Lines and Angles

Lines and Angles is a fundamental chapter in geometry. It introduces basic concepts related to lines, line segments, rays, and angles.

Key Concepts:

• Line: A straight path that extends infinitely in both directions.
• Line segment: A part of a line with two endpoints.
• Ray: A part of a line with one endpoint.
• Angle: Formed by two rays sharing a common endpoint (vertex).
• Types of angles: Acute, obtuse, right, straight, reflex, and complete angles.
• Intersecting lines: Lines that cross each other at a point.
• Parallel lines: Lines that never meet, maintaining a constant distance apart.
• Transversal: A line that intersects two or more lines.
• Angles formed by a transversal: Corresponding angles, alternate interior angles, alternate exterior angles, co-interior angles, and vertically opposite angles.  

Important Properties:

• Vertically opposite angles are equal.
• Linear pair of angles add up to 180 degrees.
• The sum of the angles of a triangle is 180 degrees.

By understanding these concepts and properties, you can solve various problems related to lines, angles, and geometric shapes.

Exercise 5.1

1. Find the complement of each of the following angles:

Ans : 

Given angles:

• (i) 20°
• (ii) 63°
• (iii) 57°

Complements:

• (i) Complement of 20° = 90° – 20° = 70°
• (ii) Complement of 63° = 90° – 63° = 27°
• (iii) Complement of 57° = 90° – 57° = 33°

2. Find the supplement of each of the following angles:

Ans : 

Given angles:

• (i) 105°
• (ii) 87°
• (iii) 154°

Supplements:

• (i) Supplement of 105° = 180° – 105° = 75°
• (ii) Supplement of 87° = 180° – 87° = 93°
• (iii) Supplement of 154° = 180° – 154° = 26°

Therefore, the supplements of the given angles are 75°, 93°, and 26° respectively.

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°

Ans :

Recall:

• Complementary angles add up to 90°.
• Supplementary angles add up to 180°.

Analysis:

• (i) 65° + 115° = 180°, so they are supplementary.
• (ii) 63° + 27° = 90°, so they are complementary.
• (iii) 112° + 68° = 180°, so they are supplementary.
• (iv) 130° + 50° = 180°, so they are supplementary.
• (v) 45° + 45° = 90°, so they are complementary.
• (vi) 80° + 10° = 90°, so they are complementary.

4. Find the angle which equal to its complement.

Ans : 

Complementary angles add up to 90 degrees.

If the two angles are equal, let’s call them x.

So, x + x = 90

2x = 90

x = 45

Therefore, the angle is 45 degrees.

5. Find the angle which is equal to its supplement.

Ans : 

An angle which is equal to its supplement is 90 degrees.

• Supplementary angles add up to 180 degrees.
• If the angle is equal to its supplement, let’s call it x.
• So, x + x = 180
• 2x = 180
• x = 90

Therefore, the angle is 90 degrees.

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.

Ans : 

In the given figure, we can observe two angles, labeled as ∠1 and ∠2. The text accompanying the image states that these angles are supplementary, meaning their sum equals 180 degrees.

7. Can two angles be supplementary if both of them are:

(i) acute?

(ii) obtuse?

(iii) right?

Ans : 

(i) acute?

• No. An acute angle is less than 90°. The sum of two acute angles will always be less than 180°.

(ii) obtuse?

• No. An obtuse angle is greater than 90°. The sum of two obtuse angles will always be greater than 180°.

(iii) right?

• Yes. A right angle is exactly 90°. So, the sum of two right angles is 90° + 90° = 180°. Therefore, two right angles are supplementary.

8. An angle is greater than 45°. Is its complementary angle greater than 45° or equal to 45° or less than 45 °?

Ans : 

If an angle is greater than 45°, its complementary angle must be less than 45°.

This is because the sum of complementary angles is always 90°. If one angle is already more than 45°, the other angle must be less than 45° to maintain their sum as 90°

9. 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) If two adjacent angles are supplementary, they form a ______ .

Ans : 

(i) If two angles are complementary, then the sum of their measures is 90°. 

(ii) If two angles are supplementary, then the sum of their measures is

180°. 

(iii) If two adjacent angles are supplementary, they form a linear pair.

10. In the given 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 but do not form a linear pair.

Ans : 

(i) Obtuse vertically opposite angles: ∠AOD and ∠BOC.

(ii) Adjacent complementary angles: ∠EOA and ∠AOB.

(iii) Equal supplementary angles: ∠EOB and ∠EOD.

(iv) Unequal supplementary angles: ∠EOA and ∠EOC.

(v) Adjacent angles that do not form a linear pair: ∠BOA and ∠BOC.

Exercise 5.2

1. 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

Ans : 

(i) If a || b, then ∠1 = ∠5

• Property: Corresponding angles property

When two parallel lines are cut by a transversal, the corresponding angles are equal.

(ii) If ∠4 = ∠6, then a || b

• Property: Alternate interior angles property

When two lines are cut by a transversal and the alternate interior angles are equal, then the lines are parallel.

(iii) If ∠4 + ∠5 = 180°, then a || b

• Property: Interior angles on the same side of transversal are supplementary

When two lines are cut by a transversal and the interior angles on the same side of the transversal are supplementary, then the lines are parallel

2. In the given figure, identify

(i) the pairs of corresponding angles.

(ii) the pairs of alternate interior angles.

(iii) the pairs of interior angles on the same side of the transversal.

Ans : 

(i) Pairs of corresponding angles:

• ∠1 and ∠5
• ∠2 and ∠6
• ∠4 and ∠8
• ∠3 and ∠7

(ii) Pairs of alternate interior angles:

• ∠3 and ∠5
• ∠2 and ∠8

(iii) Pairs of interior angles on the same side of the transversal:

• ∠3 and ∠8
• ∠2 and ∠5

3. In the given figure, p || q. Find the unknown angles.

Ans : 

1. ∠a = ∠e (corresponding angles)
2. ∠b = ∠d (vertically opposite angles)
3. ∠c = ∠f (vertically opposite angles)
4. ∠d + ∠e = 180° (linear pair)
5. ∠b + ∠c = 180° (linear pair)

Calculate the angles

Using the above equations, we can calculate the unknown angles:

1. ∠e = 55° (given)
2. ∠a = 55° (from equation 1)
3. ∠d = 125° (from equation 4)
4. ∠b = 125° (from equation 2)
5. ∠c = 55° (from equation 3)
6. ∠f = 55° (from equation 3)

4. Find the value of x in each of the following figures if l || m

Ans : 

Figure (i):

In this figure, the angle x and the given 110° angle are interior angles on the same side of the transversal t. We know that the sum of interior angles on the same side of the transversal is 180°.

Therefore, x + 110° = 180° => x = 180° – 110° => x = 70°

Figure (ii):

In this figure, the angle x and the given 70° angle are alternate interior angles. We know that alternate interior angles are equal when the lines are parallel.

Therefore, x = 70°

5. In the given figure, the arms of two angles are parallel. If ∠ABC = 70°, then find

(i) ∠DGC

(ii) ∠DEF

Ans : 

Given:

• ∠ABC = 70°
• AB || DE
• BC || EF

To find:

• ∠DGC
• ∠DEF

Finding the Angles:

1. ∠DGC:
   • Since AB || DE and BC is a transversal, ∠ABC and ∠DGC are corresponding angles.
   • Therefore, ∠DGC = ∠ABC = 70°.
2. ∠DEF:
   • Since BC || EF and DE is a transversal, ∠DGC and ∠DEF are corresponding angles.
   • We already found that ∠DGC = 70°.
   • Therefore, ∠DEF = ∠DGC = 70°.

6. In the given figure below, decide whether l is parallel to m.

Ans : 

(i)

• Given angles: 126° and 44°
• Analysis: The sum of these angles is 126° + 44° = 170°.
• Conclusion: For lines l and m to be parallel, the sum of the interior angles on the same side of the transversal should be 180°. Since 170° ≠ 180°, lines l and m are not parallel.

(ii)

• Given angles: 75° and 75°
• Analysis: The angles are alternate interior angles.
• Conclusion: Since the alternate interior angles are equal, lines l and m are parallel.

(iii)

Let the angle opposite to 57° be y.

∴ ∠y = 57° (Vertically opposite angles)

∴ Sum of interior angles on the same side of transversal

= 57° + 123° = 180°

∴ l is parallel to m.

(iv)

 Let angle opposite to 72° be z.

∴ z = 70° (Vertically opposite angle)

Sum of interior angles on the same side of transversal

= z + 98° = 72° + 98°

= 170° ≠ 180°

∴ l is not parallel to m.