To classify the triangle $\triangle TUV$ by its angles and sides, let's examine the given information:

1. Angle Information: The triangle has two angles marked as $42^\circ$ at vertices $T$ and $U$.

   • Since two angles are equal, this is an isosceles triangle (a triangle with at least two equal angles and, therefore, two equal sides).
   • To find the third angle at vertex $V$, we use the fact that the sum of angles in a triangle is always $180^\circ$.

   $42^\circ + 42^\circ + \angle V = 180^\circ$ $\angle V = 180^\circ - 84^\circ = 96^\circ$

   • The angle $V = 96^\circ$ is greater than $90^\circ$, making this an obtuse triangle (a triangle with one angle greater than $90^\circ$).

2. Classification by Sides: Since two angles are equal, the opposite sides $TV$ and $UV$ are also equal. This confirms that $\triangle TUV$ is isosceles by its sides.

Final Classification:

• By angles: Obtuse
• By sides: Isosceles

Would you like more details or have any questions about this classification?

Related Questions:

1. How do you determine if a triangle is isosceles using side lengths?
2. What are the properties of an obtuse triangle?
3. How can you calculate the area of an obtuse isosceles triangle if side lengths are known?
4. What would change in the classification if all three angles were equal?
5. How does the Pythagorean theorem apply to different types of triangles?

Tip:

In any triangle, if two angles are equal, the triangle is always isosceles. This can also help when working with unknown side lengths.