Question

1. using structure find all the unknown angle measures in the diagram. justify your answer for each angle measure.

Explanation:

Step1: Determine \( \angle 1 \)

\( \angle 1 \) is a right angle, so \( \angle 1 = 90^\circ \) (by definition of a right angle).

Step2: Determine \( \angle 2 \)

\( \angle 2 \) and the \( 30^\circ \) angle are vertical angles? Wait, no, \( \angle 2 \) and the angle adjacent to \( 30^\circ \)? Wait, actually, \( \angle 1 + \angle 2 + 30^\circ = 180^\circ \)? No, wait, looking at the diagram, \( \angle 1 \) is 90°, and the line is straight, so \( \angle 1 + \angle 2 + 30^\circ = 180^\circ \)? Wait, no, maybe \( \angle 2 \) is equal to the angle opposite? Wait, actually, \( \angle 2 \) and the angle with 30°: Wait, the vertical angle to \( \angle 2 \) would be... Wait, maybe better to see that \( \angle 1 = 90^\circ \), \( \angle 2 \): Let's see, the angle marked 30° and \( \angle 2 \) are complementary? No, wait, the horizontal and vertical lines are perpendicular, so \( \angle 1 = 90^\circ \). Then, the angle between the vertical line and the line with 30°: Wait, maybe \( \angle 2 = 90^\circ - 30^\circ = 60^\circ \)? Wait, no, let's re-examine. The diagram has a vertical line, a horizontal line, intersecting at right angles (so \( \angle 1 = 90^\circ \)). Then, there's a line making 30° with the vertical line? Wait, the blue 30° angle is between the vertical line (downward) and another line. So \( \angle 2 \) is adjacent to \( \angle 1 \) and the 30° angle. Wait, \( \angle 1 + \angle 2 + 30^\circ = 180^\circ \)? No, \( \angle 1 = 90^\circ \), so \( 90^\circ + \angle 2 + 30^\circ = 180^\circ \)? Then \( \angle 2 = 60^\circ \). Yes, that makes sense. So \( \angle 2 = 60^\circ \).

Step3: Determine \( \angle 3 \)

We know there's a 40° angle, and \( \angle 3 + 40^\circ + \angle 4 = 90^\circ \)? Wait, no, the horizontal line and vertical line are perpendicular, so the angle between them is 90°. The angle with 40° and \( \angle 3 \) and \( \angle 4 \) are in that 90° angle? Wait, \( \angle 3 + 40^\circ + \angle 4 = 90^\circ \)? Wait, also, \( \angle 4 \) and the 30° angle: Wait, maybe \( \angle 4 = 30^\circ \)? No, wait, vertical angles? Wait, the angle with 30° and \( \angle 4 \): Wait, maybe \( \angle 4 = 30^\circ \) (vertical angles). Then, \( \angle 3 + 40^\circ + 30^\circ = 90^\circ \)? Wait, \( 90^\circ - 40^\circ - 30^\circ = 20^\circ \), so \( \angle 3 = 20^\circ \). Let's check: \( \angle 3 + 40^\circ + \angle 4 = 90^\circ \), if \( \angle 4 = 30^\circ \) (vertical to the 30° angle), then \( \angle 3 = 90^\circ - 40^\circ - 30^\circ = 20^\circ \).

Step4: Determine \( \angle 4 \)

\( \angle 4 \) and the 30° angle are vertical angles, so \( \angle 4 = 30^\circ \) (vertical angles are equal).

Step5: Determine \( \angle 5 \)

\( \angle 5 \) is a straight angle with \( \angle 2 \) and the 30° angle? Wait, \( \angle 5 \) is adjacent to \( \angle 4 \) and the horizontal line? Wait, \( \angle 5 \) is a straight angle with \( \angle 2 \)? No, \( \angle 5 \) and \( \angle 2 \) are vertical angles? Wait, \( \angle 5 \) is on the horizontal line, opposite to \( \angle 2 \)? Wait, no, \( \angle 5 \) is adjacent to \( \angle 4 \) and the vertical line? Wait, \( \angle 5 = 180^\circ - \angle 2 - 30^\circ \)? No, \( \angle 5 \) is a straight angle, so \( \angle 5 = 180^\circ - 30^\circ - \angle 2 \)? Wait, \( \angle 2 = 60^\circ \), so \( 180^\circ - 30^\circ - 60^\circ = 90^\circ \)? No, that can't be. Wait, maybe \( \angle 5 = 180^\circ - \angle 4 - 90^\circ \)? No, let's start over.

Wait, let's list all angles:

- \( \angle 1 \): Right angle, so \( 90^\circ \).
- \( \angle 2 \): The angle between the horizontal line and the line with 30° below. Since the vertical line is perpendicular to horizontal, \( \angle 1 = 90^\circ \), so the angle between vertical and the line with 30° is \( 90^\circ - 30^\circ = 60^\circ \)? No, \( \angle 2 \) is adjacent to \( \angle 1 \) and the 30° angle. Wait, the straight line (horizontal) and the line with 30°: The sum of angles on a straight line is 180°, so \( \angle 1 + \angle 2 + 30^\circ = 180^\circ \). Since \( \angle 1 = 90^\circ \), then \( 90^\circ + \angle 2 + 30^\circ = 180^\circ \), so \( \angle 2 = 60^\circ \).
- \( \angle 3 \): In the upper left, the angle between vertical line and the line with 40°. The total angle between vertical and horizontal is 90°, so \( \angle 3 + 40^\circ + \angle 4 = 90^\circ \). Also, \( \angle 4 \) is equal to the 30° angle (vertical angles), so \( \angle 4 = 30^\circ \). Then \( \angle 3 = 90^\circ - 40^\circ - 30^\circ = 20^\circ \).
- \( \angle 4 \): Vertical angle to the 30° angle, so \( \angle 4 = 30^\circ \).
- \( \angle 5 \): Straight angle with \( \angle 2 \) and the 30° angle? Wait, \( \angle 5 \) is adjacent to \( \angle 4 \) and the vertical line? Wait, \( \angle 5 \) is a straight angle (180°) with \( \angle 4 \) and the angle opposite? Wait, no, \( \angle 5 \) is on the straight line (vertical? No, horizontal? Wait, the horizontal line: \( \angle 5 \) is adjacent to \( \angle 4 \) and the angle with 90°? Wait, \( \angle 5 = 180^\circ - \angle 4 - 90^\circ \)? No, \( \angle 5 \) is a straight angle, so \( \angle 5 = 180^\circ - \angle 2 = 180^\circ - 60^\circ = 120^\circ \)? Wait, no, let's use vertical angles. \( \angle 5 \) and the angle \( \angle 1 + \angle 3 \): Wait, \( \angle 5 = 180^\circ - \angle 4 = 180^\circ - 30^\circ = 150^\circ \)? No, I'm getting confused. Let's re-express:

Wait, correct approach:

1. \( \angle 1 \): Right angle, so \( 90^\circ \) (perpendicular lines).
2. \( \angle 2 \): The angle adjacent to \( \angle 1 \) and the \( 30^\circ \) angle. Since \( \angle 1 + \angle 2 + 30^\circ = 180^\circ \) (straight line), \( \angle 2 = 180^\circ - 90^\circ - 30^\circ = 60^\circ \).
3. \( \angle 4 \): Vertical angle to the \( 30^\circ \) angle, so \( \angle 4 = 30^\circ \) (vertical angles are equal).
4. \( \angle 3 \): In the 90° angle (between vertical and horizontal), \( \angle 3 + 40^\circ + \angle 4 = 90^\circ \). So \( \angle 3 = 90^\circ - 40^\circ - 30^\circ = 20^\circ \).
5. \( \angle 5 \): Straight line with \( \angle 4 \) and \( \angle 5 \), so \( \angle 4 + \angle 5 = 180^\circ \)? No, \( \angle 5 \) is adjacent to \( \angle 4 \) and the horizontal line? Wait, \( \angle 5 \) is on the straight line (horizontal), so \( \angle 5 + \angle 4 + \angle 2 = 180^\circ \)? No, \( \angle 5 = 180^\circ - \angle 4 = 180^\circ - 30^\circ = 150^\circ \)? Wait, no, \( \angle 5 \) is opposite to \( \angle 1 + \angle 3 \). \( \angle 1 + \angle 3 = 90^\circ + 20^\circ = 110^\circ \), no. Wait, \( \angle 5 \) and \( \angle 2 \): \( \angle 5 + \angle 2 = 180^\circ \)? \( 60^\circ + 120^\circ = 180^\circ \), so \( \angle 5 = 120^\circ \). Wait, let's check with \( \angle 3 + 40^\circ + \angle 4 + \angle 5 = 180^\circ \)? No, the total around the point is 360°, but we can use straight lines.

Wait, maybe better to summarize:

- \( \angle 1 = 90^\circ \) (right angle)
- \( \angle 2 = 60^\circ \) (since \( 90^\circ + 60^\circ + 30^\circ = 180^\circ \))
- \( \angle 3 = 20^\circ \) (since \( 20^\circ + 40^\circ + 30^\circ = 90^\circ \))
- \( \angle 4 = 30^\circ \) (vertical angle to the 30° angle)
- \( \angle 5 = 180^\circ - 30^\circ = 150^\circ \)? No, I think I made a mistake. Let's use vertical angles for \( \angle 5 \): \( \angle 5 \) is vertical to \( \angle 1 + \angle 3 \), so \( \angle 5 = 90^\circ + 20^\circ = 110^\circ \)? No, this is getting too confusing. Let's start over with correct angle relationships:

1. \( \angle 1 \): Perpendicular lines, so \( 90^\circ \).
2. \( \angle 2 \): The angle with 30° is adjacent to \( \angle 2 \) and \( \angle 1 \) on a straight line. So \( \angle 1 + \angle 2 + 30^\circ = 180^\circ \) → \( 90^\circ + \angle 2 + 30^\circ = 180^\circ \) → \( \angle 2 = 60^\circ \).
3. \( \angle 4 \): Vertical angle to the 30° angle, so \( \angle 4 = 30^\circ \) (vertical angles are equal).
4. \( \angle 3 \): In the right angle (90°) between vertical and horizontal, \( \angle 3 + 40^\circ + \angle 4 = 90^\circ \) → \( \angle 3 = 90^\circ - 40^\circ - 30^\circ = 20^\circ \).
5. \( \angle 5 \): Straight line with \( \angle 4 \) and \( \angle 5 \), so \( \angle 4 + \angle 5 = 180^\circ \)? No, \( \angle 5 \) is on the straight line (horizontal), so \( \angle 5 + \angle 4 + \angle 2 = 180^\circ \)? \( 30^\circ + \angle 5 + 60^\circ = 180^\circ \) → \( \angle 5 = 90^\circ \)? No, that's not right. Wait, \( \angle 5 \) is opposite to \( \angle 1 + \angle 3 \). \( \angle 1 + \angle 3 = 90^\circ + 20^\circ = 110^\circ \), so \( \angle 5 = 110^\circ \)? No, I'm really confused. Let's check standard angle problems:

Wait, the correct answers are likely:

- \( \angle 1 = 90^\circ \) (right angle)
- \( \angle 2 = 60^\circ \) (complementary with 30° in the straight line)
- \( \angle 3 = 20^\circ \) (90° - 40° - 30°)
- \( \angle 4 = 30^\circ \) (vertical angle)
- \( \angle 5 = 180^\circ - 30^\circ = 150^\circ \)? No, \( \angle 5 \) is adjacent to \( \angle 4 \) and the angle \( \angle 2 \). Wait, \( \angle 5 = 180^\circ - \angle 2 = 180^\circ - 60^\circ = 120^\circ \). Yes, that makes sense. Because \( \angle 2 = 60^\circ \), so \( \angle 5 = 180^\circ - 60^\circ = 120^\circ \).

Let's verify:

- \( \angle 1 = 90^\circ \) (right angle)
- \( \angle 2 = 60^\circ \) (since \( 90^\circ + 60^\circ + 30^\circ = 180^\circ \))
- \( \angle 3 = 20^\circ \) (since \( 20^\circ + 40^\circ + 30^\circ = 90^\circ \))
- \( \angle 4 = 30^\circ \) (vertical angle to the 30° angle)
- \( \angle 5 = 120^\circ \) (since \( \angle 5 + \angle 2 = 180^\circ \), \( 120^\circ + 60^\circ = 180^\circ \))

Yes, that works.

Answer:

\( \angle 1 = 90^\circ \), \( \angle 2 = 60^\circ \), \( \angle 3 = 20^\circ \), \( \angle 4 = 30^\circ \), \( \angle 5 = 120^\circ \)