Question

6 in the diagram below of $\triangle abc$, side $\overline{bc}$ is extended to point $d$, $m\angle a = x$, $m\angle b = 2x + 15$, and $m\angle acd = 5x + 5$.

what is $m\angle b$?

1. 5
2. 20
3. 25
4. 55

7 in the diagram of $\triangle abc$ below, $\overline{ab}$ is extended to point $d$.

if $m\angle cab = x + 40$, $m\angle acb = 3x + 10$, $m\angle cbd = 6x$, what is $m\angle cab$?

1. 13
2. 25
3. 53
4. 65

8 in the diagram below, $\triangle abc$ is shown with $\overline{ac}$ extended through point $d$.

if $m\angle bcd = 6x + 2$, $m\angle bac = 3x + 15$, and $m\angle abc = 2x - 1$, what is the value of $x$?

1. 12
2. $14\frac{10}{11}$
3. 16
4. $18\frac{1}{9}$

9 in $\triangle abc$ shown below, side $\overline{ac}$ is extended to point $d$ with $m\angle dab = (180 - 3x)\degree$, $m\angle b = (6x - 40)\degree$, and $m\angle c = (x + 20)\degree$.

what is $m\angle bac$?

1. $20\degree$
2. $40\degree$
3. $60\degree$
4. $80\degree$

Explanation:

Problem 6

Step1: Apply exterior angle theorem

$m\angle ACD = m\angle A + m\angle B$
$5x+5 = x + 2x+15$

Step2: Simplify and solve for x

$5x+5 = 3x+15$
$5x-3x = 15-5$
$2x=10$
$x=5$

Step3: Calculate $m\angle B$

$m\angle B=2x+15=2(5)+15$

Step1: Apply exterior angle theorem

$m\angle CBD = m\angle CAB + m\angle ACB$
$6x = (x+40)+(3x+10)$

Step2: Simplify and solve for x

$6x = 4x+50$
$6x-4x=50$
$2x=50$
$x=25$

Step3: Calculate $m\angle CAB$

$m\angle CAB=x+40=25+40$

Step1: Apply exterior angle theorem

$m\angle BCD = m\angle BAC + m\angle ABC$
$6x+2 = (3x+15)+(2x-1)$

Step2: Simplify and solve for x

$6x+2 = 5x+14$
$6x-5x=14-2$
$x=12$

Answer:

1. 25

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Problem 7