\#5 i GIDASMATH Listen $$ \angle L M N $$ and $$ \angle P Q R $$ are complementary angles. Find the measures of the angles when $$ m \angle L M N=(4 x-2)^{\circ} $$ and $$ m \angle P Q R=(9 x+1)^{\circ} $$. $$ m \angle L M N=\square^{\circ} $$ $$ m \angle P Q R=\square^{\circ} $$

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UpStudy AI · Accepted Answer

To determine the measures of angles $$ \angle LMN $$ and $$ \angle PQR $$ given that they are complementary (i.e., their measures add up to $$ 90^\circ $$), follow these steps:

1. **Set Up the Equation:**
   
   Since the angles are complementary:
   $$
   m \angle LMN + m \angle PQR = 90^\circ
   $$
   
   Substitute the given expressions for each angle:
   $$
   (4x - 2) + (9x + 1) = 90
   $$

2. **Combine Like Terms:**
   $$
   4x - 2 + 9x + 1 = 90
   $$
   $$
   13x - 1 = 90
   $$

3. **Solve for $$ x $$:**
   $$
   13x = 90 + 1
   $$
   $$
   13x = 91
   $$
   $$
   x = \frac{91}{13}
   $$
   $$
   x = 7
   $$

4. **Find Each Angle's Measure:**
   
   - **Measure of $$ \angle LMN $$:**
     $$
     m \angle LMN = 4x - 2 = 4(7) - 2 = 28 - 2 = 26^\circ
     $$
   
   - **Measure of $$ \angle PQR $$:**
     $$
     m \angle PQR = 9x + 1 = 9(7) + 1 = 63 + 1 = 64^\circ
     $$

5. **Final Answer:**
   $$
   m \angle LMN = 26^\circ
   $$
   $$
   m \angle PQR = 64^\circ
   $$
$$ m \angle LMN = 26^\circ $$ and $$ m \angle PQR = 64^\circ $$.

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GIDASMATH Listen
and are complementary angles. Find the measures of the angles when and
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#5 i GIDASMATH Listen and are complementary angles. Find the measures of the angles when and .