Math Problem Statement

solve

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.

Solution

To solve for A\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 180180^\circ.

The three angles in the triangle are:

  1. 84+x84^\circ + x (at vertex AA),
  2. x+54x + 54^\circ (adjacent to vertex AA),
  3. 6060^\circ.

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

Now, let's solve for xx.

84+x+x+54+60=180,84^\circ + x + x + 54^\circ + 60^\circ = 180^\circ, 2x+198=180,2x + 198^\circ = 180^\circ, 2x=180198,2x = 180^\circ - 198^\circ, 2x=18,2x = -18^\circ, x=9.x = -9^\circ.

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

Thus, A=75\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 A\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 180180^\circ when verifying solutions.

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Math Problem Analysis

Mathematical Concepts

Geometry
Triangle Angle Sum Property

Formulas

Sum of angles in a triangle = 180°

Theorems

Triangle Angle Sum Theorem

Suitable Grade Level

Grades 8-10