Question

1. find the value of x.

(image of an isosceles triangle with two sides of length 7, a vertex angle of 110°, a base of length 11.5, and a base angle labeled x°)

Explanation:

Step1: Identify triangle type

The triangle has two sides of length 7, so it's isosceles. The base angles (the angles opposite the equal sides) are equal. Wait, no—wait, the two equal sides are the legs, so the base is 11.5, and the vertex angle is \(110^\circ\). In a triangle, the sum of angles is \(180^\circ\). Let the base angles be \(x\) (since it's isosceles, the two base angles are equal). So the formula is \(110 + 2x = 180\).

Step2: Solve for x

Subtract 110 from both sides: \(2x = 180 - 110 = 70\). Then divide by 2: \(x = \frac{70}{2} = 35\). Wait, but wait—wait, maybe I made a mistake. Wait, no, in an isosceles triangle, the angles opposite equal sides are equal. The two sides of length 7 are the equal sides, so the angles opposite them are the base angles? Wait, no, the sides of length 7 are adjacent to the vertex angle of \(110^\circ\), so the base is 11.5, and the two equal sides are the legs, so the base angles (the angles at the base) are equal. So yes, the sum of angles is \(180\), so \(110 + 2x = 180\), so \(2x = 70\), so \(x = 35\). Wait, but let me check again. Wait, maybe using the Law of Cosines? Wait, no, the problem is to find the angle, and since it's isosceles, the two base angles are equal. So the sum of angles in a triangle is \(180^\circ\), so \(110 + 2x = 180\), so \(2x = 70\), so \(x = 35\).

Answer:

\(35\)