7 th maths chapter 5 || 5.1

7th Standard Mathematics — Chapter 5: Geometry

Exercise 5.1 Solutions

1. Name the pairs of adjacent angles

Name the pairs of adjacent angles from the given figure.

Adjacent angles share a common vertex and a common arm, without overlapping interiors.

• ∠ABG and ∠GBC (or ∠GBC and ∠CBD)
• ∠FCB and ∠FCE
• ∠FCE and ∠ECD
• ∠FCB and ∠ECD

2. Find the angle ∠JIL

Find the angle ∠JIL from the given figure.

Solution:

∠JIL = ∠JIK + ∠KIL

∠JIL = 27° + 38° = 65°

Answer: ∠JIL = 65°

3. Find the angle ∠GEH

Find the angle ∠GEH from the given figure.

Solution:

∠FEH = ∠FEG + ∠GEH

120° = 34° + ∠GEH

∠GEH = 120° - 34° = 86°

Answer: ∠GEH = 86°

4. Calculate the value of x° (AB is a straight line)

Given that AB is a straight line. Calculate the value of x° in the following cases.

1. 72° + x° = 180° (Linear Pair)
   x° = 180° - 72° = 108°
2. 3x° + 42° = 180° (Linear Pair)
   3x° = 180° - 42°
   3x° = 138°
   x° = 138° ÷ 3 = 46°
3. 4x° + 2x° = 180° (Linear Pair)
   6x° = 180°
   x° = 180° ÷ 6 = 30°

5. Linear Pair Property

One angle of a linear pair is a right angle. What can you say about the other angle?

Solution:

Sum of angles in a linear pair = 180°

Given one angle = 90° (Right angle)

Other angle = 180° - 90° = 90°

Answer: The other angle is also a right angle (90°).

6. Angles at a Point (Ratio 1 : 4 : 7)

If the three angles at a point are in the ratio 1 : 4 : 7, find the value of each angle?

Solution:

• Sum of angles around a point = 360°
• Let the angles be 1x, 4x, and 7x

1x + 4x + 7x = 360°

12x = 360°

x = 360° ÷ 12 = 30°

• First angle = 1 × 30° = 30°

• Second angle = 4 × 30° = 120°

• Third angle = 7 × 30° = 210°

Answer: The angles are 30°, 120°, and 210°.

7. Six Angles at a Point

There are six angles at a point. One of them is 45° and the other five angles are all equal. What is the measure of all the five angles?

Solution:

• Sum of 6 angles at a point = 360°
• Let each of the five equal angles be x

45° + 5x = 360°

5x = 360° - 45°

5x = 315°

x = 315° ÷ 5 = 63°

Answer: The measure of each of the five equal angles is 63°.

8. Identify Angles from Figure

In the given figure, identify:

1. Any two pairs of adjacent angles:
   (∠PQT, ∠TQS) and (∠SQR, ∠RQU)
2. Two pairs of vertically opposite angles:
   (∠PQT, ∠RQU) and (∠PQU, ∠RQT)

9. Largest Angle at a Point

The angles at a point are x°, 2x°, 3x°, 4x° and 5x°. Find the value of the largest angle?

Solution:

x° + 2x° + 3x° + 4x° + 5x° = 360°

15x° = 360°

x° = 360° ÷ 15 = 24°

Largest angle = 5x° = 5 × 24° = 120°

Answer: The value of the largest angle is 120°.

10. Find Missing Angle (Vertically Opposite)

From the given figure, find the missing angle x°.

Solution:

The lines intersect at point O. Vertically opposite angles are equal.

x° = 105°

Answer: x° = 105°

11. Find Angles x° and y°

Find the angles x° and y° in the figure shown.

Solution:

• By vertically opposite angles rule: x° = 3x°? No, 3x° is vertically opposite to y°, so y° = 3x°.
• x° and 3x° form a linear pair on line l:

x° + 3x° = 180°

4x° = 180°

x° = 180° ÷ 4 = 45°

Since y° is vertically opposite to 3x°:

y° = 3x° = 3 × 45° = 135°

Answer: x° = 45° and y° = 135°

12. Find Measures of Angles x° and y°

Using the figure, answer the following questions:

1. What is the measure of angle x°?
   x° and 125° are vertically opposite angles.
   x° = 125°
2. What is the measure of angle y°?
   y° and 125° form a linear pair on a straight line.
   y° + 125° = 180°
   y° = 180° - 125° = 55°