Question

use the diagram and the given angle measures to find the indicated angle measure.

1. m∠pqt = 51° and m∠tqr = 38°. find m∠pqr.
2. m∠pqr = 89° and m∠tqr=(4x + 5)° and m∠pqt=(2x)°. find x.
3. (overrightarrow{bd}) bisects ∠abc. use the diagram and m∠abd = 57° to find the m∠abc.

Explanation:

Step1: Identify angle - addition relationship

We know that $\angle PQR=\angle PQT+\angle TQR$.

Step2: Substitute given values

Given $\angle PQT = 51^{\circ}$ and $\angle TQR=38^{\circ}$, then $\angle PQR=51^{\circ}+38^{\circ}$.

Step3: Calculate the sum

$51 + 38=89^{\circ}$.

for second - part:

Step1: Set up an equation using angle - addition

Since $\angle PQR=\angle PQT+\angle TQR$, and $\angle PQR = 89^{\circ}$, $\angle TQR=(4x + 5)^{\circ}$, $\angle PQT=(2x)^{\circ}$, we have the equation $89=(2x)+(4x + 5)$.

Step2: Simplify the right - hand side of the equation

Combining like terms, $89 = 6x+5$.

Step3: Solve for $x$

Subtract 5 from both sides: $89−5=6x$, so $84 = 6x$. Then divide both sides by 6: $x=\frac{84}{6}=14$. But there is no 14 in the options. Let's re - check our work. The correct equation based on angle addition is $2x+(4x + 5)=89$. Combining like terms gives $6x+5 = 89$. Subtract 5 from both sides: $6x=89 - 5=84$. Divide by 6: $x = 14$. There may be a mis - typing in the options.

for third - part:

Step1: Use the angle - bisector property

If $\overrightarrow{BD}$ bisects $\angle ABC$, then $\angle ABC = 2\angle ABD$.

Step2: Substitute the given value

Given $\angle ABD=57^{\circ}$, then $\angle ABC=2\times57^{\circ}=114^{\circ}$.

Answer:

C. 89