Question

1. if ∠def is a straight angle, m∠deg=(23x - 3)°, and m∠gef=(12x + 8)°, find each measure.
2. if m∠tuw=(5x + 3)°, m∠wuv=(10x - 5)°, and m∠tuv=(17x - 16)°, find each measure.
3. if m∠ecd is six less than five times m∠bce, and m∠bcd = 162°, find each measure.
4. if m∠abf=(6x + 26)°, m∠ebf=(2x - 9)°, and m∠abe=(11x - 31)°, find m∠abf.
5. if (overline{bd}) bisects ∠cbe, (overline{bc}perpoverline{ba}), m∠cbd=(3x + 25)°, and m∠dbe=(7x - 19)°, find m∠abd.

Explanation:

Step1: Set up equation for problem 6

Since $\angle DEF$ is a straight - angle ($m\angle DEF = 180^{\circ}$) and $\angle DEF=\angle DEG+\angle GEF$, we have the equation $(23x - 3)+(12x + 8)=180$.
\[

$$\begin{align*} 23x-3 + 12x+8&=180\\ 35x + 5&=180\\ 35x&=180 - 5\\ 35x&=175\\ x& = 5 \end{align*}$$

\]

Step2: Find $m\angle DEG$

Substitute $x = 5$ into the expression for $m\angle DEG$: $m\angle DEG=23x-3=23\times5 - 3=115 - 3=112^{\circ}$.

Step3: Find $m\angle GEF$

Substitute $x = 5$ into the expression for $m\angle GEF$: $m\angle GEF=12x + 8=12\times5+8=60 + 8=68^{\circ}$.

Step4: Note $m\angle DEF$

Since $\angle DEF$ is a straight - angle, $m\angle DEF = 180^{\circ}$.

Step5: Set up equation for problem 7

Since $\angle TUV=\angle TUW+\angle WUV$, we have the equation $(5x + 3)+(10x - 5)=17x-16$.
\[

$$\begin{align*} 5x+3+10x - 5&=17x-16\\ 15x-2&=17x-16\\ 17x-15x&=16 - 2\\ 2x&=14\\ x&=7 \end{align*}$$

\]

Step6: Find $m\angle TUW$

Substitute $x = 7$ into the expression for $m\angle TUW$: $m\angle TUW=5x + 3=5\times7+3=35 + 3=38^{\circ}$.

Step7: Find $m\angle WUV$

Substitute $x = 7$ into the expression for $m\angle WUV$: $m\angle WUV=10x - 5=10\times7-5=70 - 5=65^{\circ}$.

Step8: Find $m\angle TUV$

Substitute $x = 7$ into the expression for $m\angle TUV$: $m\angle TUV=17x-16=17\times7-16=119 - 16=103^{\circ}$.

Step9: Set up equation for problem 8

Let $m\angle BCE=y$. Then $m\angle ECD = 5y-6$. Since $\angle BCD=\angle BCE+\angle ECD$ and $m\angle BCD = 162^{\circ}$, we have the equation $y+(5y - 6)=162$.
\[

$$\begin{align*} y+5y-6&=162\\ 6y&=162 + 6\\ 6y&=168\\ y&=28 \end{align*}$$

\]
So $m\angle BCE = 28^{\circ}$ and $m\angle ECD=5\times28-6=140 - 6=134^{\circ}$.

Step10: Set up equation for problem 9

Since $\angle ABE=\angle ABF+\angle EBF$, we have the equation $(6x + 26)+(2x - 9)=11x-31$.
\[

$$\begin{align*} 6x+26+2x-9&=11x-31\\ 8x + 17&=11x-31\\ 11x-8x&=17 + 31\\ 3x&=48\\ x&=16 \end{align*}$$

\]
Substitute $x = 16$ into the expression for $m\angle ABF$: $m\angle ABF=6x + 26=6\times16+26=96+26=122^{\circ}$.

Step11: Set up equation for problem 10

Since $\overline{BD}$ bisects $\angle CBE$, $m\angle CBD=m\angle DBE$. So $3x + 25=7x-19$.
\[

$$\begin{align*} 7x-3x&=25 + 19\\ 4x&=44\\ x&=11 \end{align*}$$

\]
$m\angle CBD=3x + 25=3\times11+25=33 + 25=58^{\circ}$. Since $\overline{BC}\perp\overline{BA}$, $m\angle ABC = 90^{\circ}$. Then $m\angle ABD=m\angle ABC+m\angle CBD=90^{\circ}+58^{\circ}=148^{\circ}$.

Answer:

Problem 6:
$x = 5$
$m\angle DEG = 112^{\circ}$
$m\angle GEF = 68^{\circ}$
$m\angle DEF = 180^{\circ}$

Problem 7:
$x = 7$
$m\angle TUW = 38^{\circ}$
$m\angle WUV = 65^{\circ}$
$m\angle TUV = 103^{\circ}$

Problem 8:
$m\angle BCE = 28^{\circ}$
$m\angle ECD = 134^{\circ}$

Problem 9:
$m\angle ABF = 122^{\circ}$

Problem 10:
$m\angle ABD = 148^{\circ}$