Question

find each measure. see example 3

1. ( mangle 1 )
2. ( x )

Explanation:

21. Find \( m\angle 1 \)

Step1: Identify Vertical Angles

The \( 123^\circ \) angle and \( \angle 1 \) are vertical angles? Wait, no, wait. Wait, the angle \( (3x + 6)^\circ \) and \( 123^\circ \) – wait, actually, the \( 123^\circ \) angle and \( \angle 1 \): Wait, no, let's look at the diagram. The angle \( 123^\circ \) and \( \angle 1 \) – wait, actually, the angle \( (3x + 6)^\circ \) and \( 123^\circ \) are supplementary? Wait, no, vertical angles. Wait, no, let's see: the \( 123^\circ \) angle and \( \angle 1 \) – wait, no, the \( 123^\circ \) angle and \( (3x + 6)^\circ \): Wait, actually, the \( 123^\circ \) angle and \( \angle 1 \) are vertical angles? Wait, no, maybe the \( 123^\circ \) angle and \( \angle 1 \) are vertical angles? Wait, no, let's think again. Wait, when two lines intersect, vertical angles are equal. Wait, the \( 123^\circ \) angle and \( \angle 1 \): Wait, no, maybe the \( 123^\circ \) angle and \( (3x + 6)^\circ \) are supplementary? Wait, no, let's check the diagram. Wait, the angle \( (3x + 6)^\circ \) and \( 123^\circ \) – wait, actually, the \( 123^\circ \) angle and \( \angle 1 \) are vertical angles? Wait, no, maybe the \( 123^\circ \) angle and \( \angle 1 \) are vertical angles, so \( m\angle 1 = 180^\circ - 123^\circ \)? Wait, no, that's supplementary. Wait, no, let's see: the angle \( 123^\circ \) and \( \angle 1 \) – if they are adjacent to a straight line, but actually, the \( 123^\circ \) angle and \( \angle 1 \) are vertical angles? Wait, no, maybe the \( 123^\circ \) angle and \( (3x + 6)^\circ \) are equal? Wait, no, let's do problem 22 first, then come back to 21.

22. Find \( x \)

Step1: Identify Supplementary Angles

The angle \( (3x + 6)^\circ \) and \( 123^\circ \) are supplementary? Wait, no, they are vertical angles? Wait, no, looking at the diagram, the \( 123^\circ \) angle and \( (3x + 6)^\circ \) – wait, actually, the \( 123^\circ \) angle and \( (3x + 6)^\circ \) are equal? Wait, no, maybe they are supplementary. Wait, no, when two lines intersect, adjacent angles are supplementary. Wait, the angle \( (3x + 6)^\circ \) and \( 123^\circ \) – wait, actually, the \( 123^\circ \) angle and \( (3x + 6)^\circ \) are equal? Wait, no, let's see: the \( 123^\circ \) angle and \( \angle 1 \) – wait, maybe the \( (3x + 6)^\circ \) angle and \( 123^\circ \) angle are equal because they are vertical angles? Wait, no, vertical angles are equal. Wait, maybe the \( (3x + 6)^\circ \) angle and \( 123^\circ \) angle are supplementary? Wait, no, let's check the diagram again. The angle \( (3x + 6)^\circ \) and \( 123^\circ \) – if they are adjacent to a straight line, but actually, the \( 123^\circ \) angle and \( (3x + 6)^\circ \) are equal? Wait, no, let's do the math. Wait, the angle \( (3x + 6)^\circ \) and \( 123^\circ \) – if they are supplementary, then \( (3x + 6) + 123 = 180 \). Wait, that makes sense. So:

Step1: Set Up Equation

\( 3x + 6 + 123 = 180 \)

Step2: Simplify Left Side

\( 3x + 129 = 180 \)

Step3: Subtract 129 from Both Sides

\( 3x = 180 - 129 \)
\( 3x = 51 \)

Step4: Divide by 3

\( x = \frac{51}{3} \)
\( x = 17 \)

Now, for problem 21: \( m\angle 1 \). The angle \( \angle 1 \) and \( 123^\circ \) – wait, \( \angle 1 \) and \( (3x + 6)^\circ \) are supplementary? Wait, no, \( \angle 1 \) and \( 123^\circ \) – wait, when \( x = 17 \), \( 3x + 6 = 3*17 + 6 = 51 + 6 = 57^\circ \). Wait, no, that can't be. Wait, maybe \( \angle 1 \) is equal to \( 180 - 123 = 57^\circ \)? Wait, no, let's check. Wait, the \( 123^\circ \) angle and \( \angle 1 \) – if they are adjacent to a straight line, but actually, the \( 123^\circ \) angle and \( \angle 1 \) are supplementary? Wait, no, \( 123 + 57 = 180 \), so \( \angle 1 = 57^\circ \). Wait, but let's confirm with \( x \). When \( x = 17 \), \( 3x + 6 = 57^\circ \), so \( \angle 1 \) is equal to \( 57^\circ \), because they are vertical angles? Wait, yes, \( \angle 1 \) and \( (3x + 6)^\circ \) are vertical angles, so they are equal. So \( m\angle 1 = 57^\circ \).

21. \( m\angle 1 \)

Step1: Use Vertical Angles or Supplementary Angles

We found \( x = 17 \), so \( 3x + 6 = 3*17 + 6 = 57^\circ \). \( \angle 1 \) and \( (3x + 6)^\circ \) are vertical angles, so \( m\angle 1 = 57^\circ \). Alternatively, \( \angle 1 \) and \( 123^\circ \) are supplementary, so \( m\angle 1 = 180 - 123 = 57^\circ \).

Answer:

(21): \( 57^\circ \)