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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.

The image shows a triangle with angles labeled as 75°, x + 64°, and x + 49°.

Learn how to solve for the unknown variable x in a triangle with angles 75°, x + 64°, and x + 49°. This problem covers important geometric and algebraic concepts, such as the angle sum property of triangles and solving linear equations. Suitable for students in Grades 8-10.

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