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, $m\angle DEG+m\angle GEF = 180^{\circ}$. So, $(23x - 3)+(12x + 8)=180$.
Combining like terms: $23x+12x-3 + 8=180$, which simplifies to $35x+5 = 180$.
Subtract 5 from both sides: $35x=180 - 5=175$.
Divide both sides by 35: $x=\frac{175}{35}=5$.
$m\angle DEG=23x - 3=23\times5-3=115 - 3 = 112^{\circ}$.
$m\angle GEF=12x + 8=12\times5+8=60 + 8 = 68^{\circ}$.
$m\angle DEF = 180^{\circ}$.

Step2: Set up equation for problem 7

Since $m\angle TUW+m\angle WUV=m\angle TUV$, we have $(5x + 3)+(10x - 5)=17x-16$.
Combining like terms: $5x+10x+3 - 5=17x-16$, which simplifies to $15x-2 = 17x-16$.
Subtract $15x$ from both sides: $-2=17x-15x-16$, so $-2 = 2x-16$.
Add 16 to both sides: $2x=-2 + 16 = 14$.
Divide both sides by 2: $x = 7$.
$m\angle TUW=5x + 3=5\times7+3=35 + 3 = 38^{\circ}$.
$m\angle WUV=10x - 5=10\times7-5=70 - 5 = 65^{\circ}$.
$m\angle TUV=17x-16=17\times7-16=119-16 = 103^{\circ}$.

Step3: Set up equation for problem 8

Let $m\angle BCE=x$. Then $m\angle ECD = 5x-6$.
Since $m\angle BCE+m\angle ECD=m\angle BCD$ and $m\angle BCD = 162^{\circ}$, we have $x+(5x-6)=162$.
Combining like terms: $x+5x-6=162$, so $6x-6 = 162$.
Add 6 to both sides: $6x=162 + 6=168$.
Divide both sides by 6: $x = 28$.
$m\angle BCE=28^{\circ}$.
$m\angle ECD=5x-6=5\times28-6=140 - 6 = 134^{\circ}$.

Step4: Set up equation for problem 9

Since $m\angle ABF+m\angle EBF=m\angle ABE$, we have $(6x + 26)+(2x-9)=11x-31$.
Combining like terms: $6x+2x+26 - 9=11x-31$, so $8x + 17=11x-31$.
Subtract $8x$ from both sides: $17=11x-8x-31$, so $17 = 3x-31$.
Add 31 to both sides: $3x=17 + 31=48$.
Divide both sides by 3: $x = 16$.
$m\angle ABF=6x + 26=6\times16+26=96 + 26 = 122^{\circ}$.

Step5: Set up equation for problem 10

Since $\overline{BD}$ bisects $\angle CBE$, $m\angle CBD=m\angle DBE$. So, $3x + 25=7x-19$.
Subtract $3x$ from both sides: $25=7x-3x-19$, so $25 = 4x-19$.
Add 19 to both sides: $4x=25 + 19=44$.
Divide both sides by 4: $x = 11$.
$m\angle CBD=3x + 25=3\times11+25=33 + 25 = 58^{\circ}$.
Since $\overline{BC}\perp\overline{BA}$, $m\angle ABC = 90^{\circ}$.
$m\angle ABD=m\angle ABC+m\angle CBD=90^{\circ}+58^{\circ}=148^{\circ}$.

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