Question

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Answer 1

`m\angle 3=30^\circ,~m\angle 8=150^\circ`

Explanation:

Angles and Lines

We must recall some properties of angles and lines:

Linear pair of angles: Two angles are linear if they are adjacent angles formed by two intersecting lines. They must add up to 180°.

Corresponding angles: They are angles that occupy the same relative position at each intersection where a straight line crosses two others. If the two lines are parallel, the corresponding angles are congruent, i.e., they have the same measure.

The figure shows two parallel lines a and b, crossed by the line m. These conditions make the following relations be true:

Angles 3 and 4 are linear pair

Angles 8 and 4 are corresponding.

The first relation leads to:

`m\angle 3+m\angle 4 = 180^\circ`

The second relation leads to:

`m\angle 4 = m\angle 8`

Since:

`m\angle 3=x`

`m\angle 8=5x`

Substituting:

`x + 5x = 180^\circ`

Simplifying:

`6x = 180^\circ`

Solving for x:

`x = 180^\circ/6`

`x = 30^\circ`

Now,

`m\angle 3=x=30^\circ`

`m\angle 8=5x=150^\circ`

`m\angle 3=30^\circ,~m\angle 8=150^\circ`