5. In the figure, $$ m \angle 4=74^{\circ} $$ and $$ m \angle 3=43^{\circ} $$. Find the measures of $$ \angle 1 $$ and $$ \angle 2 $$. (Example 4)

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To determine the measures of $$ \angle 1 $$ and $$ \angle 2 $$ given $$ m \angle 3 = 43^{\circ} $$ and $$ m \angle 4 = 74^{\circ} $$, we'll make reasonable geometric assumptions based on standard angle relationships since the figure isn't provided.

**Assumptions:**
1. **Linear Pair:** Angles that form a straight line are called a linear pair and are supplementary, meaning their measures add up to $$ 180^{\circ} $$.
2. **Adjacent Angles:** If $$ \angle 3 $$ and $$ \angle 4 $$ are adjacent and form a linear pair with $$ \angle 1 $$ and $$ \angle 2 $$ respectively, we can use the linear pair relationship.

**Step-by-Step Solution:**

1. **Determine $$ \angle 1 $$:**
   - Assume $$ \angle 1 $$ and $$ \angle 4 $$ form a linear pair.
   - Since they are supplementary:
     $$
     m \angle 1 + m \angle 4 = 180^{\circ}
     $$
   - Plug in the given measure of $$ \angle 4 $$:
     $$
     m \angle 1 + 74^{\circ} = 180^{\circ}
     $$
   - Solve for $$ m \angle 1 $$:
     $$
     m \angle 1 = 180^{\circ} - 74^{\circ} = 106^{\circ}
     $$
  
2. **Determine $$ \angle 2 $$:**
   - Assume $$ \angle 2 $$ and $$ \angle 3 $$ form a linear pair.
   - Since they are supplementary:
     $$
     m \angle 2 + m \angle 3 = 180^{\circ}
     $$
   - Plug in the given measure of $$ \angle 3 $$:
     $$
     m \angle 2 + 43^{\circ} = 180^{\circ}
     $$
   - Solve for $$ m \angle 2 $$:
     $$
     m \angle 2 = 180^{\circ} - 43^{\circ} = 137^{\circ}
     $$

**Final Answer:**
$$
\boxed{m\,\angle 1 =106^{\circ}\quad\text{and}\quad m\,\angle 2=137^{\circ}}
$$
$$ \angle 1 = 106^{\circ} $$ and $$ \angle 2 = 137^{\circ} $$.

5. In the figure, \( m \angle 4=74^{\circ} \) and \( m \angle 3=43^{\circ} \). Find the measures of \( \angle 1 \) and \( \angle 2 \). (Example 4)

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