23. Angles $$ M $$ and $$ P $$ are complementary. If $$ m \angle M=(3 x-45)^{\circ} $$ and $$ m \angle P=(2 x+15)^{\circ} $$, what is the value of $$ x $$ ? What is the measure of each angle? 24. The measure of $$ \angle C $$ is $$ (4 x-24.6)^{\circ} $$ and the measure of $$ \angle D $$ is $$ (x+11.3)^{\circ} $$. If the angles are supplementary, find the value of $$ x $$ and $$ m \angle D $$.

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**23. Complementary Angles**

Since angles $$ M $$ and $$ P $$ are complementary, we have:
$$
(3x - 45) + (2x + 15) = 90.
$$

1. Combine like terms:
   $$
   3x + 2x - 45 + 15 = 90 \quad \Longrightarrow \quad 5x - 30 = 90.
   $$

2. Solve for $$ x $$:
   $$
   5x = 90 + 30 = 120 \quad \Longrightarrow \quad x = \frac{120}{5} = 24.
   $$

3. Determine the measures of the angles:
   - $$ m\angle M = 3x - 45 = 3(24) - 45 = 72 - 45 = 27^\circ $$.
   - $$ m\angle P = 2x + 15 = 2(24) + 15 = 48 + 15 = 63^\circ $$.

---

**24. Supplementary Angles**

The angles are supplementary, so:
$$
(4x - 24.6) + (x + 11.3) = 180.
$$

1. Combine like terms:
   $$
   4x + x - 24.6 + 11.3 = 180 \quad \Longrightarrow \quad 5x - 13.3 = 180.
   $$

2. Solve for $$ x $$:
   $$
   5x = 180 + 13.3 = 193.3 \quad \Longrightarrow \quad x = \frac{193.3}{5} \approx 38.66.
   $$

3. Find the measure of $$ \angle D $$:
   $$
   m\angle D = x + 11.3 \approx 38.66 + 11.3 \approx 49.96^\circ.
   $$

Thus, the values are:
- For complementary angles: $$ x=24 $$, $$ m\angle M=27^\circ $$, $$ m\angle P=63^\circ $$.
- For supplementary angles: $$ x\approx38.66 $$ and $$ m\angle D\approx49.96^\circ $$.
**23. Complementary Angles** - $$ x = 24 $$ - $$ m\angle M = 27^\circ $$ - $$ m\angle P = 63^\circ $$ **24. Supplementary Angles** - $$ x \approx 38.66 $$ - $$ m\angle D \approx 49.96^\circ $$

Angles and are complementary. If and , what
is the value of ? What is the measure of each angle?

The measure of is and the measure of is . If the
angles are supplementary, find the value of and .

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Angles and are complementary. If and , what is the value of ? What is the measure of each angle? The measure of is and the measure of is . If the angles are supplementary, find the value of and .