Question

find the measure of the missing angles.

Explanation:

Step1: Find angle between 112° and 42°

The sum of angles around a point is 360°, but for a straight line (180°), we first find the angle between the 112° and 42° angles. Let's call the angle between them \( x \). So \( 112^\circ + x + 42^\circ = 180^\circ \) (since they are on a straight line). Solving for \( x \): \( x = 180^\circ - 112^\circ - 42^\circ = 26^\circ \). Wait, maybe better to use vertical angles and linear pairs.

First, angle \( d \): since the angle adjacent to 42° on a straight line? Wait, no. Let's look at linear pairs. The angle with 42° and the angle opposite? Wait, maybe start with angle \( f \): vertical angles with the 42°? No, wait the angle marked 42° and angle \( f \): are they vertical? Wait, no, let's see.

Wait, the angle between the two lines: the 42° angle and the angle \( f \): maybe vertical angles? Wait, no, let's check linear pairs.

First, angle \( d \): a straight line is 180°, so the angle adjacent to 42°: \( d = 180^\circ - 42^\circ = 138^\circ \)? No, wait, no. Wait, the angle with 42° and the angle between the two lines: let's see the 112° angle, the 42° angle, and the angle between them. Wait, maybe the angle between the 112° and 42° is \( 180^\circ - 112^\circ - 42^\circ = 26^\circ \), which is angle \( e \)? Wait, no.

Wait, let's list the angles:

1. Angle \( d \): vertical angle with the angle opposite? Wait, no, let's use linear pairs. The angle adjacent to 42°: since a straight line is 180°, \( d = 180^\circ - 42^\circ = 138^\circ \)? Wait, no, that's not right. Wait, the angle marked 42° and angle \( d \): are they on a straight line? Yes, because they are adjacent and form a linear pair. So \( d + 42^\circ = 180^\circ \), so \( d = 180 - 42 = 138^\circ \).

Then angle \( f \): vertical angle with the 42° angle? Wait, no, vertical angles are equal. Wait, the angle with 42° and angle \( f \): are they vertical? Yes, because they are opposite each other when two lines intersect. So \( f = 42^\circ \).

Then angle \( e \): let's see the 112° angle, angle \( e \), and angle \( f \) (42°) on a straight line? Wait, 112° + \( e \) + \( f \) = 180°? Wait, 112 + \( e \) + 42 = 180? Then \( e = 180 - 112 - 42 = 26^\circ \).

Wait, let's verify:

- Angle \( d \): linear pair with 42°, so \( d = 180 - 42 = 138^\circ \).

- Angle \( f \): vertical angle with 42°, so \( f = 42^\circ \).

- Angle \( e \): linear pair with 112° and \( f \) (42°), so \( 112 + e + 42 = 180 \) → \( e = 180 - 154 = 26^\circ \).

- Then angle opposite to 112°: let's see, the angle adjacent to \( d \) and \( e \): wait, maybe another way.

Wait, maybe the angle between the 112° and the 42° is \( 180 - 112 - 42 = 26 \), which is \( e \). Then \( f \) is vertical to 42°, so 42°, \( d \) is vertical to the angle opposite, which is \( 112 + 26 = 138 \), so \( d = 138^\circ \).

Let's confirm:

- Linear pair: \( d + 42^\circ = 180^\circ \) → \( d = 138^\circ \) (correct, since 138 + 42 = 180).

- Vertical angles: \( f = 42^\circ \) (since they are opposite each other when two lines cross).

- Linear pair: 112° + \( e \) + \( f \) = 180° → 112 + \( e \) + 42 = 180 → \( e = 26^\circ \) (correct, 112 + 26 + 42 = 180).

- Then the angle opposite to 112° + \( e \) (26°) would be \( d \) (138°), which matches, since 112 + 26 = 138, and vertical angles are equal.

So:

- \( d = 138^\circ \)

- \( e = 26^\circ \)

- \( f = 42^\circ \)

Wait, but maybe the problem is to find all missing angles (d, e, f). Let's check again.

Step 1: Find angle \( d \).

Angle \( d \) and the 42° angle form a linear pair (they are adjacent and on a straight line), so:

\( d + 42^\circ = 180^\circ \)

\( d = 180^\circ - 42^\circ = 138^\circ \)

Step 2: Find angle \( f \).

Angle \( f \) and the 42° angle are vertical angles (opposite each other when two lines intersect), so vertical angles are equal:

\( f = 42^\circ \)

Step 3: Find angle \( e \).

The angles 112°, \( e \), and \( f \) (42°) form a linear pair (they are on a straight line), so their sum is 180°:

\( 112^\circ + e + f = 180^\circ \)

Substitute \( f = 42^\circ \):

\( 112^\circ + e + 42^\circ = 180^\circ \)

\( e = 180^\circ - 112^\circ - 42^\circ = 26^\circ \)

So the missing angles are \( d = 138^\circ \), \( e = 26^\circ \), \( f = 42^\circ \).

Answer:

• \( d = 138^\circ \)
• \( e = 26^\circ \)
• \( f = 42^\circ \)