One angle of a linear pair is twice the other. What is the value of the smaller angle? (A) 120 degrees (B) 50 degrees (C) 40 degrees (D) 60 degrees.
Question
One angle of a linear pair is twice the other. What is the value of the smaller angle?
(A) $120^{\circ}$ (B) $50^{\circ}$ (C) $40^{\circ}$ (D) $60^{\circ}$.
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Answer
Question: One angle of a linear pair is twice the other. What is the value of the smaller angle?
(A) $120^{\circ}$
(B) $50^{\circ}$
(C) $40^{\circ}$
(D) $60^{\circ}$
:
The correct option is (D) $60^{\circ}$.
To find the smaller angle when one angle is twice the other in a linear pair:

Definition: A linear pair consists of two adjacent angles that sum up to $180^{\circ}$.

Let the smaller angle be $\angle ABD$. Hence, the larger angle, $\angle DBC$, is twice the smaller angle: $$\angle DBC = 2\angle ABD.$$

According to the property of a linear pair: $$\angle ABD + \angle DBC = 180^{\circ}.$$

Substituting the value of $\angle DBC$: $$\angle ABD + 2\angle ABD = 180^{\circ}.$$

Simplify: $$3\angle ABD = 180^{\circ}.$$

Solving for $\angle ABD$: $$\angle ABD = \frac{180^{\circ}}{3} = 60^{\circ}.$$

Therefore, the larger angle is: $$\angle DBC = 2 \times 60^{\circ} = 120^{\circ}.$$
Thus, the value of the smaller angle is $\mathbf{60^{\circ}}$.
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