Question

8 fill in the blank 1 point find ( m angle y q r ). ( m angle y q r= )

Explanation:

Step1: Use the Angle - Bisector Theorem

Since \( QY = QY\) (common side), \( QR = QP\) (given, as the segments are marked equal), and \(\angle QRY=\angle QPY = 90^{\circ}\), by the Hypotenuse - Leg (HL) congruence criterion for right - triangles \(\triangle QRY\cong\triangle QPY\). Then, by the property of congruent triangles, \(\angle RQY=\angle PQY\). So, \(16x - 27=5x + 6\).

Step2: Solve the equation for \(x\)

Subtract \(5x\) from both sides: \(16x-5x - 27=5x-5x + 6\), which gives \(11x-27 = 6\).
Add \(27\) to both sides: \(11x-27 + 27=6 + 27\), so \(11x=33\).
Divide both sides by \(11\): \(x=\frac{33}{11}=3\).

Step3: Find \(m\angle YQR\)

Substitute \(x = 3\) into the expression for \(m\angle YQR\).
\(m\angle YQR=(16x - 27)+(5x + 6)\)
\(=16x-27 + 5x+6\)
\(=21x-21\)
Now, substitute \(x = 3\) into \(21x-21\): \(21\times3-21\)
\(=63 - 21\)
\(=42\)

Answer:

\(42\)