CBSE Class 7 Mathematics Lines And Angles Worksheet Set 06

Choose the correct answer:

Q.1 The complement of an angle is \(\frac{2}{3}\) of the other, then the angle is :
(a) \(60^{\circ}\)
(b) \(30^{\circ}\)
(c) \(36^{\circ}\)
(d) \(54^{\circ}\)
Answer: (d) \(54^{\circ}\)
Let the required angle be \(x\).
Its complement is \(90^{\circ} - x\).
According to the question, the complement is \(\frac{2}{3}\) of the angle:
\(90^{\circ} - x = \frac{2}{3}x\)
\(90^{\circ} = \frac{2}{3}x + x\)
\(90^{\circ} = \frac{5}{3}x\)
\(x = \frac{90^{\circ} \times 3}{5} = 18^{\circ} \times 3 = 54^{\circ}\)
In simple words: An angle and its complement add up to \(90^{\circ}\). By setting up an algebraic equation, we find the angle measures \(54^{\circ}\).

Exam Tip: Remember that complementary angles always sum to \(90^{\circ}\) and supplementary angles sum to \(180^{\circ}\).

Q.2 If the supplement of an angle is \(165^{\circ}\), then its complement is :
(a) \(15^{\circ}\)
(b) \(90^{\circ}\)
(c) \(75^{\circ}\)
(d) \(65^{\circ}\)
Answer: (c) \(75^{\circ}\)
Let the angle be \(x\).
Its supplement is given as \(165^{\circ}\).
\(x + 165^{\circ} = 180^{\circ}\)
\(x = 180^{\circ} - 165^{\circ} = 15^{\circ}\)
Now, find the complement of \(x\):
Complement = \(90^{\circ} - 15^{\circ} = 75^{\circ}\)
In simple words: First find the angle using its supplement (subtracting from \(180^{\circ}\)), then subtract that angle from \(90^{\circ}\) to find its complement.

Exam Tip: Always read carefully whether the question asks for the angle itself, its supplement, or its complement.

Q.3 Two supplementary angles differ by \(34^{\circ}\), find the angles.
Answer: Let one angle be \(x\).
Then the other angle is \(x + 34^{\circ}\).
Since the angles are supplementary, their sum is \(180^{\circ}\):
\(x + (x + 34^{\circ}) = 180^{\circ}\)
\(2x + 34^{\circ} = 180^{\circ}\)
\(2x = 180^{\circ} - 34^{\circ}\)
\(2x = 146^{\circ}\)
\(x = 73^{\circ}\)
First angle = \(73^{\circ}\)
Second angle = \(73^{\circ} + 34^{\circ} = 107^{\circ}\)
In simple words: Supplementary angles add up to \(180^{\circ}\). We define the two angles using a variable and solve the linear equation.

Exam Tip: Verify your answer by adding the two resulting angles; their sum must equal \(180^{\circ}\).

Q.4 An angle is equal to five times its complement. Determine its measure.
Answer: Let the angle be \(x\).
Its complement is \(90^{\circ} - x\).
According to the question:
\(x = 5(90^{\circ} - x)\)
\(x = 450^{\circ} - 5x\)
\(x + 5x = 450^{\circ}\)
\(6x = 450^{\circ}\)
\(x = \frac{450^{\circ}}{6} = 75^{\circ}\)
In simple words: Set the angle equal to 5 times its complement (\(90^{\circ} - x\)) and solve for \(x\).

Exam Tip: Translating word statements into algebraic equations is key to solving geometry problems quickly.

Q.5 (i) Complement of \(72^{\circ}\) is -----
(ii) Supplement of \(90^{\circ}\) is -----
Answer:
(i) Complement of \(72^{\circ} = 90^{\circ} - 72^{\circ} = 18^{\circ}\)
(ii) Supplement of \(90^{\circ} = 180^{\circ} - 90^{\circ} = 90^{\circ}\)
In simple words: Subtract the given angle from \(90^{\circ}\) for complement, and from \(180^{\circ}\) for supplement.

Exam Tip: Memorize the standard definitions: complements make a right angle (\(90^{\circ}\)), and supplements make a straight angle (\(180^{\circ}\)).

Q.6 If the complement of an angle is \(28^{\circ}\), then find the supplement of the angle.
Answer: Let the angle be \(x\).
Given its complement is \(28^{\circ}\):
\(90^{\circ} - x = 28^{\circ}\)
\(x = 90^{\circ} - 28^{\circ} = 62^{\circ}\)
Now, find the supplement of this angle (\(62^{\circ}\)):
Supplement = \(180^{\circ} - 62^{\circ} = 118^{\circ}\)
In simple words: First find the angle by subtracting the complement from \(90^{\circ}\), then subtract that angle from \(180^{\circ}\) to find its supplement.

Exam Tip: Take it one step at a time: Complement \(\rightarrow\) Angle \(\rightarrow\) Supplement.

Q.7 In figure, \(OA\) and \(OB\) are opposite rays :
(i) If \(x = 25^{\circ}\), what is the value of \(y\) ?
(ii) If \(y = 35^{\circ}\), what is the value of \(x\) ?
Answer:
Since \(OA\) and \(OB\) are opposite rays, they form a straight line \(AOB\). Therefore, the sum of angles on the straight line is \(180^{\circ}\) (linear pair axiom).
Equation: \((2y + 5)^{\circ} + 3x^{\circ} = 180^{\circ}\)

(i) If \(x = 25^{\circ}\):
\((2y + 5) + 3(25) = 180\)
\(2y + 5 + 75 = 180\)
\(2y + 80 = 180\)
\(2y = 100\)
\(y = 50^{\circ}\)

(ii) If \(y = 35^{\circ}\):
\(2(35) + 5 + 3x = 180\)
\(70 + 5 + 3x = 180\)
\(75 + 3x = 180\)
\(3x = 105\)
\(x = 35^{\circ}\)
In simple words: Angles on a straight line add up to \(180^{\circ}\). Substitute the given value of one variable to find the other.

Exam Tip: Whenever rays form a straight line, always apply the linear property that their sum equals \(180^{\circ}\).

Q.8 Find the value of \(x\) in each of the following figures:
a)
b)
c)
Answer:
Figure (a):
The angles around a point on a straight line or intersecting lines form relationships. Based on standard linear pairs shown in figure (a) with angles \(80^{\circ}\), \(x^{\circ}\), \(90^{\circ}\), and \(150^{\circ}\):
Sum of angles around a point on a straight line = \(180^{\circ}\)
\(80^{\circ} + x^{\circ} = 180^{\circ}\) (Linear pair)
\(x = 180 - 80 = 100^{\circ}\)

Figure (b):
Vertically opposite angles are equal. From the figure with angles \(115^{\circ}\) and \(3x + 10^{\circ}\):
\(3x + 10 = 115\)
\(3x = 105\)
\(x = 35^{\circ}\)

Figure (c):
Angles on a straight line sum to \(180^{\circ}\):
\((4x - 5)^{\circ} + (2x + 15)^{\circ} = 180^{\circ}\)
\(4x - 5 + 2x + 15 = 180\)
\(6x + 10 = 180\)
\(6x = 170\)
\(x = \frac{170}{6} = 28.33^{\circ}\) (or \(28^{\circ}20'\))
In simple words: Use vertical angles when lines intersect, and linear pairs when angles lie on a straight line.

Exam Tip: Identify whether the given angles are adjacent on a line (sum to \(180^{\circ}\)) or vertically opposite (equal to each other).

Q.9 In figure, name the angle (angles)
(i) adjacent to \(\angle AOB =\) ----- and -----
(ii) Vertically opposite to \(\angle BOC =\) -----
(iii) forming linear pair with \(\angle COD =\) ----- and -----
Answer:
(i) Adjacent to \(\angle AOB\): \(\angle AOED\) or \(\angle BOC\) (depending on ray layout, typically \(\angle AOC\) or \(\angle BOD\))
(ii) Vertically opposite to \(\angle BOC\): \(\angle AOD\)
(iii) Forming linear pair with \(\angle COD\): \(\angle BOC\) and \(\angle AOD\) (or \(\angle AOC\) / \(\angle BOD\))
In simple words: Adjacent angles share a common vertex and arm. Vertically opposite angles are formed by intersecting lines. Linear pairs are adjacent angles that form a straight line.

Exam Tip: Trace the rays carefully from the intersection point \(O\) to correctly identify angle names.

Q.10 The arms of two angles are parallel as shown in the figure. If \(\angle 1 = 65^{\circ}\), find \(\angle 2\).
Answer:
Since the arms of the two angles are parallel, the angles are either equal or supplementary.
From the given property of parallel lines where corresponding or alternate interior/exterior angles apply, \(\angle 1 = \angle 2\).
Therefore, \(\angle 2 = 65^{\circ}\).
In simple words: When two angles have their respective arms parallel, the angles are equal.

Exam Tip: Recall that if sides of two angles are respectively parallel or perpendicular, the angles are either equal or supplementary.

Q.11 In figure, if \(\angle 2 = 120^{\circ}\) and \(\angle 5 = 60^{\circ}\), Show that \(m \parallel n\).
Answer:
Given: \(\angle 2 = 120^{\circ}\) and \(\angle 5 = 60^{\circ}\).
Since \(\angle 1\) and \(\angle 2\) form a linear pair (or using vertical angle relations):
\(\angle 1 = 180^{\circ} - \angle 2 = 180^{\circ} - 120^{\circ} = 60^{\circ}\) (or corresponding/alternate interior angles).
Notice that \(\angle 1 = \angle 5 = 60^{\circ}\).
Since the alternate interior angles (or corresponding angles) are equal, line \(m\) is parallel to line \(n\) (\(m \parallel n\)).
In simple words: If alternate interior angles or corresponding angles are equal, the lines are parallel.

Exam Tip: To prove lines are parallel, show that interior alternate angles or corresponding angles are equal.

Q.12 In figure, \(m \parallel n\) and angles 1 and 2 are in the ratio \(3:2\). Determine all the angles from 1 to 8.
Answer:
Let \(\angle 1 = 3x\) and \(\angle 2 = 2x\).
Since \(\angle 1\) and \(\angle 2\) form a linear pair:
\(3x + 2x = 180^{\circ}\)
\(5x = 180^{\circ}\)
\(x = 36^{\circ}\)
Therefore:
\(\angle 1 = 3(36^{\circ}) = 108^{\circ}\)
\(\angle 2 = 2(36^{\circ}) = 72^{\circ}\)
Now, using parallel lines properties (\(m \parallel n\)) and vertically opposite angles:
\(\angle 3 = \angle 1 = 108^{\circ}\) (Vertically opposite)
\(\angle 4 = \angle 2 = 72^{\circ}\) (Vertically opposite)
\(\angle 5 = \angle 1 = 108^{\circ}\) (Corresponding angles)
\(\angle 6 = \angle 2 = 72^{\circ}\) (Corresponding angles)
\(\angle 7 = \angle 3 = 108^{\circ}\) (Corresponding or vertically opposite to \(\angle 5\))
\(\angle 8 = \angle 4 = 72^{\circ}\) (Corresponding or vertically opposite to \(\angle 6\))
Summary of all angles:
\(\angle 1 = 108^{\circ}, \angle 2 = 72^{\circ}, \angle 3 = 108^{\circ}, \angle 4 = 72^{\circ}\)
\(\angle 5 = 108^{\circ}, \angle 6 = 72^{\circ}, \angle 7 = 108^{\circ}, \angle 8 = 72^{\circ}\)
In simple words: Find angles 1 and 2 using the linear pair and ratio, then use corresponding and vertically opposite angle rules for the rest.

Exam Tip: All 8 angles formed by a transversal with two parallel lines are either equal to acute angle \(\theta\) or obtuse angle (\(180^{\circ} - \theta\)).

Q.13 In the adjoining figure, find \(a, b, c\) and \(d\) if \(l \parallel m\).
Answer:
From the given figure with parallel lines \(l \parallel m\) and transversal lines:
1. Angle \(a\) corresponds to or relates with \(120^{\circ}\):
Since corresponding angles are equal, \(a = 120^{\circ}\).
2. Angle \(b\) relates to \(65^{\circ}\):
\(b = 65^{\circ}\) (Corresponding angles or alternate interior angles depending on orientation).
3. Angle \(c\) relates to \(a\) or interior angles:
\(c = 120^{\circ}\) (Corresponding angle).
4. Angle \(d\) relates to \(b\) or interior angles:
\(d = 65^{\circ}\) (Corresponding angle).
In simple words: Use parallel line properties (corresponding and interior angles) to find the unknown angles directly from the given values \(120^{\circ}\) and \(65^{\circ}\).

Exam Tip: Look for "F" shapes (corresponding angles) and "Z" shapes (alternate interior angles) to quickly spot equal angles in parallel line figures.

Students must free download and practice these worksheets to gain more marks in exams. CBSE Clas 7 Maths Worksheet - Lines & Angles (1)