ML Aggarwal Class 7 Maths Solutions Chapter 10 Lines and Angles

Question 1. (i) Can two right angles be complementary?
Answer: No. When you add two right angles together, you get 90° + 90° = 180°. Since this sum is 180° and not 90°, they cannot be called complementary.
In simple words: Two right angles add up to 180°, not 90°, so they are not complementary.

Exam Tip: Remember that complementary angles must sum to exactly 90° - two right angles sum to 180°, making them supplementary instead.

Question 1. (ii) Can two right angles be supplementary?
Answer: Yes. Two right angles add up to 90° + 90° = 180°. Since supplementary angles must sum to 180°, two right angles are supplementary.
In simple words: Two right angles make 180° together, which is what supplementary angles do.

Exam Tip: Supplementary means the angles add to 180° - two 90° angles fit this definition perfectly.

Question 1. (iii) Can two adjacent angles be complementary?
Answer: Yes. Two adjacent angles can be complementary. They can share a common vertex and a common arm, and their angle measures can add up to 90°. For instance, one angle of 40° and another of 50° sitting next to each other would be complementary adjacent angles.
In simple words: Two angles side by side can add to 90° if they share a vertex and an arm.

Exam Tip: Adjacent angles sharing a vertex and common arm can have any pair of measures that sum to 90° - this is a key property to remember.

Question 1. (iv) Can two adjacent angles be supplementary?
Answer: Yes. Two adjacent angles can be supplementary. When they share a common vertex and a common arm, and their measures add up to 180°, they form what is called a linear pair. This is a common relationship in geometry.
In simple words: Two angles next to each other can add up to 180° and form a linear pair.

Exam Tip: A linear pair is the most common example of supplementary adjacent angles - they sit on a straight line.

Question 1. (v) Can two obtuse angles be adjacent?
Answer: Yes. Two obtuse angles can be adjacent. They can share a common vertex and a common arm. What matters is that their non-common arms (the arms that do not touch) are positioned on different sides of the shared arm. This arrangement allows two obtuse angles to sit next to each other.
In simple words: Two wide angles can be next to each other if they share a vertex and arm, with their other arms pointing in different directions.

Exam Tip: Adjacent angles do not require the measures to add to any particular value - they just need a shared vertex and arm.

Question 1. (vi) Can an acute angle be adjacent to an obtuse angle?
Answer: Yes. An acute angle and an obtuse angle can be adjacent. They can share a common vertex and a common arm. When positioned this way, the acute angle (less than 90°) sits next to the obtuse angle (greater than 90°), and they qualify as adjacent angles.
In simple words: A small angle and a large angle can be next to each other if they share a vertex and an arm.

Exam Tip: The definition of adjacent angles depends only on sharing a vertex and arm - the actual angle sizes do not matter.

Question 1. (vii) Can two right angles form a linear pair?
Answer: Yes. Two right angles can form a linear pair. A linear pair requires two angles that sit on a straight line and sum to 180°. Since 90° + 90° = 180°, and their non-common arms can be opposite rays (forming a straight line), two right angles satisfy both conditions of a linear pair.
In simple words: Two right angles add to 180° and can have their outer arms form a straight line, making a linear pair.

Exam Tip: For a linear pair, you need the sum to be 180° and the non-common arms to form opposite rays on a straight line.

Question 2. Find the complement of each of the following angles:
(i) 25°
(ii) 63°
(iii) 57°
Answer: To find the complement of an angle, subtract it from 90°.

(i) Complement of 25° = 90° - 25° = 65°. The complement is 65°.

(ii) Complement of 63° = 90° - 63° = 27°. The complement is 27°.

(iii) Complement of 57° = 90° - 57° = 33°. The complement is 33°.
In simple words: To find a complement, take 90° and subtract the angle you are given.

Exam Tip: Always verify by adding the angle and its complement - they must sum to exactly 90°.

Question 3. Find the supplement of each of the following angles:
(i) 105°
(ii) 87°
(iii) 142°
Answer: To find the supplement of an angle, subtract it from 180°.

(i) Supplement of 105° = 180° - 105° = 75°. The supplement is 75°.

(ii) Supplement of 87° = 180° - 87° = 93°. The supplement is 93°.

(iii) Supplement of 142° = 180° - 142° = 38°. The supplement is 38°.
In simple words: To find a supplement, take 180° and subtract the angle.

Exam Tip: Double-check by adding the angle to its supplement - the total must be 180°.

Question 4. Identify which of the following pairs of angles are complementary and which are supplementary:
(i) 55°, 125°
(ii) 34°, 56°
(iii) 137°, 43°
(iv) 112°, 68°
(v) 45°, 45°
(vi) 72°, 18°
Answer: Two angles are complementary if they add to 90° and supplementary if they add to 180°.

(i) 55° + 125° = 180°. The angles are supplementary.

(ii) 34° + 56° = 90°. The angles are complementary.

(iii) 137° + 43° = 180°. The angles are supplementary.

(iv) 112° + 68° = 180°. The angles are supplementary.

(v) 45° + 45° = 90°. The angles are complementary.

(vi) 72° + 18° = 90°. The angles are complementary.
In simple words: Add the two angles together and check: if the sum is 90°, they are complementary; if the sum is 180°, they are supplementary.

Exam Tip: Be careful with the two key totals - 90° for complementary and 180° for supplementary - and always show the addition in your working.

Question 5. (i) Find the angle which is equal to its complement.
Answer: Let the angle be x. If it equals its complement, then x and its complement are the same angle. This means x = 90° - x. Solving: x + x = 90°, so 2x = 90°, which gives x = 45°. The angle is 45°.
In simple words: An angle that equals its own complement must be 45°, because 45° + 45° = 90°.

Exam Tip: When an angle equals its complement, set up the equation angle = 90° - angle and solve for the unknown.

Question 5. (ii) Find the angle which is equal to its supplement.
Answer: Let the angle be x. If it equals its supplement, then x = 180° - x. Solving: x + x = 180°, so 2x = 180°, which gives x = 90°. The angle is 90°.
In simple words: An angle that equals its own supplement must be 90°, because 90° + 90° = 180°.

Exam Tip: Set the angle equal to its supplement (angle = 180° - angle) to find which single angle satisfies this condition.

Question 6. Two complementary angles are (x + 4)° and (2x - 7)°, find the value of x.
Answer: Since the angles are complementary, their sum is 90°.

(x + 4)° + (2x - 7)° = 90°

3x° - 3° = 90°

3x° = 93°

x° = 31°

Therefore, x = 31.
In simple words: Combine the two angle expressions and set their sum to 90°, then solve for x.

Exam Tip: Always verify your answer by substituting back: (31 + 4)° + (2(31) - 7)° = 35° + 55° = 90°.

Question 7. Two supplementary angles are in the ratio of 2 : 7, find the angles.
Answer: Since the angles are in the ratio 2 : 7, let the angles be 2x and 7x. Because they are supplementary, their sum is 180°.

2x + 7x = 180°

9x = 180°

x = 20°

So the angles are 2x = 2 × 20° = 40° and 7x = 7 × 20° = 140°. The angles are 40° and 140°.
In simple words: Express the angles as 2x and 7x, add them to get 180°, solve for x, then multiply to get each angle.

Exam Tip: Always check that your final angles have the correct ratio (40 : 140 = 2 : 7) and sum to 180°.

Question 8. Among two supplementary angles, the measure of the longer angle is 44° more than the measure of the smaller angle. Find their measures.
Answer: Let the smaller angle be x. Then the longer angle is x + 44°. Since they are supplementary, they add to 180°.

x + (x + 44°) = 180°

2x + 44° = 180°

2x = 136°

x = 68°

So the smaller angle is 68° and the longer angle is 68° + 44° = 112°. The angles are 68° and 112°.
In simple words: Set up one angle as x and the other as x + 44°, then use the fact that they sum to 180°.

Exam Tip: Verify: 68° + 112° = 180° (supplementary) and 112° - 68° = 44° (difference check).

Question 9. If an angle is half of its complement, find the measure of angles.
Answer: Let the angle be x. Its complement is 90° - x. Given that the angle is half its complement:

x = \( \frac{1}{2} \)(90° - x)

2x = 90° - x

3x = 90°

x = 30°

So the angle is 30° and its complement is 90° - 30° = 60°. The angles are 30° and 60°.
In simple words: If one angle is half of its complement, multiply both sides by 2 and then solve.

Exam Tip: Check: 30° is half of 60° (its complement), and 30° + 60° = 90°.

Question 10. Two adjacent angles are in the ratio 5 : 3 and they together form an angle of 128°, find these angles.
Answer: Since the angles are in the ratio 5 : 3, let them be 5x and 3x. Together they form 128°.

5x + 3x = 128°

8x = 128°

x = 16°

So the angles are 5x = 5 × 16° = 80° and 3x = 3 × 16° = 48°. The angles are 80° and 48°.
In simple words: Use the ratio to write the angles as 5x and 3x, add them to 128°, find x, then multiply.

Exam Tip: Verify the ratio: 80 : 48 simplifies to 5 : 3, and 80° + 48° = 128°.

Question 11. Find the value of x in each of the following diagrams:
Answer:

(i) The sum of all angles around a point is 360°.

41° + x + 105° + 130° = 360°

x + 276° = 360°

x = 360° - 276° = 84°

Therefore, x = 84°.

(ii) The sum of angles at a point on one side of a straight line is 180°.

3x + x + 40° = 180°

4x = 140°

x = 35°

Therefore, x = 35°.

(iii) The sum of angles at a point on one side of a straight line is 180°.

(2x + 10°) + (3x - 10°) + 40° = 180°

5x + 40° = 180°

5x = 140°

x = 28°

Therefore, x = 28°.
In simple words: Angles around a point sum to 360°; angles on one side of a line sum to 180°.

Exam Tip: Identify whether angles are around a point (360°) or on one side of a line (180°), then set up and solve the equation.

Question 12. Find the values of x, y and z in each of the following diagrams:
Answer:

(i) From the figure, x and 135° form a linear pair on a straight line.

x + 135° = 180°

x = 180° - 135° = 45°

Vertically opposite angles are equal, so:

y = 135° (vertically opposite to 135°)

z = x = 45° (vertically opposite to x)

Hence, x = 45°, y = 135° and z = 45°.

(ii) From the figure, one angle is a right angle (90°).

x = 31° (vertically opposite to 31°)

31° + y + 90° = 180° (linear pair)

y = 180° - 90° - 31° = 59°

z = y = 59° (vertically opposite to y)

Hence, x = 31°, y = 59° and z = 59°.

(iii) Using the property of vertically opposite angles:

x = 44° (vertically opposite to 44°)

z = 51° (vertically opposite to 51°)

Angles x, y and z lie on a straight line, so:

x + y + z = 180° (linear pair)

44° + y + 51° = 180°

y = 180° - 44° - 51° = 85°

Hence, x = 44°, y = 85° and z = 51°.
In simple words: Use linear pairs (angles on a line sum to 180°) and vertically opposite angles (which are equal) to find all unknowns.

Exam Tip: Always identify vertically opposite angles first - they are always equal. Then use linear pairs to find remaining angles.

Question 13. In the adjoining figure, lines AB and CD intersect at F. If ∠EFA = ∠AFD and ∠CFB = 50°, find ∠EFC.
Answer: When lines AB and CD cross at point F, angles ∠AFD and ∠CFB are vertically opposite angles, so they are equal. Since ∠CFB = 50°, we have ∠AFD = 50°. We are told that ∠EFA = ∠AFD, which means ∠EFA = 50°. Now, angles ∠AFC and ∠CFB form a linear pair along the straight line AB, so their sum is 180°. This gives ∠AFC = 180° - 50° = 130°. Finally, ∠EFC is calculated as the difference: ∠EFC = ∠AFC - ∠EFA = 130° - 50° = 80°.
In simple words: Two angles that sit opposite each other when two lines cross are always equal. Using this fact and adding angles, we can find that the angle EFC equals 80°.

Exam Tip: Always identify vertically opposite angles first and use the linear pair property when angles sit on a straight line - these two techniques solve most geometry problems of this type.

Exercise 10.2

Question 1. Identify each of the given pair of angles as alternate interior angles, co-interior angles or corresponding angles or none of these in the given figure:
(i) ∠2, ∠6
(ii) ∠1, ∠6
(iii) ∠3, ∠5
(iv) ∠2, ∠7
(v) ∠3, ∠6
(vi) ∠4, ∠8
Answer:
(i) ∠2 and ∠6 are positioned identically at their two intersections, making them corresponding angles.
(ii) ∠1 and ∠6 do not occupy any special position, so they are none of these.
(iii) ∠3 and ∠5 are interior angles on opposite sides of the transversal, making them alternate interior angles.
(iv) ∠2 and ∠7 are exterior angles on the same side of the transversal, so they are none of these.
(v) ∠3 and ∠6 are interior angles on the same side of the transversal, making them co-interior angles.
(vi) ∠4 and ∠8 are positioned identically at their two intersections, making them corresponding angles.
In simple words: When two lines are cut by a third line, angles in the same position at each intersection are called corresponding angles. Angles between the lines on opposite sides are alternate interior angles, and angles between the lines on the same side are co-interior angles.

Exam Tip: Draw and label all eight angles clearly at both intersections - visualizing the position of each angle pair makes identification much easier.

Question 2. 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
Answer:
(i) When corresponding angles are equal, this property holds - it states that corresponding angles formed by a transversal cutting two parallel lines are always equal to each other.
(ii) When two alternate interior angles are equal, the lines being cut must be parallel to one another.
(iii) When two co-interior angles add up to 180°, the lines are parallel.
In simple words: These are three key rules: matching angles mean lines are parallel, opposite-side angles equal means parallel lines, and two same-side angles adding to 180° means parallel lines.

Exam Tip: Memorize all three conditions for parallel lines - questions often ask you to identify which property was used in a given statement.

Question 3. In each of the following figures, a pair of parallel lines is cut by a transversal. Find the value of x:
(i) The 100° angle and x are corresponding angles.

Answer: Since these angles are in matching positions where the transversal crosses the two parallel lines, they must be equal. Therefore, x = 100°.
In simple words: When a line crosses two parallel lines, angles in the same position at each crossing are always equal.

Exam Tip: Always check which type of angle pair you have before solving - corresponding angles are equal, while co-interior angles sum to 180°.

Question 3. (ii) The angle x and 110° are co-interior angles, so they are supplementary.
Answer: Since x and 110° are interior angles on the same side of the transversal, they add up to 180°. This gives us x + 110° = 180°, so x = 70°.
In simple words: When angles are on the same side between two parallel lines, they always add up to 180°.

Exam Tip: Co-interior angles always sum to 180° - use this fact immediately when you identify this angle pair.

Question 3. (iii)
Answer: From the figure, the angle 110° and ∠AOB form a linear pair on the top line, so ∠AOB + 110° = 180°, which gives ∠AOB = 70°. Now, ∠AOB and x are corresponding angles, meaning they are in the same position at their respective intersections. Since corresponding angles are equal when a transversal cuts parallel lines, x = 70°.
In simple words: First find the angle on the top line using the straight-line rule. Then use the corresponding angles property to find x.

Exam Tip: When an angle is not directly in the matching position, always use the linear pair property to find the angle you need first.

Question 4. In the following figures, a pair of parallel lines are cut by a transversal. Find the value of x in each figure.
(i) The angles 2x + 6° and 3x + 54° are co-interior angles, so they are supplementary.

Answer: Since the two angles are on the same side of the transversal and between the parallel lines, they must add to 180°. Setting up the equation: 2x + 6° + 3x + 54° = 180°. Combining like terms gives 5x + 60° = 180°. Subtracting 60° from both sides: 5x = 120°. Dividing by 5: x = 24°.
In simple words: Add the two angles together and set the sum equal to 180°. Then solve for x by using basic algebra.

Exam Tip: Always collect all x terms and numbers separately before solving - this prevents careless errors in combining like terms.

Question 4. (ii)
Answer: Looking at the figure, ∠DBF and ∠CBO are vertically opposite angles, so ∠CBO = 3x + 30°. Next, ∠AOB and ∠CBO are co-interior angles, meaning they sum to 180°. Setting up the equation: (2x + 15°) + (3x + 30°) = 180°. Simplifying: 5x + 45° = 180°. Subtracting 45°: 5x = 135°. Dividing by 5: x = 27°.
In simple words: First use the vertically opposite angles rule to find one angle, then use the co-interior angles rule to write an equation and solve for x.

Exam Tip: Watch for vertically opposite angles in multi-line figures - they give you a quick way to find unknown angles without extra steps.

Question 5. In the following figures (i) to (vi), a pair of parallel lines are cut by a transversal. Find the size of each lettered angle.
(i) From the figure, x = 60° (vertically opposite to the 60° angle). Since y and x are corresponding angles, y = x = 60°.

Answer: x = 60° and y = 60°.
In simple words: Opposite angles at an intersection are equal. Matching angles at the two intersections are also equal.

Exam Tip: Always start by finding angles using the vertically opposite property, as this is the fastest step before moving to parallel line properties.

Question 5. (ii)
Answer: Looking at the diagram, q = 135° (vertically opposite to the 135° angle). Since p and q are co-interior angles, they sum to 180°. Therefore, p + 135° = 180°, giving p = 45°.
In simple words: Opposite angles are equal. Same-side interior angles add to 180°.

Exam Tip: When you find q from the vertically opposite property, immediately use the co-interior angles rule to find p.

Question 5. (iii)
Answer: From the figure, a = 70° (alternate interior angles are equal when a transversal cuts parallel lines). Since a and b form a linear pair along the straight line, b = 180° - 70° = 110°.
In simple words: Angles on opposite sides between parallel lines are equal. Angles on a straight line add to 180°.

Exam Tip: Alternate interior angles are easy to spot - they are on opposite sides of the transversal and between the parallel lines.

Question 5. (iv)
Answer: From the figure, x = 180° - 128° = 52° (linear pair with 128°). Next, z = 128° (corresponding angles are equal). Since y and z form a linear pair, y = 180° - 128° = 52°.
In simple words: Use the straight-line rule to find x. Use matching angles to find z. Use the straight-line rule again to find y.

Exam Tip: In problems with multiple unknowns, work step-by-step using one angle property at a time to avoid confusion.

Question 5. (v)
Answer: From the figure, b = 75° (vertically opposite to 75°). Since a and b form a linear pair, a = 180° - 75° = 105°. Next, c = 75° (corresponding angles are equal). Since d and c form a linear pair, d = 180° - 75° = 105°.
In simple words: Opposite angles are equal. Matching angles are equal. Angles on a straight line add to 180°.

Exam Tip: Notice the pattern - angles alternate between 75° and 105° as you move through the figure.

Question 5. (vi)
Answer: From the figure, p = 62° (vertically opposite to 62°). Since p and q form a linear pair, q = 180° - 62° = 118°. Next, r and p are co-interior angles, so r = 180° - 62° = 118°. Since r and s form a linear pair, s = 180° - 118° = 62°.
In simple words: Opposite angles are equal. Angles on a line add to 180°. Same-side interior angles also add to 180°.

Exam Tip: Watch how the co-interior angles property gives a quick way to find r without needing to reference the matching angle.

Question 6. In the adjoining diagram, lines AB, CD and EF are parallel. Calculate the values of x and y. Hence, find the reflex angle ECA.
Answer: Since EF and CD are parallel lines and EC is a transversal cutting both, the angles ∠FEC and ∠ECD (which is x) are co-interior angles. Therefore, ∠FEC + x = 180°, which gives 120° + x = 180°, so x = 60°. Next, since CD and AB are parallel lines and CA is a transversal cutting both, the angles ∠DCA (which is y) and ∠CAB are co-interior angles. Therefore, y + ∠CAB = 180°. Substituting the given value: y + 140° = 180°, so y = 40°. The angle ∠ECA is found by adding x and y: ∠ECA = 60° + 40° = 100°. The reflex angle is the larger angle formed on the other side: Reflex ∠ECA = 360° - 100° = 260°.
In simple words: Use the co-interior angles rule twice - once for the top parallel lines and once for the bottom ones. Add the two angles to get the angle ECA, then subtract from 360° to get the reflex angle.

Exam Tip: Always remember that a reflex angle is found by subtracting the normal angle from 360° - this is a common finishing step in parallel line problems.

Question 7. In the adjoining figure, l ∥ m. Find the values of x, y and z.
Answer: Looking at the triangle formed between the two parallel lines, the interior angles of the triangle are related to x, y, and z. The angle at the top-left vertex inside the triangle corresponds to the 45° angle at the bottom (alternate interior angles), so it equals 45°. Similarly, the angle at the top-right vertex inside the triangle equals 55° (alternate interior angles with the bottom-right angle). Since the three angles of a triangle sum to 180°, the angle at the top vertex equals 180° - 45° - 55° = 80°. This means y = 80°. The angles x and z are supplementary to the angles on either side of y on the line l. Since the angles on a straight line sum to 180°, x = 180° - 45° = 135° and z = 180° - 55° = 125°.
In simple words: Use the alternate interior angles property to find the angles inside the triangle. Use the triangle angle sum rule to find y. Use the straight-line rule to find x and z.

Exam Tip: In problems with a triangle between two parallel lines, apply alternate interior angles first to find the triangle's interior angles, then use the 180° rule for the line.

Question 8. Calculate the measure of each lettered angle in the following figure (parallel lines, segment or rays are denoted by thick matching arrows):
Answer:
(i) Looking at the figure, we can use the property that when two parallel lines are cut by a transversal, certain angle relationships hold.

Since x and 143° form co-interior angles (also called same-side interior angles):
\( x + 143° = 180° \)
\( x = 37° \)

Since z and 60° form co-interior angles:
\( z + 60° = 180° \)
\( z = 120° \)

Because x, y, and z all lie on one side of a straight line, they must add up to 180°:
\( x + y + z = 180° \)
\( 37° + y + 120° = 180° \)
\( y = 180° - 157° = 23° \)

Therefore, x = 37°, y = 23°, and z = 120°.

(ii) From the figure, angle a equals 55° because they are corresponding angles formed by parallel lines and a transversal.

Using the angle sum property of triangles, the three interior angles must total 180°:
\( 72° + a + b = 180° \)
\( 72° + 55° + b = 180° \)
\( b = 180° - 127° = 53° \)

Angles b, c, and 55° all sit on a straight line, so they sum to 180°:
\( b + c + 55° = 180° \)
\( 53° + c + 55° = 180° \)
\( c = 180° - 108° = 72° \)

Therefore, a = 55°, b = 53°, and c = 72°.

(iii) Working through this configuration:

Angle b equals 75° (alternate interior angles are equal when lines are parallel):
\( b = 75° \)

Since a and b form a linear pair (supplementary angles on a straight line):
\( a + 75° = 180° \)
\( a = 180° - 75° = 105° \)

Angle c equals 75° (corresponding angles are equal):
\( c = 75° \)

Angle d equals b, which is 75° (corresponding angles are equal):
\( d = 75° \)

Therefore, a = 105°, b = 75°, c = 75°, and d = 75°.
In simple words: Use the properties of parallel lines to find unknown angles. When lines are parallel, alternate interior angles are equal, corresponding angles are equal, and co-interior angles add to 180°. Angles on a straight line also add to 180°.

Exam Tip: Label all angle relationships clearly (co-interior, corresponding, alternate, linear pair) before solving. Always verify your answers by checking that angles on a straight line sum to 180° and angles at a point sum correctly.

Question 9. In the figures given below, state whether the lines l and m are parallel or not.
Answer:
(i) The two angles marked are 106° and 64°, and they are co-interior angles (on the same side of the transversal, between the two lines).

For two lines to be parallel when cut by a transversal, co-interior angles must be supplementary (add up to 180°). However:
\( 106° + 64° = 170° \neq 180° \)

Since the co-interior angles do not sum to 180°, the lines are not parallel.

(ii) From the figure, we identify the angles at the intersection points.

The angles marked 75° at point O are vertically opposite to angle BOA. Using the vertically opposite angles property:
\( \angle COA = \angle BOP = 75° \)

Now, angles COA (75°) and DAO (75°) form a pair of co-interior angles. For the lines to be parallel, their sum must equal 180°:
\( 75° + 75° = 150° \neq 180° \)

Since these co-interior angles do not sum to 180°, lines l and m are not parallel.

(iii) Examining this configuration, we find the following angle pair:

Using vertically opposite angles:
\( \angle POS = \angle AOB = 57° \)

Now, angles QSO and POS are co-interior angles. Let's check:
\( 123° + 57° = 180° \)

Since the co-interior angles are supplementary (sum to 180°), the lines are parallel.
In simple words: Co-interior angles are on the same side of the transversal, between two lines. If they add to 180°, the lines are parallel. If they don't add to 180°, the lines are not parallel.

Exam Tip: Always identify which angle relationship applies (corresponding, alternate interior, or co-interior) before concluding whether lines are parallel. Show the sum or comparison calculation explicitly to earn full credit.

Question 1. 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) Supplement of an obtuse angle is ....
(iv) Two angles forming a linear pair are ....
(v) If two adjacent angles are supplementary, then they form a ....
(vi) Angles of a linear pair are ..... as well as ..... .
(vii) Adjacent angles have a common vertex, a common ..... and no common ..... .
(viii) Angles formed by two intersecting lines having no common arms are called .....
(ix) If two lines intersect and if one pair of vertically opposite angles are acute angles, then the other pair of vertically opposite angles are ....
(x) Two lines in a plane which never meet are called ..... .
(xi) Alternate interior angles have one common ..... .
(xii) Corresponding angles are on the ..... side of transversal.
(xiii) Alternate interior angles are on the ..... side of transversal.
(xiv) If two lines are cut by a transversal such that a pair of corresponding angles are not equal, then the lines are ....
Answer:
(i) 90°
(ii) 180°
(iii) an acute angle
(iv) supplementary
(v) linear pair
(vi) adjacent as well as supplementary
(vii) common arm and no common interior points
(viii) vertically opposite angles
(ix) obtuse
(x) parallel lines
(xi) arm
(xii) same
(xiii) opposite
(xiv) not parallel
In simple words: Complementary angles add to 90°, supplementary angles add to 180°. Vertically opposite angles are always equal. Parallel lines have special angle properties when cut by a transversal.

Exam Tip: These fill-in-the-blank answers are core definitions. Learn each one precisely - examiners often test these basic facts in objective questions.

Question 2. In the adjoining figure, AB is a straight line and OD \( \perp \) AB. Observe the figure and fill in the following blanks:
(i) ∠AOC and ∠COE form a pair of .... angles.
(ii) ∠AOC and ∠COB are .... angles.
(iii) ∠AOC is ..... of ∠COD.
(iv) ∠BOE is ..... of ∠EOA.
Answer:
(i) adjacent
(ii) supplementary
(iii) complement
(iv) supplement
In simple words: Adjacent angles share a common arm. Supplementary angles add to 180°. Complements add to 90°, and supplements add to 180°.

Exam Tip: Pay careful attention to which angles are named in the question - the diagram is essential for identifying the correct relationship between each pair.

Question 3. State whether the following statements are true (T) or false (F):
(i) Two obtuse angles can be supplementary.
(ii) Two acute angles can form a linear pair.
(iii) Two obtuse angles can form a linear pair.
(iv) Two adjacent angles always form a linear pair.
(v) Pair of vertically opposite angles are always supplementary.
(vi) 30° is one-half of its complement.
(vii) If two lines are cut by a transversal, then each pair of corresponding angles are equal.
(viii) If two lines are cut by a transversal, then each pair of alternate interior angles are equal.
Answer:
(i) False. Each obtuse angle measures more than 90°, so adding two of them gives a sum greater than 180°.
(ii) False. Each acute angle measures less than 90°, so their sum is less than 180° and cannot make a linear pair.
(iii) False. Each obtuse angle measures more than 90°, so their combined measure exceeds 180°.
(iv) False. Two adjacent angles form a linear pair only when their non-common arms lie on opposite rays of a straight line.
(v) False. Vertically opposite angles are always equal in measure, but they are not necessarily supplementary.
(vi) True. The complement of 30° is \( 90° - 30° = 60° \), and \( \frac{1}{2} \times 60° = 30° \).
(vii) False. Corresponding angles are equal only if the two lines being cut are parallel.
(viii) False. Alternate interior angles are equal only when the two lines being cut are parallel.
In simple words: Remember that angles in a linear pair must sum to exactly 180°, and angle relationships with transversals only hold when lines are parallel.

Exam Tip: For each false statement, be ready to explain why it is false with a concrete example or counterexample. This shows deeper understanding beyond just stating true or false.

Question 4. A pair of complementary angles is
(a) 130°, 50°
(b) 35°, 55°
(c) 25°, 75°
(d) 27°, 53°
Answer: (b) 35°, 55°
In simple words: Two angles are complementary when they add up to exactly 90°. Checking option (b): 35° + 55° = 90°.

Exam Tip: Quickly add each pair to check the sum. The correct pair must total 90° for complementary angles.

Question 5. A pair of supplementary angles is
(a) 55°, 115°
(b) 65°, 125°
(c) 47°, 133°
(d) 40°, 50°
Answer: (c) 47°, 133°
In simple words: Two angles are supplementary when they add to exactly 180°. Checking: 47° + 133° = 180°.

Exam Tip: Always verify by adding the pair. Supplementary pairs must sum to 180°, not 90°.

Question 6. If an angle is one-third of its supplement, then the measure of the angle is
(a) 45°
(b) 30°
(c) 135°
(d) 150°
Answer: (a) 45°
In simple words: Let the angle be x. Its supplement is 180° - x. We're told x equals one-third of its supplement, so set up the equation x = (180° - x)/3. Solving: 3x = 180° - x, which gives 4x = 180°, so x = 45°.

Exam Tip: Read carefully - "one-third of its supplement" means the angle equals (1/3) times the supplement, not the other way around. Set up the equation correctly before solving.

Question 7. If an angle measures 10° more than its complement, then the measure of the angle is
(a) 40°
(b) 55°
(c) 35°
(d) 50°
Answer: (d) 50°
In simple words: Let the angle be x. Its complement is 90° - x. The angle is 10° more than the complement, so x = (90° - x) + 10°. Simplifying: 2x = 100°, giving x = 50°.

Exam Tip: Translate "10° more than" into a mathematical expression carefully. Write the equation as (angle) = (complement) + 10° to avoid mistakes.

Question 8. If one angle of a linear pair is acute, then the other angle is
(a) acute
(b) obtuse
(c) right
(d) straight
Answer: (b) obtuse
In simple words: When two angles form a linear pair, they always add up to 180°. If one is acute (smaller than 90°), the other must be bigger than 90° - that makes it obtuse.

Exam Tip: Remember that linear pairs always sum to 180°. Once you know one angle is acute, the other must be obtuse without any calculation needed.

Question 9. In the adjoining figure, the value of x that will make AOB a straight line is
(a) x = 40
(b) x = 35
(c) x = 30
(d) x = 25
Answer: (b) x = 35
In simple words: For a straight line, all angles at one point must add to 180°. Add both expressions together and set them equal to 180°, then solve for x.

Exam Tip: Always set up the equation by adding all angles at a point and equating to 180° for a straight line. Check your answer by substituting back.

Question 10. If two lines are intersected by a transversal, then the number of pairs of interior angles on the same side of transversal is
(a) 1
(b) 2
(c) 3
(d) 4
Answer: (b) 2
In simple words: When a transversal cuts two lines, there are two sets of angles between the lines on the same side. Each set forms one pair, giving you 2 pairs total.

Exam Tip: Co-interior angles (also called same-side interior angles) always appear in exactly 2 pairs when a transversal intersects two lines - one pair on each side of the transversal.

Question 11. In the adjoining figure, if l ∥ m then the value of x is
(a) x = 50
(b) x = 60
(c) x = 70
(d) x = 45
Answer: (a) x = 50
In simple words: The 110° angle and the angle (3x - 40)° are on opposite sides of the transversal and between the parallel lines, so they are equal. Solve: 3x - 40 = 110, which gives x = 50.

Exam Tip: When lines are parallel and a transversal cuts them, alternate interior angles are always equal - use this property to set up your equation directly.

Question 12. In the adjoining figure, if l ∥ m then the value of x is
(a) x = 75°
(b) x = 95°
(c) x = 105°
(d) x = 115°
Answer: (c) x = 105°
In simple words: The angles marked 75° and x are on the same side of the transversal and inside the parallel lines, making them co-interior angles. They must add up to 180°, so x = 180 - 75 = 105.

Exam Tip: Co-interior angles (same-side interior angles) formed by a transversal cutting parallel lines are always supplementary - they sum to 180°.

Question 13. In the adjoining figure, AB ∥ CD. If ∠APQ = 50° and ∠PRD = 130°, then ∠QPR is
(a) 30°
(b) 50°
(c) 80°
(d) 130°
Answer: (c) 80°
In simple words: First, find ∠PRQ by subtracting ∠PRD from 180° (they form a linear pair). Then use the fact that AB ∥ CD to find ∠PQR using alternate angles. Finally, use the triangle angle sum to find ∠QPR.

Exam Tip: Break multi-step problems into smaller parts: find all angles using angle properties, then apply the triangle angle sum rule at the end.

Question 14. In the adjoining figure, PA ∥ BC ∥ DQ and AB ∥ DC. Then the values of x and y are respectively:
(a) 50°, 120°
(b) 50°, 130°
(c) 60°, 120°
(d) 60°, 130°
Answer: (b) 50°, 130°
In simple words: Since PA ∥ BC, the angle x equals 50° by the alternate interior angle rule. Then since AB ∥ DC, the angles x and ∠BCD are co-interior, so ∠BCD = 130°. Finally, since BC ∥ DQ, the angle y also equals 130° by alternate interior angles.

Exam Tip: In problems with multiple parallel lines, work step-by-step using one pair of parallel lines at a time, and track which angle properties apply to each pair.

Question 15. Statement I: Two acute angles can complement each other. Statement II: Two acute angles can supplement each other.
(a) Statement I is true but statement II is false.
(b) Statement I is false but statement II is true.
(c) Both Statement I and statement II are true.
(d) Both Statement I and statement II are false.
Answer: (a) Statement I is true but statement II is false.
In simple words: Two acute angles can add to 90° (like 30° and 60°), so they can be complementary. But since every acute angle is less than 90°, two acute angles always add to less than 180°, so they cannot be supplementary.

Exam Tip: Test complementary and supplementary claims with examples: for complementary, find two acute angles that sum to 90°; for supplementary, try to find two acute angles that sum to 180° - you cannot.

Question 16. Statement I: In the adjoining figure, l ∥ m and p is a transversal. If ∠1 = 50°, then ∠2 = 150° Statement II: If a transversal cuts two parallel lines, each pair of corresponding angles are equal.
(a) Statement I is true but statement II is false.
(b) Statement I is false but statement II is true.
(c) Both Statement I and statement II are true.
(d) Both Statement I and statement II are false.
Answer: (b) Statement I is false but statement II is true.
In simple words: Using vertically opposite angles, ∠3 = ∠1 = 50°. Since l ∥ m, angles ∠3 and ∠2 are co-interior angles and must sum to 180°, so ∠2 = 130°, not 150°. Statement II is correct - corresponding angles are always equal when lines are parallel.

Exam Tip: When evaluating statements about parallel lines, always use the diagram to identify which angle pair is being described, then apply the correct angle property.

Check Your Progress

Question 1. Find the supplementary angle of each of the following angles:
(i) \( \frac{1}{2} \) of 90°
(ii) \( \frac{3}{7} \) of 280°
Answer:
(i) The angle = \( \frac{1}{2} \times 90° = 45° \).

Supplement = 180° - 45° = 135°.

The supplementary angle is 135°.

(ii) The angle = \( \frac{3}{7} \times 280° = 120° \).

Supplement = 180° - 120° = 60°.

The supplementary angle is 60°.
In simple words: Multiply the fraction by the given angle to find the actual angle first, then subtract it from 180° to get its supplement.

Exam Tip: Always perform the fraction multiplication before finding the supplement - do not skip this step or you will get the wrong answer.

Question 2. How many degrees are there in an angle which is one-fifth of its complement?
Answer: Let the angle be x, then its complement = 90° - x.

Since the angle is one-fifth of its complement,
\[ x = \frac{1}{5}(90° - x) \]

\[ \implies 5x = 90° - x \]

\[ \implies 6x = 90° \]

\[ \implies x = 15° \]

The required angle is 15°.
In simple words: Set up an equation where the angle equals one-fifth of its complement. Solve by multiplying both sides by 5 and then combining like terms.

Exam Tip: When an angle relationship involves fractions, multiply through to clear the fraction immediately, then gather all terms with the variable on one side.

Question 3. If two angles are supplementary and one angle is 5° more than four-times the other, find the angles.
Answer: Let the angle be x, then the other angle = 4x + 5°.

Since the angles are supplementary, their sum is 180°.

\[ \implies x + (4x + 5°) = 180° \]

\[ \implies 5x + 5° = 180° \]

\[ \implies 5x = 175° \]

\[ \implies x = 35° \]

The angles are 35° and 4 × 35° + 5° = 145°.

The angles are 35° and 145°.
In simple words: Write the second angle in terms of the first, then use the fact that supplementary angles sum to 180° to set up and solve the equation.

Exam Tip: Always identify which angle is expressed in terms of the other, substitute into the sum equation, and simplify to isolate the variable.

Question 4. The adjoining diagram shows two intersecting straight lines. Find the values of x, y and z.
Answer: Since (3x - 10)° and (2x + 15)° are vertically opposite angles,

\[ \implies (3x - 10)° = (2x + 15)° \]

\[ \implies 3x - 10 = 2x + 15 \]

\[ \implies 3x - 2x = 15 + 10 \]

\[ \implies x = 25 \]

So, the angle = (2x + 15)° = (2 × 25 + 15)° = 65°.

Since y and (2x + 15)° form a linear pair,

\[ \implies y + 65° = 180° \]

\[ \implies y = 115° \]

\[ \implies z = y = 115° \text{ (vertically opposite angles are equal)} \]

Therefore, x = 25°, y = 115° and z = 115°.
In simple words: Start by equating vertically opposite angles to find x. Then use linear pair angles to find y. Finally, use vertically opposite angles again to find z.

Exam Tip: When two straight lines intersect, work systematically: first use vertically opposite angles, then use linear pairs for adjacent angles, and verify your answers before finalizing.

Question 5. In the adjoining diagram, lines AB and CD intersect at O. If ∠1 + ∠3 = 78°, find the size of ∠2.
Answer: Since ∠1 and ∠3 are vertically opposite angles, they are equal.

\[ \implies \angle1 = \angle3 \]

Given, ∠1 + ∠3 = 78°

\[ \implies 2\angle1 = 78° \]

\[ \implies \angle1 = 39° \]

Since ∠1 and ∠2 form a linear pair,

\[ \implies \angle1 + \angle2 = 180° \]

\[ \implies 39° + \angle2 = 180° \]

\[ \implies \angle2 = 141° \]
In simple words: Since vertically opposite angles are equal, if their sum is 78°, then each angle is 39°. The angle next to it forms a linear pair and must be 180 - 39 = 141°.

Exam Tip: Remember that vertically opposite angles are always equal - use this to convert a sum of two equal angles into a single equation before solving.

Question 6. (a) In the fig. (i) given below, PQ ∥ RS. Find the value of x + y.
Answer: Using the angle sum property of triangle PQR, we find that ∠PRQ = 180° - 45° - 55° = 80°. Since PRT is a straight line, the angles at R must total 180°. Therefore, x + y + 80° = 180°, which gives us x + y = 100°.
In simple words: All angles in a triangle add up to 180°. The angles on a straight line also add up to 180°. Using these two facts, we can figure out that x + y = 100°.

Exam Tip: Always identify angle relationships like linear pairs and triangle angle sums first - these are your starting points for solving parallel line problems.

Question 6. (b) In the fig. (ii) given below, PQ ∥ RS. Find the value of x + y.
Answer: From the figure, y and 150° create a linear pair, so y = 180° - 150° = 30°. Since PQ ∥ RS and a slanting line acts as a transversal, the alternate angles are equal, making x = 75°. Adding these together: x + y = 75° + 30° = 105°.
In simple words: When two lines are parallel and a third line cuts through them, certain angle pairs are always the same. Using this rule, we get x + y = 105°.

Exam Tip: Mark all angle relationships on the figure - linear pairs, alternate angles, corresponding angles - before writing any equations.

Question 6. (c) In the fig. (iii) given below, QP ∥ SR and QR ∥ ST. Find the value of x + 2y.
Answer: Since QP ∥ SR and QR acts as a transversal, alternate angles are equal, so x = 32°. Since QR ∥ ST and RS serves as a transversal, alternate angles are again equal, giving y = x = 32°. Therefore, x + 2y = 32° + 2(32°) = 32° + 64° = 96°.
In simple words: When you have two pairs of parallel lines, you can use the alternate angle rule twice. First you find x, then you find y, and finally you add them up the way the question asks.

Exam Tip: In problems with multiple parallel lines, track each parallel pair separately and apply angle rules to each one in turn.

Question 7. (a) In the fig. (i) given below, l ∥ m. If ∠5 = 65°, find all other angles.
Answer: Since l ∥ m and a transversal cuts through them with ∠5 = 65°, we can find all the other angles using angle relationships. Using corresponding angles: ∠1 = ∠5 = 65°. Using vertically opposite angles: ∠3 = ∠1 = 65° and ∠7 = ∠5 = 65°. For the remaining angles, since ∠2 and ∠3 form a linear pair: ∠2 = 180° - 65° = 115°. The same applies to the other parallel line: ∠6 = 180° - 65° = 115°. Since vertically opposite angles are equal: ∠2 = ∠4 = ∠6 = ∠8 = 115°. Therefore, ∠1 = ∠3 = ∠7 = 65° and ∠2 = ∠4 = ∠6 = ∠8 = 115°.
In simple words: When parallel lines are cut by a transversal, you can find all 8 angles by using three rules: angles that are the same position are equal, angles across from each other are equal, and angles on a straight line add up to 180°.

Exam Tip: Draw and label all 8 angles clearly before finding them - this prevents mistakes and shows examiners your method.

Question 7. (b) In the fig. (ii) given below, l ∥ m. Find the values of x, y and z.
Answer: Since l ∥ m, x and the 40° angle are alternate interior angles, so x = 40°. The 105° angle and z are co-interior angles, so z + 105° = 180°, giving z = 75°. Since the angles on line l form a linear pair: 40° + y + z = 180°, which means 40° + y + 75° = 180°, so y = 65°. Therefore, x = 40°, y = 65° and z = 75°.
In simple words: Use the rules for parallel lines to find x and z first. Then use the fact that angles on a straight line add up to 180° to find y.

Exam Tip: Identify which angle relationship applies at each step - alternate, co-interior, or linear pair - before writing your equation.

Question 7. (c) In the fig. (iii) given below, l ∥ m and p ∥ q. Find the values of x, y and z.
Answer: Since p ∥ q and l is a transversal, x and 105° form a linear pair, so x = 180° - 105° = 75°. Since l ∥ m and q is a transversal, the 105° angle and y are corresponding angles, making y = 105°. Since p ∥ q and m is a transversal, y and z are co-interior angles, so y + z = 180°, which gives z = 180° - 105° = 75°. Therefore, x = 75°, y = 105° and z = 75°.
In simple words: When you have two sets of parallel lines meeting a third line, use angle rules for each pair separately to build up all the answers.

Exam Tip: With multiple parallel line pairs, organize your working by dealing with one parallel pair at a time, then combine the results.

Question 8. Calculate the measure of each lettered angle in the following figures (parallel line segments/rays are denoted by thick matching arrows):
Answer:
(i) From the figure, using the alternate angle property since the upper ray is parallel to the ray containing angle p, we get p = 135°. Since the rays meet at a point, the angles around that point sum to 360°, so p + r + 110° = 360°. This gives 135° + r + 110° = 360°, which means r = 115°. Since the ray containing angle r is parallel to the ray containing angle q, these two angles are co-interior angles, so r + q = 180°. Therefore, q = 180° - 115° = 65°. Hence, p = 135°, q = 65° and r = 115°.

(ii) From the figure, since AB ∥ HI and HB is a transversal, s = 110° by alternate angles. Since ∠BCD and ∠ECH are vertically opposite, ∠BCD = ∠ECH = p. Since ∠ECH and 110° are co-interior angles on parallel lines, ∠ECH + 110° = 180°, so p = 70°. Since HB ∥ EF, we have CB ∥ EF, making q and p corresponding angles, so q = 70°. Since HB ∥ GD, we have CB ∥ GD, making r and p alternate angles, so r = 70°. Hence, p = 70°, q = 70°, r = 70° and s = 110°.
In simple words: Break the figure into smaller sections. For each section, identify which angle rules apply - alternate angles, corresponding angles, angles around a point, or angles on a straight line - then use those rules step by step to find every unknown angle.

Exam Tip: In multi-part angle figures, solve angles in a logical order - find angles that depend on fewer other angles first, then use those results to find the rest.