Question: Find the value of $x$.
a) A
c) C
b) B
d) D
7.
T
C
$9x + 6$
A
B
$3x + 2$
$70^\circ$
A) 6
B) 4
C) 11
D) 5
top-left: A triangle is shown in the center of the page inside a rectangular box, with vertices labeled $A$ at bottom-left, $B$ at bottom-right, and $C$ near the upper middle; side $AC$ is extended past $C$ to a point marked $T$ with an arrow, the angle at $B$ is labeled $70^\circ$, the angle near $C$ is labeled $9x + 6$, and the angle at $A$ along the base is labeled $3x + 2$. Find the value of $x$.
a) A
c) C
b) B
d) D
7.
A
C
T
$9x + 6$
A
$3x + 2$
$70^\circ$
B
A) 6
B) 4
C) 11
D) 5
position_hint: center
Solve for $x$
1. **Problem Statement:** We have a triangle $ABC$ with an exterior point $T$ on the extension of side $AC$. The angles are given as follows:
- Angle at $B$ is $70^\circ$.
- Angle at $C$ (exterior angle) is $9x + 6$.
- Angle at $A$ is $3x + 2$.
We need to find the value of $x$.
2. **Key Formula:** The exterior angle theorem states that the exterior angle is equal to the sum of the two opposite interior angles.
$$\text{Exterior angle} = \text{Angle at } A + \text{Angle at } B$$
3. **Apply the theorem:**
$$9x + 6 = (3x + 2) + 70$$
4. **Simplify the right side:**
$$9x + 6 = 3x + 2 + 70$$
$$9x + 6 = 3x + 72$$
5. **Isolate $x$ terms:**
$$9x + 6 - 3x = 72$$
$$6x + 6 = 72$$
6. **Subtract 6 from both sides:**
$$6x + \cancel{6} - \cancel{6} = 72 - 6$$
$$6x = 66$$
7. **Divide both sides by 6:**
$$\frac{6x}{\cancel{6}} = \frac{66}{\cancel{6}}$$
$$x = 11$$
**Final answer:** $\boxed{11}$ which corresponds to option C.