Question

use the diagram and the given angle measures to find the indicated angle measure. 8. if (mangle pqr = 113^{circ}) and (mangle tqr = 30.25^{circ}). find (mangle pqt). (mangle pqt=2x + 4^{circ}) and (mangle tqr = 2x^{circ}). if (mangle pqr = 104^{circ}), find x.

Explanation:

Step1: Identify angle - addition relationship

From the diagram, $m\angle PQR=m\angle PQT + m\angle TQR$.

Step2: Solve for $m\angle PQT$ in the first - part

Given $m\angle PQR = 113^{\circ}$ and $m\angle TQR=30.25^{\circ}$, we substitute into the formula: $m\angle PQT=m\angle PQR - m\angle TQR$.
$m\angle PQT = 113^{\circ}-30.25^{\circ}=82.75^{\circ}$

Step3: Solve for $x$ in the second - part

Given $m\angle PQT = 2x + 4^{\circ}$, $m\angle TQR = 2x^{\circ}$, and $m\angle PQR = 104^{\circ}$. Substitute into $m\angle PQR=m\angle PQT + m\angle TQR$.
$104^{\circ}=(2x + 4)^{\circ}+2x^{\circ}$
$104=2x + 4+2x$
$104=4x + 4$
$4x=104 - 4$
$4x=100$
$x = 25$

Answer:

1. $m\angle PQT = 82.75^{\circ}$
2. $x = 25$