We are given a diagram with two lines intersecting and another line intersecting one of them. We are given that one angle is $120^\circ$ and another is $70^\circ$. We need to find the measures of angles $\angle 1$, $\angle 2$, $\angle 3$, $\angle 4$, $\angle 5$, and $\angle 6$.

1. Problem Description

We are given a diagram with two lines intersecting and another line intersecting one of them. We are given that one angle is $120^\circ$ and another is $70^\circ$. We need to find the measures of angles $\angle 1$, $\angle 2$, $\angle 3$, $\angle 4$, $\angle 5$, and $\angle 6$.

2. Solution Steps

a) $\angle 1$:

$\angle 1$ and the $120^\circ$ angle form a straight line. Therefore, their sum is $180^\circ$.

$\angle 1 + 120^\circ = 180^\circ$

$\angle 1 = 180^\circ - 120^\circ = 60^\circ$

b) $\angle 2$:

$\angle 2$ and the $120^\circ$ angle are vertical angles, so they are equal.

$\angle 2 = 120^\circ$

c) $\angle 3$:

$\angle 3$ and $\angle 1$ are vertical angles, so they are equal.

$\angle 3 = \angle 1 = 60^\circ$

Alternatively, $\angle 3$ and $\angle 2$ form a straight line.

$\angle 3 + \angle 2 = 180^\circ$

$\angle 3 + 120^\circ = 180^\circ$

$\angle 3 = 180^\circ - 120^\circ = 60^\circ$

d) $\angle 4$:

$\angle 4$ and the $70^\circ$ angle form a straight line. Therefore, their sum is $180^\circ$.

$\angle 4 + 70^\circ = 180^\circ$

$\angle 4 = 180^\circ - 70^\circ = 110^\circ$

e) $\angle 5$:

$\angle 5$ and the $70^\circ$ are vertical angles, so they are equal.

$\angle 5 = 70^\circ$

f) $\angle 6$:

$\angle 6$ and $\angle 4$ are vertical angles, so they are equal.

$\angle 6 = \angle 4 = 110^\circ$

Alternatively, $\angle 6$ and $\angle 5$ form a straight line.

$\angle 6 + \angle 5 = 180^\circ$

$\angle 6 + 70^\circ = 180^\circ$

$\angle 6 = 180^\circ - 70^\circ = 110^\circ$

3. Final Answer

a) $\angle 1 = 60^\circ$

b) $\angle 2 = 120^\circ$

c) $\angle 3 = 60^\circ$

d) $\angle 4 = 110^\circ$

e) $\angle 5 = 70^\circ$

f) $\angle 6 = 110^\circ$