8. In $$ riangle P Q R $$, if $$ m \angle P=x-12, m \angle Q=5 x-27 $$, and $$ m \angle R=2 x+3 $$, find $$ x $$ and the measure of each angle.

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8. In $$ 	riangle P Q R $$, if $$ m \angle P=x-12, m \angle Q=5 x-27 $$, and $$ m \angle R=2 x+3 $$, find $$ x $$ and the measure of each angle.

UpStudy AI · Accepted Answer

To solve for $$ x $$ and the measures of each angle in $$ \triangle PQR $$, follow these steps:

1. **Given:**
   - $$ m \angle P = x - 12 $$
   - $$ m \angle Q = 5x - 27 $$
   - $$ m \angle R = 2x + 3 $$

2. **Sum of the Angles in a Triangle:**
   The sum of the interior angles in any triangle is $$ 180^\circ $$. Therefore:
   $$
   (x - 12) + (5x - 27) + (2x + 3) = 180
   $$

3. **Combine Like Terms:**
   $$
   x + 5x + 2x - 12 - 27 + 3 = 180
   $$
   $$
   8x - 36 = 180
   $$

4. **Solve for $$ x $$:**
   $$
   8x = 180 + 36
   $$
   $$
   8x = 216
   $$
   $$
   x = \frac{216}{8} = 27
   $$

5. **Find Each Angle:**
   - **Angle P:**
     $$
     m \angle P = x - 12 = 27 - 12 = 15^\circ
     $$
   - **Angle Q:**
     $$
     m \angle Q = 5x - 27 = 5(27) - 27 = 135 - 27 = 108^\circ
     $$
   - **Angle R:**
     $$
     m \angle R = 2x + 3 = 2(27) + 3 = 54 + 3 = 57^\circ
     $$

6. **Verification:**
   $$
   15^\circ + 108^\circ + 57^\circ = 180^\circ
   $$
   The angles sum up to $$ 180^\circ $$, confirming the solution is correct.

**Final Answer:**
$$
x = 27^\circ
$$
$$
m \angle P = 15^\circ,\quad m \angle Q = 108^\circ,\quad m \angle R = 57^\circ
$$
$$ x = 27^\circ $$, $$ m \angle P = 15^\circ $$, $$ m \angle Q = 108^\circ $$, $$ m \angle R = 57^\circ $$.

8. In \( \triangle P Q R \), if \( m \angle P=x-12, m \angle Q=5 x-27 \), and \( m \angle R=2 x+3 \), find \( x \) and the measure of each angle.

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