9th-grade geometry. need help with a question : r/ReuseSchoolwork - Free Printable

To find the value of $x$, we need to use the Exterior Angle Theorem.

Step 1: Understand the Exterior Angle Theorem
The Exterior Angle Theorem states that the measure of an exterior angle of a triangle is equal to the sum of the measures of its two remote interior angles.
* Exterior Angle: The angle outside the triangle at vertex $B$, which is labeled $20x + 5$.
* Remote Interior Angles: The two angles inside the triangle that are not adjacent to the exterior angle. These are angle $C$ ($40^\circ$) and angle $D$ ($9x - 2$).

Step 2: Set up the equation
According to the theorem:
$$ \text{Exterior Angle} = \text{Interior Angle } D + \text{Interior Angle } C $$

Substitute the given expressions into the equation:
$$ 20x + 5 = (9x - 2) + 40 $$

Step 3: Solve for $x$
First, simplify the right side of the equation by combining the constant numbers ($-2$ and $40$):
$$ 20x + 5 = 9x + 38 $$

Next, subtract $9x$ from both sides to get the $x$ terms on one side:
$$ 20x - 9x + 5 = 38 $$
$$ 11x + 5 = 38 $$

Then, subtract $5$ from both sides to isolate the term with $x$:
$$ 11x = 38 - 5 $$
$$ 11x = 33 $$

Finally, divide by $11$ to solve for $x$:
$$ x = \frac{33}{11} $$
$$ x = 3 $$

Verification:
If $x = 3$:
* Angle $D = 9(3) - 2 = 27 - 2 = 25^\circ$
* Exterior Angle at $B = 20(3) + 5 = 60 + 5 = 65^\circ$
* Check: Does $25^\circ + 40^\circ = 65^\circ$? Yes, it does.

Final Answer:
3