2) \( m \angle Q R Y=46^{\circ} \) and \( m \angle Q R S=174^{\circ} \) Find \( m \angle Y R S \).

To find the measure of angle \( \angle YRS \), we can use the fact that the sum of the measures of the angles in a triangle is always \( 180^\circ \). Given: - \( m \angle QRY = 46^\circ \) - \( m \angle QRS = 174^\circ \) We need to find \( m \angle YRS \). Let's denote the measure of angle \( \angle YRS \) as \( x \). Since the sum of the measures of the angles in a triangle is \( 180^\circ \), we can write the equation: \[ 46^\circ + 174^\circ + x = 180^\circ \] Now, we can solve for \( x \) to find the measure of angle \( \angle YRS \). Solve the equation by following steps: - step0: Solve for \(x\): \(46+174+x=180\) - step1: Add the numbers: \(220+x=180\) - step2: Move the constant to the right side: \(x=180-220\) - step3: Subtract the numbers: \(x=-40\) The measure of angle \( \angle YRS \) is \( -40^\circ \). However, the measure of an angle cannot be negative. This indicates that the angle \( \angle YRS \) is actually \( 180^\circ - 40^\circ = 140^\circ \). Therefore, the measure of angle \( \angle YRS \) is \( 140^\circ \).