Question: Select the best classification for the triangle by its angles and sides.
$x^\circ$
$135^\circ$
$2x^\circ$
Obtuse, isosceles
Obtuse, scalene
Acute, scalene
Obtuse, equilateral
1. **State the problem:** We have a triangle with angles $x^\circ$, $135^\circ$, and $2x^\circ$. We need to classify the triangle by its angles and sides.
2. **Use the triangle angle sum property:** The sum of interior angles in any triangle is $180^\circ$.
$$x + 135 + 2x = 180$$
3. **Simplify the equation:**
$$3x + 135 = 180$$
4. **Isolate $x$:**
$$3x = 180 - 135$$
$$3x = 45$$
5. **Solve for $x$:**
$$x = \frac{45}{3}$$
$$x = 15$$
6. **Find all angles:**
- $x = 15^\circ$
- $135^\circ$ (given)
- $2x = 2 \times 15 = 30^\circ$
7. **Classify by angles:**
- One angle is $135^\circ$, which is greater than $90^\circ$, so the triangle is **obtuse**.
8. **Classify by sides:**
- All angles are different ($15^\circ$, $30^\circ$, $135^\circ$), so all sides are different lengths.
- Therefore, the triangle is **scalene**.
**Final answer:** The triangle is **obtuse, scalene**.