EXAMPLE 3 If $$ m \angle 6=142^{\circ} $$, find each measure. Glve your reasoning. a. $$ m \angle 1= $$ b. $$ m \angle 2= $$ c. $$ m \angle 3= $$ d. $$ m \angle 4= $$ e. $$ m \angle 5= $$ f. $$ m \angle 7= $$ g. $$ m \angle 8= $$

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To solve this problem, we need to understand the relationships between the angles based on the given information that $$ m \angle 6 = 142^{\circ} $$.

Let's assume that the angles are arranged in a way that allows us to use properties of angles such as vertical angles, supplementary angles, and corresponding angles.

### Step-by-step reasoning:

1. **Identify Relationships**:
   - If $$ m \angle 6 = 142^{\circ} $$, we can find the measures of the other angles based on their relationships with angle 6.

2. **Find $$ m \angle 1 $$**:
   - If angle 1 is vertical to angle 6, then $$ m \angle 1 = m \angle 6 = 142^{\circ} $$.

3. **Find $$ m \angle 2 $$**:
   - If angle 2 is supplementary to angle 6, then:
   $$
   m \angle 2 = 180^{\circ} - m \angle 6 = 180^{\circ} - 142^{\circ} = 38^{\circ}
   $$

4. **Find $$ m \angle 3 $$**:
   - If angle 3 is vertical to angle 2, then $$ m \angle 3 = m \angle 2 = 38^{\circ} $$.

5. **Find $$ m \angle 4 $$**:
   - If angle 4 is supplementary to angle 1, then:
   $$
   m \angle 4 = 180^{\circ} - m \angle 1 = 180^{\circ} - 142^{\circ} = 38^{\circ}
   $$

6. **Find $$ m \angle 5 $$**:
   - If angle 5 is vertical to angle 4, then $$ m \angle 5 = m \angle 4 = 38^{\circ} $$.

7. **Find $$ m \angle 7 $$**:
   - If angle 7 is supplementary to angle 6, then:
   $$
   m \angle 7 = 180^{\circ} - m \angle 6 = 180^{\circ} - 142^{\circ} = 38^{\circ}
   $$

8. **Find $$ m \angle 8 $$**:
   - If angle 8 is vertical to angle 7, then $$ m \angle 8 = m \angle 7 = 38^{\circ} $$.

### Summary of Angle Measures:
- $$ m \angle 1 = 142^{\circ} $$
- $$ m \angle 2 = 38^{\circ} $$
- $$ m \angle 3 = 38^{\circ} $$
- $$ m \angle 4 = 38^{\circ} $$
- $$ m \angle 5 = 38^{\circ} $$
- $$ m \angle 7 = 38^{\circ} $$
- $$ m \angle 8 = 38^{\circ} $$

This reasoning assumes a typical arrangement of angles where vertical angles and supplementary angles are used. If the arrangement is different, the relationships may change accordingly.
$$ m \angle 1 = 142^{\circ} $$, $$ m \angle 2 = 38^{\circ} $$, $$ m \angle 3 = 38^{\circ} $$, $$ m \angle 4 = 38^{\circ} $$, $$ m \angle 5 = 38^{\circ} $$, $$ m \angle 7 = 38^{\circ} $$, $$ m \angle 8 = 38^{\circ} $$.

EXAMPLE 3 If \( m \angle 6=142^{\circ} \), find each measure. Glve your reasoning. a. \( m \angle 1= \) b. \( m \angle 2= \) c. \( m \angle 3= \) d. \( m \angle 4= \) e. \( m \angle 5= \) f. \( m \angle 7= \) g. \( m \angle 8= \)

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