7TH MATHS TERM-1 EX 5.1

Exercise 5.1 — Geometry

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1. Name the pairs of adjacent angles.

Correct Answer: ∠ABG and ∠GBC; ∠BCF and ∠FCE; ∠FCE and ∠ECD; ∠ACF and ∠FCE; ∠ACF and ∠ECD.

Explanation: Adjacent angles have a common vertex, one common arm, and no common interior points. At B, ∠ABG and ∠GBC share BG. At C, the listed pairs share a common arm without overlapping interiors.

2. Find the angle ∠JIL from the given figure.

Correct Answer: ∠JIL = 65°.

Explanation: The ray IK lies inside ∠JIL. Therefore, ∠JIL = ∠JIK + ∠KIL = 27° + 38° = 65°.

3. Find the angle ∠GEH from the given figure.

Correct Answer: ∠GEH = 86°.

Explanation: The whole angle ∠HEF is 120°. It is divided into ∠HEG and ∠GEF, where ∠GEF = 34°. Hence ∠GEH = 120° − 34° = 86°.

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

Correct Answer: (i) x = 108°; (ii) x = 46°; (iii) x = 30°.

Explanation: Angles forming a linear pair on a straight line add to 180°. (i) 72° + x = 180°, so x = 108°. (ii) 3x + 42° = 180°, so 3x = 138° and x = 46°. (iii) 4x + 2x = 180°, so 6x = 180° and x = 30°.

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

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

Explanation: Angles in a linear pair are supplementary, so their sum is 180°. If one angle is 90°, the other is 180° − 90° = 90°. Therefore, the other angle is also a right angle.

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

Correct Answer: 30°, 120°, and 210°.

Explanation: Angles at a point total 360°. Let the angles be x, 4x and 7x. Then x + 4x + 7x = 360°, so 12x = 360° and x = 30°. Therefore the angles are 30°, 120° and 210°.

7. 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?

Correct Answer: Each of the five equal angles is 63°.

Explanation: The sum of angles at a point is 360°. Let each of the five equal angles be x°. Then 45° + 5x = 360°. Thus 5x = 315° and x = 63°.

8. In the given figure, identify (i) any two pairs of adjacent angles. (ii) two pairs of vertically opposite angles.

Correct Answer: (i) Adjacent: ∠PQU and ∠PQT; ∠TQS and ∠SQR. (ii) Vertically opposite: ∠PQU and ∠TQR; ∠PQT and ∠UQR.

Explanation: Adjacent angles share one arm and one vertex without overlapping interiors. Vertically opposite angles are formed by two intersecting straight lines and lie opposite each other; each pair has equal measure.

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

Correct Answer: 120°.

Explanation: Angles at a point sum to 360°. Thus x + 2x + 3x + 4x + 5x = 360°, giving 15x = 360° and x = 24°. The largest angle is 5x = 5 × 24° = 120°.

10. From the given figure, find the missing angle.

Correct Answer: x = 105°.

Explanation: The two intersecting lines form vertically opposite angles. The angle marked 105° is vertically opposite to the angle marked x°. Vertically opposite angles are equal, so x = 105°.

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

Correct Answer: x = 45° and y = 135°.

Explanation: The angles x° and 3x° form a linear pair, so x + 3x = 180°. Hence 4x = 180° and x = 45°. The angle y° is supplementary to x°, so y = 180° − 45° = 135°. This also agrees with the vertically opposite pair 3x° and y°: 3 × 45° = 135°.

12. Using the figure, answer the following questions. (i) What is the measure of angle x°? (ii) What is the measure of angle y°?

Correct Answer: (i) x = 125°; (ii) y = 55°.

Explanation: The angle x° is vertically opposite the 125° angle, so x = 125°. The angles x° and y° form a linear pair, so x + y = 180°. Therefore y = 180° − 125° = 55°.