The image shows a triangle with angles labeled as follows:

1. The angle at vertex $A$ is $75^\circ$.
2. One of the other angles is labeled as $x + 64$.
3. The remaining angle is labeled as $x + 49$.

We know that the sum of all interior angles of a triangle is always $180^\circ$. Thus, we can set up the following equation to solve for $x$:

$75^\circ + (x + 64) + (x + 49) = 180^\circ$

Now, let's simplify and solve for $x$:

$75 + x + 64 + x + 49 = 180$ $75 + 64 + 49 + 2x = 180$ $188 + 2x = 180$ $2x = 180 - 188$ $2x = -8$ $x = -4$

Now that we know $x = -4$, we can substitute this value back into the expressions for the other angles to find their measures:

• $x + 64 = -4 + 64 = 60^\circ$
• $x + 49 = -4 + 49 = 45^\circ$

Thus, the three angles of the triangle are:

• $75^\circ$,
• $60^\circ$,
• $45^\circ$.

These angles indeed sum up to $180^\circ$, confirming the solution is correct.

Would you like more details on any step? Here are 5 related questions:

1. How do you solve for $x$ when given different types of angle relationships in polygons?
2. What are some common angle properties of triangles that are useful for problem-solving?
3. Can a triangle have a negative angle? If not, why do we allow negative solutions like $x = -4$?
4. How do the properties of an isosceles triangle differ from this type of triangle?
5. What is the importance of the sum of interior angles in polygons?

Tip: When solving for angles, always remember that the sum of interior angles for a triangle is $180^\circ$, regardless of the triangle type.