Question

find the measure of the missing angles.
answer attempt 2 out of 2
d = \boxed{^\circ} \quad e = \boxed{^\circ} \quad f = \boxed{^\circ}

Explanation:

Step1: Find angle \( f \)

Angle \( f \) is a straight angle, so \( f = 180^\circ \) (since a straight line forms a \( 180^\circ \) angle). Wait, no, actually, looking at the vertical angles or linear pairs. Wait, first, let's identify the angles. The angle with \( 49^\circ \) and the vertical angle? Wait, no, let's see: the angle \( f \) is adjacent to the \( 49^\circ \) and \( e \), but also, the angle with \( 29^\circ \) and the vertical angle. Wait, maybe first, angle \( d \): the angle with \( 29^\circ \) and the vertical angle? Wait, no, let's correct.

Wait, angle \( f \) is a straight angle? No, wait, the two vertical lines (the straight lines) are intersecting, so angle \( f \) is a straight angle? Wait, no, the lines are intersecting, so vertical angles. Wait, let's start with angle \( f \): the angle \( f \) is a straight angle? No, wait, the angle between the two vertical lines (the up and down arrows) is a straight line, so \( f = 180^\circ \)? No, that can't be. Wait, no, the diagram: there are two intersecting lines (the vertical one and the one with \( 29^\circ \) and the other two angles). Wait, maybe angle \( f \) is a right angle? No, wait, let's look at the angles around point \( f \).

Wait, the angle with \( 49^\circ \), \( e \), and the vertical line. Wait, maybe angle \( f \) is \( 90^\circ \)? No, wait, let's use linear pairs and vertical angles.

First, angle \( d \): the angle opposite to the \( 49^\circ \) angle? No, wait, the angle with \( 29^\circ \) and the vertical line. Wait, maybe angle \( d \) is equal to \( 49^\circ \)? No, wait, let's see: the angle between the two lines (the one with \( 29^\circ \) and the vertical line) and the other line. Wait, maybe angle \( f \) is \( 180^\circ - 49^\circ - e \), but also, angle \( d \) is vertical to the angle with \( 49^\circ \) and \( e \)? No, maybe I should start with angle \( f \). Wait, the angle \( f \) is a straight angle? No, the diagram shows that the two vertical lines (up and down) are a straight line, so angle \( f \) is \( 180^\circ \)? No, that's not right. Wait, no, the angle \( f \) is adjacent to the \( 49^\circ \) angle and \( e \), and also, the angle with \( 29^\circ \) and \( d \). Wait, maybe angle \( f \) is \( 90^\circ \)? No, let's think again.

Wait, the angle \( e \): since the angle with \( 29^\circ \) and \( e \) are vertical angles? Wait, no, the angle with \( 29^\circ \) and \( e \) are vertical angles, so \( e = 29^\circ \). Wait, is that right? Because vertical angles are equal. So the angle with \( 29^\circ \) and \( e \) are vertical, so \( e = 29^\circ \).

Then, angle \( d \): since the angle with \( 49^\circ \), \( e \), and \( d \) form a straight line? Wait, no, the vertical line (up and down) and the line with \( 49^\circ \), \( e \), and the other angle. Wait, the angle \( d \) is equal to \( 49^\circ \)? No, wait, the angle \( d \) and the angle with \( 49^\circ \) and \( e \) form a straight line? Wait, let's calculate angle \( d \): the sum of \( 49^\circ \), \( e \), and \( d \) should be \( 180^\circ \)? No, wait, the vertical line (up and down) is a straight line, so the angles on one side of it should sum to \( 180^\circ \). Wait, the angle \( 49^\circ \), \( e \), and \( d \) are on one side? No, maybe the angle \( d \) is equal to \( 49^\circ \) because they are vertical angles? Wait, no, let's see: the angle \( d \) and the angle with \( 49^\circ \) are vertical? No, maybe the angle \( f \) is \( 90^\circ \)? Wait, no, let's do step by step.

Step1: Find \( e \)

Angle \( e \) and the \( 29^\circ \) angle are vertical angles (opposite angles formed by intersecting lines), so they are equal. Thus, \( e = 29^\circ \).

Step2: Find \( d \)

Angle \( d \) and the \( 49^\circ \) angle are vertical angles? Wait, no, the angle \( d \) and the angle with \( 49^\circ \) are vertical? Wait, no, the angle \( d \) is equal to \( 49^\circ \) because they are vertical angles. Wait, let's check: the vertical line (up and down) and the line with \( 49^\circ \), \( e \), and the other angle. Wait, the angle \( d \) is vertical to the \( 49^\circ \) angle? Yes, because they are opposite each other when two lines intersect. So \( d = 49^\circ \).

Step3: Find \( f \)

Angle \( f \) is a straight angle? No, wait, the sum of \( 49^\circ \), \( e \), and \( f \) should be \( 180^\circ \)? Wait, no, the angle \( f \) is adjacent to \( d \) and the \( 29^\circ \) angle? Wait, no, let's see: the angles around point \( f \) should sum to \( 180^\circ \) (since it's a straight line). Wait, the angle \( d \) is \( 49^\circ \), \( e \) is \( 29^\circ \), so \( f = 180^\circ - 49^\circ - 29^\circ = 102^\circ \)? Wait, no, that can't be. Wait, maybe angle \( f \) is \( 180^\circ - 49^\circ - e \), but \( e = 29^\circ \), so \( 180 - 49 - 29 = 102 \)? Wait, no, maybe I made a mistake.

Wait, let's re-examine the diagram. The vertical line (up and down) and the line with \( 49^\circ \), \( e \), and the other angle. The angle \( f \) is a straight angle? No, the angle \( f \) is between the two lines (the one with \( 49^\circ \) and \( e \), and the vertical line). Wait, maybe angle \( f \) is \( 180^\circ - 49^\circ - e \), but \( e = 29^\circ \), so \( 180 - 49 - 29 = 102 \). Wait, but let's check another way: the angle \( d \) is \( 49^\circ \), \( e \) is \( 29^\circ \), so \( f = 180 - 49 - 29 = 102^\circ \). Wait, but maybe angle \( f \) is \( 90^\circ \)? No, that doesn't make sense. Wait, maybe the vertical line is perpendicular? No, the diagram doesn't show that. Wait, let's start over.

1. **Angle \( e \)**: Vertical angles are equal. The angle with \( 29^\circ \) and \( e \) are vertical, so \( e = 29^\circ \).

2. **Angle \( d \)**: Vertical angles are equal. The angle with \( 49^\circ \) and \( d \) are vertical, so \( d = 49^\circ \).

3. **Angle \( f \)**: The sum of angles on a straight line is \( 180^\circ \). So \( f = 180^\circ - d - e = 180 - 49 - 29 = 102^\circ \).

Wait, but let's check again. If \( d = 49^\circ \), \( e = 29^\circ \), then \( f = 180 - 49 - 29 = 102^\circ \). That seems correct.

Answer:

\( d = 49^\circ \), \( e = 29^\circ \), \( f = 102^\circ \)