GSEB Class 7 Maths Solutions Chapter 5 Lines and Angles Exercise 5.1

Question 1. Find the complement of each of the following angles:
(i) 20°
(ii) 63°
(iii) 57°
Answer:
(i) The complement of \( 20^\circ \) is found by subtracting it from \( 90^\circ \), so \( 90^\circ - 20^\circ = 70^\circ \).
(ii) The complement of \( 63^\circ \) is calculated as \( 90^\circ - 63^\circ = 27^\circ \).
(iii) The complement of \( 57^\circ \) is determined by \( 90^\circ - 57^\circ = 33^\circ \).
In simple words: To find the complement of an angle, you subtract it from 90 degrees. The two angles together make 90 degrees.

Exam Tip: Remember that complementary angles add up to \( 90^\circ \). Always perform the subtraction carefully to avoid calculation mistakes.

Question 2. Find the supplement of each of the following angles:
(i) 105°
(ii) 87°
(iii) 154°
Answer:
(i) The supplement of \( 105^\circ \) is obtained by subtracting it from \( 180^\circ \), which gives \( 180^\circ - 105^\circ = 75^\circ \).
(ii) The supplement of \( 87^\circ \) is found by calculating \( 180^\circ - 87^\circ = 93^\circ \).
(iii) The supplement of \( 154^\circ \) is determined as \( 180^\circ - 154^\circ = 26^\circ \).
In simple words: To find the supplement of an angle, you subtract it from 180 degrees. The two angles together make 180 degrees.

Exam Tip: Supplementary angles always add up to \( 180^\circ \). Ensure you use \( 180^\circ \) for supplementary and \( 90^\circ \) for complementary angles.

Question 3. Identify which of the following pairs of angles are complementary and which are supplementary.
(i) 65°, 115°
(ii) 63°, 27°
(iii) 112°, 68°
(iv) 130°, 50°
(v) 45°, 45°
(vi) 80°, 10°
Answer:
(i) For \( 65^\circ \) and \( 115^\circ \), their sum is \( 65^\circ + 115^\circ = 180^\circ \). Therefore, these are supplementary angles.
(ii) For \( 63^\circ \) and \( 27^\circ \), their sum is \( 63^\circ + 27^\circ = 90^\circ \). Therefore, these are complementary angles.
(iii) For \( 112^\circ \) and \( 68^\circ \), their sum is \( 112^\circ + 68^\circ = 180^\circ \). Therefore, these are supplementary angles.
(iv) For \( 130^\circ \) and \( 50^\circ \), their sum is \( 130^\circ + 50^\circ = 180^\circ \). Therefore, these are supplementary angles.
(v) For \( 45^\circ \) and \( 45^\circ \), their sum is \( 45^\circ + 45^\circ = 90^\circ \). Therefore, these are complementary angles.
(vi) For \( 80^\circ \) and \( 10^\circ \), their sum is \( 80^\circ + 10^\circ = 90^\circ \). Therefore, these are complementary angles.
In simple words: Add the two angles together. If they make 90 degrees, they are complementary. If they make 180 degrees, they are supplementary.

Exam Tip: The key difference between complementary and supplementary angles is the sum: \( 90^\circ \) for complementary and \( 180^\circ \) for supplementary. Practice summing different pairs of angles quickly.

Question 4. Find the angle which is equal to its complement.
Answer:
Let the required angle be \( x \).
Since it is equal to its complement, we can write the equation:
\( x = 90^\circ - x \)
To solve for \( x \), we move \( -x \) from the right side to the left side:
\( x + x = 90^\circ \)
\( 2x = 90^\circ \)
Now, we divide both sides by 2 to find the value of \( x \):
\( \frac{2x}{2} = \frac{90^\circ}{2} \)
\( x = 45^\circ \)
Thus, the angle that equals its complement is \( 45^\circ \).
In simple words: If an angle is the same size as its complement, it must be 45 degrees, because 45 plus 45 equals 90.

Exam Tip: When a question asks for an angle equal to its complement or supplement, set up an equation where the angle (x) equals its complement (\( 90^\circ - x \)) or supplement (\( 180^\circ - x \)).

Question 5. Find the angle which is equal to its supplement.
Answer:
Let the required angle be \( m \).
Its supplement is \( (180^\circ - m) \).
Since the angle is equal to its supplement, we can set up the equation:
\( m = 180^\circ - m \)
To solve for \( m \), we move \( -m \) from the right side to the left side:
\( m + m = 180^\circ \)
\( 2m = 180^\circ \)
Now, we divide both sides by 2 to find the value of \( m \):
\( \frac{2m}{2} = \frac{180^\circ}{2} \)
\( m = 90^\circ \)
Thus, the angle that equals its supplement is \( 90^\circ \).
In simple words: If an angle is the same size as its supplement, it must be 90 degrees, because 90 plus 90 equals 180.

Exam Tip: Always remember that the sum of an angle and its supplement is \( 180^\circ \). A \( 90^\circ \) angle is the only angle that is equal to its own supplement.

Question 6. In the given figure, ∠1 and ∠2 are supplementary angles. If ∠1 is decreased, what changes should take place in ∠2 so that both the angles still remain supplementary.
Answer:
If \( \angle 1 \) is decreased by a certain amount, then \( \angle 2 \) must be increased by the exact same amount. This ensures that the sum of both angles still remains \( 180^\circ \), keeping them supplementary.
In simple words: If one angle gets smaller, the other angle has to get bigger by the same amount to keep their total at 180 degrees.

Exam Tip: For supplementary angles, if one angle changes, the other must change in the opposite direction by the same magnitude to maintain their sum of \( 180^\circ \).

Question 7. Can two angles be supplementary if both of them are:
(i) acute?
(ii) obtuse
(iii) right?
Answer:
(i) No, two acute angles cannot be supplementary. An acute angle is less than \( 90^\circ \). If you add two angles that are both less than \( 90^\circ \), their sum will always be less than \( 180^\circ \).
(ii) No, two obtuse angles cannot be supplementary. An obtuse angle is greater than \( 90^\circ \). If you add two angles that are both greater than \( 90^\circ \), their sum will always be more than \( 180^\circ \).
(iii) Yes, two right angles can be supplementary. A right angle is exactly \( 90^\circ \). If you add two right angles, their sum is \( 90^\circ + 90^\circ = 180^\circ \), which means they are supplementary.
In simple words: Two small angles can't make 180 degrees. Two big angles also can't make 180 degrees. Only two 90-degree angles can add up to 180 degrees.

Exam Tip: Understand the definitions: acute (less than \( 90^\circ \)), obtuse (greater than \( 90^\circ \)), and right (exactly \( 90^\circ \)). Use these definitions to test the sum for supplementary angles (\( 180^\circ \)).

Question 8. An angle is greater than 45°. Is its complementary angle greater than 45° or equal to 45° or less than 45"?
Answer:
If an angle is greater than \( 45^\circ \), its complementary angle will be less than \( 45^\circ \). This is because the sum of complementary angles must be \( 90^\circ \). If one angle takes up more than half of \( 90^\circ \), the other must take up less than half.
In simple words: If one angle is bigger than 45 degrees, its complement must be smaller than 45 degrees so that they still add up to 90 degrees.

Exam Tip: For complementary angles, if one angle is \( 45^\circ \), its complement is also \( 45^\circ \). If one is larger than \( 45^\circ \), the complement must be smaller, and vice-versa.

Question 9. In the adjoining figure:
(i) Is \( \angle 1 \) adjacent to \( \angle 2 \)?
(ii) Is \( \angle AOC \) adjacent to \( \angle AOE \)?
(iii) Do \( \angle COE \) and \( \angle EOD \) form a linear pair?
(iv) Are \( \angle BOD \) and \( \angle DOA \) supplementary?
(v) Is \( \angle 1 \) vertically opposite to \( \angle 4 \)?
(vi) Which is the vertically opposite angle of \( \angle 5 \)?
Answer:
(i) Yes, \( \angle 1 \) and \( \angle 2 \) are adjacent angles. They share a common vertex O and a common arm OC.
(ii) No, \( \angle AOC \) is not adjacent to \( \angle AOE \). This is because \( \angle AOC \) is a part of \( \angle AOE \), and for angles to be adjacent, they must share a common side and vertex but not overlap.
(iii) Yes, \( \angle COE \) and \( \angle EOD \) form a linear pair. This is because they are adjacent angles that form a straight line \( \overleftrightarrow{CD} \).
(iv) Yes, \( \angle BOD \) and \( \angle DOA \) are supplementary. They form a linear pair along the straight line \( \overleftrightarrow{AB} \), so their sum is \( 180^\circ \).
(v) Yes, \( \angle 1 \) is vertically opposite to \( \angle 4 \). This is because they are formed by the intersection of two straight lines \( \overleftrightarrow{AB} \) and \( \overleftrightarrow{CD} \).
(vi) The vertically opposite angle of \( \angle 5 \) is \( \angle BOC \) (or \( \angle COB \)). This angle is formed directly across the vertex O from \( \angle 5 \).
In simple words: Look at the picture carefully. Adjacent angles share a side. A linear pair makes a straight line. Vertically opposite angles are across from each other when two lines cross.

Exam Tip: For questions involving diagrams, always visually identify the common vertex, common arm, and straight lines to correctly determine adjacent, linear pair, and vertically opposite angles.

Question 10. Indicate which pairs of angles are:
(i) Vertically opposite angles.
(ii) Linear pairs.
Answer:
(i) The vertically opposite angles in the figure are:
\( \angle 1 \) and \( \angle 4 \)
\( \angle 5 \) and \( (\angle 2 + \angle 3) \)
(ii) The linear pairs in the figure are:
\( \angle 4 \) and \( \angle 5 \)
\( \angle 1 \) and \( \angle 5 \)
\( \angle 1 \) and \( (\angle 3 + \angle 2) \)
\( \angle 4 \) and \( (\angle 1 + \angle 2) \)
In simple words: Vertically opposite angles are across from each other when lines cross. Linear pairs are adjacent angles that form a straight line.

Exam Tip: Remember that vertically opposite angles are equal, and angles forming a linear pair add up to \( 180^\circ \). These properties help verify your identifications.

Question 11. In the adjoining figure, is \( \angle 1 \) adjacent to \( \angle 2 \)? Give reasons.
Answer:
No, \( \angle 1 \) and \( \angle 2 \) are not adjacent angles. The reason is that they do not have a common vertex. For angles to be adjacent, they must share a common vertex and a common arm, and their non-common arms must be on opposite sides of the common arm.
In simple words: No, they aren't adjacent because they don't share the same corner point.

Exam Tip: The definition of adjacent angles requires a common vertex and a common arm. If either of these conditions is not met, the angles are not adjacent.

Question 12. Find the value of the angles x, y and z in each of the following:
(i)
(ii)
Answer:
(i) From the figure, we observe the following:
Since \( x \) and \( 55^\circ \) are vertically opposite angles, they are equal.
\( x = 55^\circ \)
Also, \( 55^\circ \) and \( y \) form a linear pair on a straight line, so their sum is \( 180^\circ \).
\( 55^\circ + y = 180^\circ \)
To find \( y \), we subtract \( 55^\circ \) from \( 180^\circ \):
\( y = 180^\circ - 55^\circ \)
\( y = 125^\circ \)
Furthermore, \( z \) and \( y \) are vertically opposite angles, which means they are equal.
Since \( y = 125^\circ \), then \( z = 125^\circ \).
Thus, the values are \( x = 55^\circ \), \( y = 125^\circ \), and \( z = 125^\circ \).
(ii) From the figure, we observe the following:
Since \( 40^\circ \) and \( z \) are vertically opposite angles, they are equal.
\( z = 40^\circ \)
Also, \( y \) and \( 40^\circ \) form a linear pair on a straight line, so their sum is \( 180^\circ \).
\( y + 40^\circ = 180^\circ \)
To find \( y \), we transpose \( 40^\circ \) to the right side:
\( y = 180^\circ - 40^\circ \)
\( y = 140^\circ \)
Additionally, \( y \) and \( (x + 25^\circ) \) are vertically opposite angles, so they are equal.
\( (x + 25^\circ) = y \)
Since \( y = 140^\circ \), we have:
\( x + 25^\circ = 140^\circ \)
To find \( x \), we transpose \( 25^\circ \) to the right side:
\( x = 140^\circ - 25^\circ \)
\( x = 115^\circ \)
Thus, the values are \( x = 115^\circ \), \( y = 140^\circ \), and \( z = 40^\circ \).
In simple words: Use the rules for vertical angles (they are equal) and linear pairs (they add up to 180 degrees) to find all the unknown angles. Work step-by-step.

Exam Tip: When solving for multiple angles, start by identifying the easiest relationships (e.g., vertically opposite angles or linear pairs) to find initial values, then use those to find the remaining angles.

Question 13. Fill in the blanks:
(i) If two angles are complementary, then the sum of their measures is ______.
(ii) If two angles are supplementary, then the sum of their measures is ______.
(iii) Two angles forming a linear pair are ______.
(iv) If two adjacent angles are supplementary, they form a ______.
(v) If two lines intersect at a point, then the vertically opposite angles are always ______.
(vi) If two lines intersect at a point, and if one pair of vertically opposite angles are acute angles, then the other pair of vertically opposite angles are ______.
Answer:
(i) If two angles are complementary, then the sum of their measures is \( \underline{90^\circ} \).
(ii) If two angles are supplementary, then the sum of their measures is \( \underline{180^\circ} \).
(iii) Two angles forming a linear pair are \( \underline{supplementary} \).
(iv) If two adjacent angles are supplementary, they form a \( \underline{linear \ pair} \).
(v) If two lines intersect at a point, then the vertically opposite angles are always \( \underline{equal} \).
(vi) If two lines intersect at a point, and if one pair of vertically opposite angles are acute angles, then the other pair of vertically opposite angles are \( \underline{obtuse \ angles} \).
In simple words: Complementary angles sum to 90 degrees, supplementary to 180 degrees. Linear pairs are supplementary. Vertically opposite angles are equal. If one pair of vertical angles is acute, the other pair is obtuse.

Exam Tip: Memorize the definitions and properties of complementary angles, supplementary angles, linear pairs, and vertically opposite angles. These are fundamental concepts.

Question 14. In the adjoining figure, name the following pairs of angles:
(i) Obtuse vertically opposite angles.
(ii) Adjacent complementary angles.
(iii) Equal supplementary angles.
(iv) Unequal supplementary angles.
(v) Adjacent angles that do not form a linear pair.
Answer:
(i) The obtuse vertically opposite angles are \( \angle BOC \) and \( \angle AOD \). These angles are greater than \( 90^\circ \) and are opposite each other.
(ii) The adjacent complementary angles are \( \angle AOB \) and \( \angle AOE \). These angles are next to each other and add up to \( 90^\circ \).
(iii) The equal supplementary angles are \( \angle BOE \) and \( \angle EOD \). These angles are both \( 90^\circ \) (implied, as they are equal and supplementary) and add up to \( 180^\circ \).
(iv) The unequal supplementary angles are \( \angle AOE \) and \( \angle EOC \). These angles add up to \( 180^\circ \) but are not equal in measure.
(v) The adjacent angles that do not form a linear pair are \( \angle AOB \) and \( \angle BOC \). While adjacent, their sum does not form a straight line, so they do not add up to \( 180^\circ \).
In simple words: Look at the diagram. Find pairs that are big and opposite (obtuse vertical). Find pairs next to each other that make 90 degrees (adjacent complementary). Find pairs that are equal and make 180 degrees (equal supplementary, implies two 90-degree angles). Find pairs that make 180 degrees but are different sizes (unequal supplementary). Find pairs next to each other that don't make a straight line.

Exam Tip: For angle identification, carefully examine the lines and rays to determine relationships. Pay attention to whether angles share a common vertex, a common arm, or form a straight line.