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Question

If $$\angle 1 = 120^\circ,$$ find the measures of: $$\angle 2, \angle 3, \angle 4, \angle 5, \angle 6, \angle 7$$ and $$\angle 8.$$ Give reasons.

Solution
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Given
$$l \parallel m$$ and p is their transversal and
$$\angle 1 = 120^\circ$$
$$\angle 1 + \angle 2 = 180^\circ$$ [Straight line angle]
$$120^\circ + \angle 2 = 180^\circ$$
$$\angle 2 = 180^\circ - 120^\circ$$
We get
$$\angle 2 = 60^\circ$$
Therefore, $$\angle 2 = 60^\circ$$
But $$\angle 1$$ and $$\angle 3$$ [Vertically opposite angles]
Hence, $$\angle 3 = \angle 1 = 120^\circ$$
Similarly,
$$\angle 4 = \angle 2$$ [Vertically opposite angles]
$$\angle 4 = 60^\circ$$
$$\angle 5 = \angle 1$$ [Corresponding angles]
Hence, $$\angle 5 = 120^\circ$$
Similarly,
$$\angle 6 = \angle 2$$ [Corresponding angles]
$$\angle 6 = 60^\circ$$
$$\angle 7 = \angle 5$$ [Vertically opposite angles]
Hence, $$\angle 7 = 120^\circ$$
and $$\angle 8 = \angle 6$$ [Vertically opposite angles]
Hence, $$\angle 8 = 60^\circ$$
Therefore, the measures of angles are,
$$\angle 2 = 60^\circ$$
$$\angle 3 = 120^\circ$$
$$\angle 4 = 60^\circ$$
$$\angle 5 = 120^\circ$$
$$\angle 6 = 60^\circ$$
$$\angle 7 = 120^\circ$$ and
$$\angle 8 = 60^\circ$$

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