solve this---**Question Stem:**
In $\triangle ABC$ above, what is the value of $x$?
**Options:**
(A) 25
(B) 30
(C) 35
(D) 40
(E) 60
**Chart/Diagram Description:**
* Type: Geometric figure (Triangle with an exterior angle).
* Main Elements:
* Points: Points labeled A, B, and C are shown.
* Lines: Straight line segments form a triangle ABC. A horizontal line passes through points A and C, extending to the right of C.
* Angles:
* Angle at vertex A within triangle ABC is labeled as 25°.
* Angle at vertex B within triangle ABC is labeled as x°.
* The exterior angle at vertex C, formed by side BC and the extension of side AC, is labeled as 60°.
* Labels and Annotations: Points A, B, C are labeled. Angles are labeled as 25°, x°, and 60°.
* Relative Position and Direction: Points A and C are on a horizontal line. B is above this line. C is between A and the point where the exterior angle is measured. The line segment AC extends to the right.
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In this problem, we need to find the value of x in triangle ABC. We're given that the angle at vertex A is 25 degrees, and the exterior angle at vertex C is 60 degrees. Let's recall that the exterior angle of a triangle is equal to the sum of the two opposite interior angles.
According to the exterior angle theorem, the exterior angle of a triangle is equal to the sum of the two opposite interior angles. In our triangle, the exterior angle at C is 60 degrees. The opposite interior angles are at vertices A and B, which are 25 degrees and x degrees respectively. So we can write: 60 degrees equals 25 degrees plus x degrees. Solving for x, we get: x equals 60 minus 25, which is 35 degrees.
Therefore, the value of x is 35 degrees. Looking at our options, the correct answer is C, 35 degrees. To summarize, we used the exterior angle theorem, which states that the exterior angle of a triangle equals the sum of the two opposite interior angles. In our case, the exterior angle at C is 60 degrees, which equals the sum of the angle at A (25 degrees) and the angle at B (x degrees). Solving the equation 60 equals 25 plus x, we found that x equals 35 degrees.
Let's summarize what we've learned. The exterior angle of a triangle equals the sum of the two opposite interior angles. This is known as the exterior angle theorem. In our problem, we had the exterior angle at C as 60 degrees, and one of the opposite interior angles at A as 25 degrees. Using the theorem, we wrote the equation: 60 degrees equals 25 degrees plus x degrees. Solving for x, we got x equals 35 degrees. Therefore, the correct answer is option C, 35 degrees. This problem demonstrates a direct application of the exterior angle theorem in geometry.