CBSE Class 7 Mathematics Lines And Angles Worksheet Set 04

Question 1. Write the complements of the following angles:
\( 15^\circ, 28^\circ, 30^\circ, 45^\circ, 59^\circ, 67^\circ, 75^\circ, 87^\circ, 90^\circ, x^\circ \)
Answer:
To find the complement of any angle, we subtract it from \( 90^\circ \).
- Complement of \( 15^\circ = 90^\circ - 15^\circ = 75^\circ \)
- Complement of \( 28^\circ = 90^\circ - 28^\circ = 62^\circ \)
- Complement of \( 30^\circ = 90^\circ - 30^\circ = 60^\circ \)
- Complement of \( 45^\circ = 90^\circ - 45^\circ = 45^\circ \)
- Complement of \( 59^\circ = 90^\circ - 59^\circ = 31^\circ \)
- Complement of \( 67^\circ = 90^\circ - 67^\circ = 23^\circ \)
- Complement of \( 75^\circ = 90^\circ - 75^\circ = 15^\circ \)
- Complement of \( 87^\circ = 90^\circ - 87^\circ = 3^\circ \)
- Complement of \( 90^\circ = 90^\circ - 90^\circ = 0^\circ \)
- Complement of \( x^\circ = (90 - x)^\circ \)
In simple words: Subtract the angle from \( 90^\circ \) to find its complement.

Exam Tip: Check your work by making sure both angles add up to exactly \( 90^\circ \).

Question 2. Write the supplements of the following angles:
\( 25^\circ, 36^\circ, 42^\circ, 65^\circ, 90^\circ, 110^\circ, 120^\circ, 125^\circ, 138^\circ, 147^\circ, 175^\circ, x^\circ \)
Answer:
To find the supplement of any angle, we subtract it from \( 180^\circ \).
- Supplement of \( 25^\circ = 180^\circ - 25^\circ = 155^\circ \)
- Supplement of \( 36^\circ = 180^\circ - 36^\circ = 144^\circ \)
- Supplement of \( 42^\circ = 180^\circ - 42^\circ = 138^\circ \)
- Supplement of \( 65^\circ = 180^\circ - 65^\circ = 115^\circ \)
- Supplement of \( 90^\circ = 180^\circ - 90^\circ = 90^\circ \)
- Supplement of \( 110^\circ = 180^\circ - 110^\circ = 70^\circ \)
- Supplement of \( 120^\circ = 180^\circ - 120^\circ = 60^\circ \)
- Supplement of \( 125^\circ = 180^\circ - 125^\circ = 55^\circ \)
- Supplement of \( 138^\circ = 180^\circ - 138^\circ = 42^\circ \)
- Supplement of \( 147^\circ = 180^\circ - 147^\circ = 33^\circ \)
- Supplement of \( 175^\circ = 180^\circ - 175^\circ = 5^\circ \)
- Supplement of \( x^\circ = (180 - x)^\circ \)
In simple words: Subtract the angle from \( 180^\circ \) to find its supplement.

Exam Tip: Be careful with subtraction when finding the supplement of angles close to \( 180^\circ \).

Question 3. Write a pair of equal angles which are
i) Complementary
ii) Supplementary
Answer:
i) Let the two equal complementary angles be \( x \).
\( x + x = 90^\circ \implies 2x = 90^\circ \implies x = 45^\circ \)
The pair of equal complementary angles is \( 45^\circ \) and \( 45^\circ \).

ii) Let the two equal supplementary angles be \( y \).
\( y + y = 180^\circ \implies 2y = 180^\circ \implies y = 90^\circ \)
The pair of equal supplementary angles is \( 90^\circ \) and \( 90^\circ \).
In simple words: Equal complementary angles are \( 45^\circ \) each, and equal supplementary angles are \( 90^\circ \) each.

Exam Tip: Remember that complementary angles must be acute, while equal supplementary angles are always right angles.

Question 4. In the following figures, find the values of x, y and z

Answer:
i) The angles \( 45^\circ \) and \( z^\circ \) form a linear pair:
\( z = 180^\circ - 45^\circ = 135^\circ \)
The angles \( x^\circ \) and \( 45^\circ \) are vertically opposite:
\( x = 45^\circ \)
The angles \( y^\circ \) and \( z^\circ \) are vertically opposite:
\( y = z = 135^\circ \)

ii) Vertically opposite angles are equal:
\( z = 40^\circ \)
The angles \( y^\circ \) and \( z^\circ \) form a linear pair:
\( y = 180^\circ - 40^\circ = 140^\circ \)
The sum of angles on a straight line is \( 180^\circ \):
\( 40^\circ + x + 25^\circ = 180^\circ \implies x + 65^\circ = 180^\circ \implies x = 115^\circ \)

iii) The angles \( x^\circ \) and \( 55^\circ \) form a linear pair:
\( x = 180^\circ - 55^\circ = 125^\circ \)

iv) The right-angle marker shows a \( 90^\circ \) angle:
\( x + 41^\circ = 90^\circ \implies x = 49^\circ \)
Since \( y \) forms a right angle on the straight line:
\( y = 90^\circ \)
Angle \( z \) is vertically opposite to \( y \):
\( z = 90^\circ \)

v) The adjacent angles on the straight line form a linear pair:
\( (x + 40) + (x - 10) = 180^\circ \implies 2x + 30 = 180 \implies 2x = 150 \implies x = 75^\circ \)
Now find \( y \) and \( z \) using vertically opposite angles:
\( y = x + 40^\circ = 75^\circ + 40^\circ = 115^\circ \)
\( z = x - 10^\circ = 75^\circ - 10^\circ = 65^\circ \)

vi) The angle \( 2x \) is vertically opposite to \( 60^\circ \):
\( 2x = 60^\circ \implies x = 30^\circ \)
The angles \( y \), \( x \), and \( 2x \) lie on a straight line:
\( y + 30^\circ + 60^\circ = 180^\circ \implies y = 90^\circ \)
Angle \( z \) is vertically opposite to \( y \):
\( z = 90^\circ \)
In simple words: Match opposite angles first, then subtract from \( 180^\circ \) to find angles next to each other on a straight line.

Exam Tip: Label the vertically opposite angles first since they do not require any calculation to solve.

Question 5. Two Complementary angles are such that the measure of one is twice the measure of the other. Find the angles
Answer:
Let the smaller angle be \( x \).
The larger angle is twice as big, so it is \( 2x \).
Since they are complementary, their sum is \( 90^\circ \):
\( x + 2x = 90^\circ \implies 3x = 90^\circ \implies x = 30^\circ \)
The smaller angle is \( 30^\circ \) and the larger angle is \( 2 \times 30^\circ = 60^\circ \).
In simple words: The two angles are \( 30^\circ \) and \( 60^\circ \), which add up to a right angle of \( 90^\circ \).

Exam Tip: Set up the algebraic equation with \( x \) and \( 2x \) to show a clear step-by-step mathematical method.

Question 6. In the figure, l // m and ∠1 = 50° Find the measures of the other Seven angles

Answer:
Given \( \angle 1 = 50^\circ \) and \( l \parallel m \):
- \( \angle 3 = \angle 1 = 50^\circ \) (Vertically opposite angles)
- \( \angle 2 = 180^\circ - \angle 1 = 180^\circ - 50^\circ = 130^\circ \) (Linear pair)
- \( \angle 4 = \angle 2 = 130^\circ \) (Vertically opposite angles)
Using parallel line rules, we find the angles on line \( m \):
- \( \angle 5 = \angle 1 = 50^\circ \) (Corresponding angles)
- \( \angle 6 = \angle 2 = 130^\circ \) (Corresponding angles)
- \( \angle 7 = \angle 3 = 50^\circ \) (Corresponding angles)
- \( \angle 8 = \angle 4 = 130^\circ \) (Corresponding angles)
In simple words: The angles at the same relative corners are equal, meaning four angles are \( 50^\circ \) and four are \( 130^\circ \).

Exam Tip: Mention the name of the angle property (e.g., "corresponding angles" or "vertically opposite") for each calculated value.

Question 7. check whether the lines l and m are parallel or not

Answer:
i) The given interior angle on line \( l \) is \( 47^\circ \). The corresponding exterior angle on line \( m \) is \( 123^\circ \).
If the lines were parallel, these corresponding angles would be equal. Since \( 47^\circ \neq 123^\circ \), lines \( l \) and \( m \) are **not parallel**.

ii) The consecutive interior angles on the same side of the transversal are \( 50^\circ \) and \( 130^\circ \).
Let us add them: \( 50^\circ + 130^\circ = 180^\circ \).
Since their sum is supplementary (\( 180^\circ \)), lines \( l \) and \( m \) **are parallel**.

iii) The corresponding angles on lines \( l \) and \( m \) are both \( 120^\circ \).
Since corresponding angles are equal, lines \( l \) and \( m \) **are parallel**.
In simple words: Lines are parallel if their corresponding angles are equal, or if their consecutive interior angles add up to \( 180^\circ \).

Exam Tip: State the exact property used (e.g., "co-interior angles are supplementary") to justify your parallel lines answer.

Question 8. In the figure, l // m and t, s are transversals. Find x, y, z

Answer:
i) Let us find the angles step-by-step:
- On line \( l \), the interior angle and \( 120^\circ \) form a linear pair:
\( \text{Interior angle} = 180^\circ - 120^\circ = 60^\circ \)
- In the triangle formed by the intersecting lines:
\( x + 55^\circ + 60^\circ = 180^\circ \implies x + 115^\circ = 180^\circ \implies x = 65^\circ \)
- Since \( l \parallel m \):
\( y = x = 65^\circ \) (Alternate interior angles)
\( z = 60^\circ \) (Alternate interior angles)

ii) Using parallel line properties:
- The adjacent angles \( x \) and \( 100^\circ \) form a linear pair:
\( x = 180^\circ - 100^\circ = 80^\circ \)
- Since lines are parallel:
\( y = x = 80^\circ \) (Corresponding angles)
\( z = 100^\circ \) (Alternate interior angles with the given \( 100^\circ \) angle)
In simple words: Look for triangle shapes to find internal angles, then match parallel lines to solve the rest.

Exam Tip: Clearly write down which transversal line you are using when calculating alternate or corresponding angles.

Question 9. In the adjacent figure, lines AB and CD are intersected by a transversal PQ at E and F. Name two pairs of
i) Corresponding angles.
ii) alternate interior angles.
iii) alternate exterior angles.
iv) interior angles on the same side of the Transversal (Consecutive int. angles)
v) exterior angles on the same side of the transversal
Answer:
Based on the intersection of lines \( AB \) and \( CD \) by transversal \( PQ \):

i) Corresponding angles:
- Pair 1: \( \angle PEB \) and \( \angle EFD \)
- Pair 2: \( \angle AEP \) and \( \angle CFE \)

ii) Alternate interior angles:
- Pair 1: \( \angle AEF \) and \( \angle EFD \)
- Pair 2: \( \angle FEB \) and \( \angle CFE \)

iii) Alternate exterior angles:
- Pair 1: \( \angle AEP \) and \( \angle DFQ \)
- Pair 2: \( \angle PEB \) and \( \angle CFQ \)

iv) Interior angles on the same side of the transversal:
- Pair 1: \( \angle AEF \) and \( \angle CFE \)
- Pair 2: \( \angle FEB \) and \( \angle EFD \)

v) Exterior angles on the same side of the transversal:
- Pair 1: \( \angle AEP \) and \( \angle CFQ \)
- Pair 2: \( \angle PEB \) and \( \angle DFQ \)
In simple words: These terms describe pairs of angles positioned at matching or alternating corners of the intersections.

Exam Tip: Use three-letter angle notation (e.g., \( \angle AEF \)) instead of single letters to avoid confusion.

Question 10. In the adj. figure l // m // n Find the values of x and y
Answer:
Since lines \( l \), \( m \), and \( n \) are parallel to each other:
- For the transversal line crossing \( l \) and \( m \), the alternate interior angles are equal:
\( x = 48^\circ \)
- For the transversal line crossing \( m \) and \( n \), the alternate interior angles are equal:
\( y = 25^\circ \)
In simple words: The angles \( x \) and \( y \) match the given values directly because of parallel line rules.

Exam Tip: Look for the "Z" shape to locate alternate interior angles when working with multiple parallel lines.