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 $m\angle dab = (180 - 3x)^\circ$, $m\angle b = (6x - 40)^\circ$, and $m\angle c = (x + 20)^\circ$.

what is $m\angle bac$?

1. $20^\circ$
2. $40^\circ$
3. $60^\circ$
4. $80^\circ$

Explanation:

Step1: Solve Question 6 (Exterior Angle Thm)

The exterior angle of a triangle equals the sum of the two remote interior angles: $m\angle ACD = m\angle A + m\angle B$
$\implies 5x + 5 = x + (2x + 15)$
Simplify: $5x + 5 = 3x + 15$
$\implies 5x - 3x = 15 - 5$
$\implies 2x = 10 \implies x=5$
$m\angle B = 2x +15 = 2(5)+15=25$

Step2: Solve Question7 (Exterior Angle Thm)

$m\angle CBD = m\angle CAB + m\angle ACB$
$\implies 6x = (x+40)+(3x+10)$
Simplify: $6x=4x+50$
$\implies 2x=50 \implies x=25$
$m\angle CAB = x+40=25+40=65$

Step3: Solve Question8 (Exterior Angle Thm)

$m\angle BCD = m\angle BAC + m\angle ABC$
$\implies 6x+2=(3x+15)+(2x-1)$
Simplify: $6x+2=5x+14$
$\implies 6x-5x=14-2 \implies x=12$

Step4: Solve Question9 (Linear Pair + Triangle Sum)

$m\angle BAC + m\angle DAB=180^\circ \implies m\angle BAC=180-(180-3x)=3x$
Triangle sum: $m\angle BAC + m\angle B + m\angle C=180^\circ$
$\implies 3x + (6x-40)+(x+20)=180$
Simplify: $10x-20=180$
$\implies 10x=200 \implies x=20$
$m\angle BAC=3x=3(20)=60^\circ$

Answer:

1. 3) 25
2. 4) 65
3. 1) 12
4. 3) $60^\circ$