34. The measure of one angle is $$ 3^{\circ} $$ more than $$ \frac{1}{2} $$ the measure of its supplement. 35. Two angles form a linear pair. The measure of one angle is $$ 15^{\circ} $$ less than $$ \frac{2}{3} $$ the measure of the other angle.

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**Problem 34:**
*The measure of one angle is $$ 3^{\circ} $$ more than $$ \frac{1}{2} $$ the measure of its supplement.*

**Solution:**

1. **Define Variables:**
   - Let $$ x $$ be the measure of the first angle.
   - Its supplement is $$ 180^{\circ} - x $$.

2. **Set Up the Equation:**
   $$
   x = \frac{1}{2}(180^{\circ} - x) + 3^{\circ}
   $$

3. **Multiply Both Sides by 2 to Eliminate the Fraction:**
   $$
   2x = 180^{\circ} - x + 6^{\circ}
   $$
   $$
   2x = 186^{\circ} - x
   $$

4. **Solve for $$ x $$:**
   $$
   2x + x = 186^{\circ}
   $$
   $$
   3x = 186^{\circ}
   $$
   $$
   x = 62^{\circ}
   $$

5. **Find the Supplementary Angle:**
   $$
   180^{\circ} - x = 180^{\circ} - 62^{\circ} = 118^{\circ}
   $$

**Answer:**
One angle measures $$ 62^{\circ} $$ and its supplement measures $$ 118^{\circ} $$.

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**Problem 35:**
*Two angles form a linear pair. The measure of one angle is $$ 15^{\circ} $$ less than $$ \frac{2}{3} $$ the measure of the other angle.*

**Solution:**

1. **Define Variables:**
   - Let $$ x $$ be the measure of the first angle.
   - Its supplementary angle is $$ 180^{\circ} - x $$.

2. **Set Up the Equation:**
   $$
   x = \frac{2}{3}(180^{\circ} - x) - 15^{\circ}
   $$

3. **Multiply Both Sides by 3 to Eliminate the Fraction:**
   $$
   3x = 2(180^{\circ} - x) - 45^{\circ}
   $$
   $$
   3x = 360^{\circ} - 2x - 45^{\circ}
   $$
   $$
   3x = 315^{\circ} - 2x
   $$

4. **Solve for $$ x $$:**
   $$
   3x + 2x = 315^{\circ}
   $$
   $$
   5x = 315^{\circ}
   $$
   $$
   x = 63^{\circ}
   $$

5. **Find the Supplementary Angle:**
   $$
   180^{\circ} - x = 180^{\circ} - 63^{\circ} = 117^{\circ}
   $$

6. **Verification:**
   - Check if $$ 63^{\circ} = \frac{2}{3}(117^{\circ}) - 15^{\circ} $$:
   $$
   \frac{2}{3} \times 117^{\circ} = 78^{\circ}
   $$
   $$
   78^{\circ} - 15^{\circ} = 63^{\circ}
   $$
   - The condition holds true.

**Answer:**
The two angles measure $$ 63^{\circ} $$ and $$ 117^{\circ} $$.

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**Problem 34:** One angle is $$ 62^{\circ} $$ and its supplement is $$ 118^{\circ} $$. **Problem 35:** The two angles are $$ 63^{\circ} $$ and $$ 117^{\circ} $$.

34. The measure of one angle is \( 3^{\circ} \) more than \( \frac{1}{2} \) the measure of its supplement. 35. Two angles form a linear pair. The measure of one angle is \( 15^{\circ} \) less than \( \frac{2}{3} \) the measure of the other angle.

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