Answer:
Step-by-step explanation:
Let's solve each part of the problem step by step:
### Part (a):
If \( m\angle 1 = 63^\circ \), find \( m\angle 2 \) and \( m\angle 3 \).
From the diagram, angles \( \angle 1 \) and \( \angle 2 \) are corresponding angles, and angles \( \angle 1 \) and \( \angle 3 \) are alternate interior angles. Because the lines are parallel, corresponding and alternate interior angles are equal.
So,
\[ m\angle 2 = m\angle 1 = 63^\circ \]
\[ m\angle 3 = m\angle 1 = 63^\circ \]
### Part (b):
If \( m\angle 1 = 74^\circ \) and \( m\angle 4 = 3x - 18^\circ \), write an equation and find \( x \).
From the diagram, angles \( \angle 1 \) and \( \angle 4 \) are alternate interior angles. Since the lines are parallel, alternate interior angles are equal.
So,
\[ m\angle 1 = m\angle 4 \]
\[ 74^\circ = 3x - 18^\circ \]
Solve for \( x \):
\[ 74 + 18 = 3x \]
\[ 92 = 3x \]
\[ x = \frac{92}{3} \]
\[ x = 30.67^\circ \] (approximately)
### Part (c):
If \( m\angle 2 = 3x - 9^\circ \) and \( m\angle 1 = x + 25^\circ \), write an equation to find \( x \). Then find \( m\angle 2 \).
From the diagram, angles \( \angle 1 \) and \( \angle 2 \) are corresponding angles. Since the lines are parallel, corresponding angles are equal.
So,
\[ m\angle 1 = m\angle 2 \]
\[ x + 25^\circ = 3x - 9^\circ \]
Solve for \( x \):
\[ x + 25 = 3x - 9 \]
\[ 25 + 9 = 3x - x \]
\[ 34 = 2x \]
\[ x = \frac{34}{2} \]
\[ x = 17^\circ \]
Now, find \( m\angle 2 \):
\[ m\angle 2 = 3x - 9 \]
\[ m\angle 2 = 3(17) - 9 \]
\[ m\angle 2 = 51 - 9 \]
\[ m\angle 2 = 42^\circ \]
So the solutions are:
- Part (a): \( m\angle 2 = 63^\circ \), \( m\angle 3 = 63^\circ \)
- Part (b): \( x = 30.67 \)
- Part (c): \( x = 17^\circ \), \( m\angle 2 = 42^\circ \)
