Question

qs bisects ∠pqr. ( mangle pqs = 11x + 24 ) and ( mangle rqs = 36 + 9x ). select the correct geometric set up and the correct algebraic set up to determine ( mangle pqr ). pro tip: sketch it out! ( 11x + 24 = 36 + 9x ); ( square mangle pqr = 20x + 60 ); ( square mangle pqr = 11x + 24 ); ( square 36 + 9x = mangle pqr + 11x + 24 ); ( square mangle pqs = 36 + 9x ); ( square 11x + 24 = mangle pqr + 36 + 9x ); ( square mangle pqs = mangle rqs ); ( square mangle pqr = mangle pqs + mangle rqs ); ( square mangle pqr = mangle pqs ); ( square mangle rqs = mangle pqr + mangle pqs ); ( square mangle pqr = mangle rqs ); ( square mangle pqs = mangle pqr + mangle rqs )

Explanation:

Step1: Recall Angle Bisector Definition

Since \( QS \) bisects \( \angle PQR \), by the definition of an angle bisector, \( m\angle PQS = m\angle RQS \) and \( m\angle PQR = m\angle PQS + m\angle RQS \).

Step2: Analyze Algebraic Set Up

From the angle bisector property, \( m\angle PQS = m\angle RQS \), so substitute the given expressions: \( 11x + 24 = 36 + 9x \).

Step3: Analyze Geometric Set Up

Since \( QS \) divides \( \angle PQR \) into two equal angles (\( \angle PQS \) and \( \angle RQS \)), the measure of \( \angle PQR \) is the sum of the measures of \( \angle PQS \) and \( \angle RQS \), so \( m\angle PQR = m\angle PQS + m\angle RQS \). Also, substituting the expressions for \( m\angle PQS \) and \( m\angle RQS \), we get \( m\angle PQR=(11x + 24)+(36 + 9x)=20x + 60 \).

Answer:

• Correct Algebraic Set Up: \( 11x + 24 = 36 + 9x \) (because \( m\angle PQS = m\angle RQS \) from angle bisector)
• Correct Geometric Set Up: \( m\angle PQR = m\angle PQS + m\angle RQS \) (or \( m\angle PQR = 20x + 60 \) after substitution)