Q.1. If P, Q, and R are three collinear points, then name all the line segments determined by them.
Ans.
We can have the following line segments:
Q.2. In the adjoining figure, identify at least four collinear points.
Ans. The four collinear points are A, B, C, and R.
Q.3. Find the complement of 36°.
Ans. ∵ 36° + [Complement of 36°] = 90°
⇒ Complement of 36°= 90° – 36° = 54°
Q.4. Find the supplement of 105°.
Ans. ∵ 105° + [Supplement of 105°] = 180°
⇒ Supplement of 105° = 180° – 105° = 75°
Q.5. Angles ∠ P and 100° form a linear pair. What is the measure of ∠ P?
Ans. ∵ The sum of the angles of a linear pair equal to 180°.
∴ ∠ P + 100° = 180° ⇒ ∠ P = 180° – 100 = 80°.
Q.6. In the adjoining figure, what is the measure of p?Ans. ∵ p and 120° form a linear pair.
∴ p + 120° = 180° ⇒ p = 180° – 120° = 60°
Q.7. In the adjoining figure, AOB is a straight line. Find the value of x.Ans. ∵ AOB is a straight line.
∴ ∠AOC + ∠COB = 180°
⇒ 63° + x = 180° ⇒ x = 180° – 63° = 117°
Q.8. In the given figure, AB, CD, and EF are three lines concurrent at O. Find the value of y.Ans. ∵ ∠AOE and ∠BOF and vertically opposite angles.
∴ ∠AOE = ∠BOF = 5y ….. (1)
Now, CD is a straight line,
⇒ ∠COE + ∠EOA + ∠AOD = 180°
⇒ 2y + 5y + 2y = 180° [From (1)]
⇒ 9y = 180°⇒ y = (180°/2)= 20°
Thus, the required value of y is 20°.
Q.9. In the adjoining figure, AB || CD and PQ is transversal. Find x.Ans. ∵ AB || CD and PQ is a transversal.
∴ ∠ BOQ = ∠ CQP [∵ Alternate angles are equal]
⇒ x = 110° [∵ ∠ CQP = 110°]
Q.10. Find the measure of an angle that is 26° more than its complement.
Ans. Let the measure of the required angle be x.
∴ Measure of the complement of x° = (90° – x)
⇒ x° – (90° – x) = 26°
⇒ x – 90° + x = 26°
⇒ 2x = 26° + 90° = 116°
⇒ x = (116°/2) = 58°
Thus, the required measure = 58°.
Q.11. Find the measure of an angle if four times its complement is 10° less than twice its supplement.
Ans. Let the measure of the required angle be x.
∴ Its complement = (90° – x) and Its supplement = (180° – x)
According to the condition:
4(Complement of x) = 2(Supplement of x) – 10
⇒ 4(90° – x) = 2(180° – x) – 10°
⇒ 360° – 4x = 360° – 2x – 10°
⇒ 4x – 2x = 360° – 360° + 10°
⇒ 2x = 10° ⇒ x = (10°/2)= 5°
Thus, the measure of the required angle is 5°.
Q.12. Two supplementary angles are in the ratio 3:2. Find the angles.
Ans. Let the measure of the two angles be 3x and 2x.
∵ They are supplementary angles.
∴ 3x + 2x = 180°
⇒ 5x = 180° ⇒ x = (180°/5) = 36°
∴ 3x = 3 x 36° = 108° and 2x = 2 x 36° = 72°
Thus, the required angles are 108° and 72°.
Q.13. In the adjoining figure, AOB is a straight line.
Ans. ∵ AOB is a straight line.
∴ ∠ AOC + ∠ BOC = 180°
⇒ (3x + 10°) + (2x – 30°) = 180° [Linear pair]
⇒ 3x + 2x + 10° – 30° = 180°
⇒ 5x – 20° = 180°
⇒ 5x = 180° + 20° = 200° ⇒ x = (200°/5) = 40°
Thus, the required value of x is 40°.
Q.14. In the adjoining figure, find ∠ AOC and ∠ BOD.Ans. ∵ AOB is a straight line.
∴ ∠AOC + ∠COD + ∠DOB =180°
⇒ x + 70° + (2x – 25°) = 180°
⇒ x + 2x = 180° + 25° – 70°
⇒ 3x = 205° – 70° = 135° ⇒ x = (135°/3) = 45°
∴ ∠ AOC = 45°
⇒ ∠ BOD = 2x – 25° = 2 (45°) – 25° = 90° – 25° = 65°
Q.15. In the adjoining figure, AB || CD. Find the value of x.Ans. Let us draw EF || AB and pass through point O.
∴ EF || CD and CO is a transversal.
⇒ ∠ 1 = 25° [Alternate angles]
Similarly, ∠ 2 = 35°
Adding, ∠ 1 + ∠ 2 = 25° + 35° ⇒ x = 60°
Thus, the required value of x is 60°.