We are given a triangle with two angles marked as $31^\circ$ and $75^\circ$, and we are asked to find the missing angle, denoted by $x$, and determine the type of triangle.

Step 1: Calculate the missing angle

In a triangle, the sum of all interior angles is always $180^\circ$. Therefore, we can find the unknown angle $x$ by using the equation:

$x + 31^\circ + 75^\circ = 180^\circ$

Simplifying this:

$x = 180^\circ - (31^\circ + 75^\circ) = 180^\circ - 106^\circ = 74^\circ$

So, $x = 74^\circ$.

Step 2: Determine the type of triangle

Since all the angles in the triangle are less than $90^\circ$, the triangle is acute (i.e., all angles are acute angles).

Final Answer:

• $x = 74^\circ$
• Type of triangle: Acute triangle

Would you like more details or have any questions?

Here are 5 related questions to expand on this concept:

1. What is the sum of angles in a quadrilateral?
2. Can a triangle have more than one obtuse angle? Why or why not?
3. What defines a right triangle in terms of its angles?
4. How do we classify triangles based on their sides and angles?
5. What is the exterior angle theorem for triangles?

Tip: Always double-check the sum of angles in polygons when solving for missing angles to avoid calculation errors.