\#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 \( m \angle P Q R=(9 x+1)^{\circ} \). \( m \angle L M N=\square^{\circ} \) \( m \angle P Q R=\square^{\circ} \)

To find the measures of the angles \( \angle L M N \) and \( \angle P Q R \), we start with the fact that these angles are complementary, meaning their sum equals \( 90^\circ \). So, we can set up the equation: \[ (4x - 2) + (9x + 1) = 90 \] This simplifies to: \[ 13x - 1 = 90 \] Adding 1 to both sides gives: \[ 13x = 91 \] Dividing by 13, we find: \[ x = 7 \] Now we can substitute \( x \) back into the expressions for the angles: \[ m \angle L M N = 4(7) - 2 = 28 - 2 = 26^\circ \] \[ m \angle P Q R = 9(7) + 1 = 63 + 1 = 64^\circ \] Thus, the measures of the angles are: \( m \angle L M N = 26^\circ \) \( m \angle P Q R = 64^\circ \) Now you know the values, let’s dive into some cool facts! As angles that complement each other, \( \angle L M N \) and \( \angle P Q R \) showcase a beautiful balance in geometry—like the yin and yang of the angle world! This relationship plays a vital role in many fields, including architecture and art, where harmony is vital for design integrity. In some cases, a common mistake when dealing with complementary angles is forgetting to set the equation equal to \( 90^\circ \). Always remember to check if the angles truly add up to \( 90^\circ \), and you’ll avoid the pitfall of incorrect calculations—a lesson every budding math magician should carry!