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To solve for \( \angle A \) in the given triangle, we can use the triangle angle sum property, which states that the sum of the angles in any triangle is \( 180^\circ \).

The three angles in the triangle are:
1. \( 84^\circ + x \) (at vertex \( A \)),
2. \( x + 54^\circ \) (adjacent to vertex \( A \)),
3. \( 60^\circ \).

The sum of the angles is:
\[
(84^\circ + x) + (x + 54^\circ) + 60^\circ = 180^\circ.
\]

Now, let's solve for \( x \).

\[
84^\circ + x + x + 54^\circ + 60^\circ = 180^\circ,
\]
\[
2x + 198^\circ = 180^\circ,
\]
\[
2x = 180^\circ - 198^\circ,
\]
\[
2x = -18^\circ,
\]
\[
x = -9^\circ.
\]

Now, substitute \( x = -9^\circ \) into the expression for \( \angle A \):
\[
\angle A = 84^\circ + x = 84^\circ + (-9^\circ) = 75^\circ.
\]

Thus, \( \angle A = 75^\circ \).

Would you like more details or have any other questions?

Here are 5 related questions:
1. How can you confirm the solution by checking all angles of the triangle?
2. How does the triangle angle sum property apply to different types of triangles?
3. What would happen if \( \angle A \) had a different expression? How would that change the solution?
4. Can you solve for the other two angles of the triangle?
5. What is the relationship between this method and the exterior angle theorem?

**Tip:** Always ensure the total sum of interior angles of a triangle equals \( 180^\circ \) when verifying solutions.

Find angle A in a triangle where one angle is 60 degrees, another angle is (x + 54) degrees, and the third angle is (84 + x) degrees.

Learn how to find angle A in a triangle where the angles are given as expressions in terms of x. This problem uses the triangle angle sum property to solve for the unknown variable x. Suitable for students in Grades 8-10.

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