Answer :

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): \( x = 17^\circ \), \( m\angle 2 = 42^\circ \)

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