^\circ\)
\(x = 180^\circ - 85^\circ = 95^\circ\)
The angle \(y\) is vertically opposite to the combined angle \((60^\circ + z)\) or can be found using linear pair with \(x\):
\(y = 180^\circ - x = 180^\circ - 95^\circ = 85^\circ\)
In simple words: Use vertically opposite angles to find equal angles, and linear pairs (sum equal to \(180^\circ\)) to find unknown angles on a straight line.

Exam Tip: Identify intersecting lines first to spot equal vertically opposite angles before applying linear pair rules.

Question 4. Find x, if \(l \parallel m\) : (a)                                  (b)
Answer:
(a) Given lines \(l \parallel m\) cut by a transversal:
The alternate exterior or corresponding angle properties show that the interior angle adjacent to \(105^\circ\) and \(x\) relates as follows:
The angle corresponding to \(105^\circ\) on line \(m\) is \(105^\circ\).
Since \(x\) and this corresponding angle form a linear pair:
\(x + 105^\circ = 180^\circ\)
\(x = 180^\circ - 105^\circ = 75^\circ\)
(b) Given parallel lines with transversal lines \(a\) and \(b\):
Using alternate interior angles and corresponding angles given by \(110^\circ\) and \(70^\circ\):
The two parallel transversals cut line \(a\), and the angle \(110^\circ\) and \(x\) lie on the same side of line \(a\) between the two transversals, so they are co-interior (consecutive interior) angles and add up to \(180^\circ\):
\(x + 110^\circ = 180^\circ\)
\(x = 180^\circ - 110^\circ = 70^\circ\)
In simple words: Use properties of parallel lines (alternate interior, corresponding, and consecutive interior angles) to equate or relate angles.

Exam Tip: Always extend the transversal lines mentally to clearly visualize alternate interior and corresponding angle pairs.

Revision Notes on Lines and Angles

Point: A point is a geometrical element which has no dimensions

Line: A line is a straight path which has no endpoints.

 

3. Adjacent Angles: It is the pair of two angles which are placed next to each other. Adjacent angles have-

• A common vertex.

• A common arm.

• A non-common arm could be on either side of the common arm.

4. Linear Pair: A pair of adjacent angles whose non-common arm makes a single line i.e. they are the opposite rays.

A linear pair is also a pair of supplementary angles as their sum is 180°.

The above pair of angles is –

•  Adjacent, as they have one common arm.

•  Supplementary, as the sum of two angles, is 180°.

•  The linear pair, as the sum is 180° and the non – common arms are opposite rays.

5. Vertically Opposite Angles: When two lines intersect each other then they form four angles. So that

• ∠a and ∠b is pair of vertically opposite angles.

• ∠n and ∠m is pair of vertically opposite angles.

Vertically opposite angles are equal.

Pairs of Lines

1. Intersecting Lines If two lines touch each other in such a way that there is a point in common then these lines are called intersecting lines.

That common point is called a Point of Intersection.

Here, line l and m intersect each other at point C.

2. Transversal: If a line intersects two or more lines at different points then that line is called Transversal Line.

3. Angles made by a transversal: When a transversal intersects two lines then they make 8 angles.