Question

1. determine the value of x.

a. ( x = 32^{circ} ) b. ( x = 180^{circ} )
c. ( x = 74^{circ} ) d. ( x = 148^{circ} )

Explanation:

Step1: Identify the triangle type

The triangle has two equal - length sides (marked with the same tick marks), so it is an isosceles triangle. In an isosceles triangle, the base - angles are equal.

Step2: Use the angle - sum property of a triangle

The sum of the interior angles of a triangle is \(180^{\circ}\). Let the two equal angles be \(32^{\circ}\) and \(x\). Then, using the formula \(x + 32^{\circ}+32^{\circ}=180^{\circ}\).

$$x=180^{\circ}-(32^{\circ} + 32^{\circ})$$

$$x = 180^{\circ}-64^{\circ}$$

$$x=116^{\circ}$$

(Wait, no, wrong. Wait, no, the two equal angles: the side - mark shows that the side opposite to \(32^{\circ}\) and the side opposite to the un - named angle are equal. So the angle opposite to the side with the mark (the un - named angle) is \(32^{\circ}\). Then \(x+32^{\circ}+32^{\circ}=180^{\circ}\) is wrong. Wait, no, the correct is: Since the triangle is isosceles (two equal sides), the angles opposite to the equal sides are equal. The side opposite to \(32^{\circ}\) and the side opposite to the angle adjacent to \(x\) (the un - named angle) are equal. So the un - named angle is \(32^{\circ}\). Then \(x = 180^{\circ}-(32^{\circ}+74^{\circ})\) (no, wait, another approach. The sum of angles in a triangle is \(180^{\circ}\). Let's use the property again.
Since the triangle has two equal sides (isosceles), the base angles are equal. The given non - equal angle: Let's re - check. Wait, the two equal sides: so the angles opposite to them are equal. The side with one mark: assume the triangle has two sides with one mark. Then the angles opposite to them are equal. One of the angles is \(32^{\circ}\), so the other non - \(x\) angle is \(32^{\circ}\). Then \(x=180-(32 + 32)\) (no, that's \(116\), but not in the options. Wait, no, wrong. Wait, the correct formula: The sum of angles in a triangle \(A + B + C=180^{\circ}\). If two sides are equal (isosceles), then two angles are equal. Let's assume the two equal angles are \(32^{\circ}\) and the other angle \(x\). No, no, wait, the side - mark: if two sides are equal (the two sides that are not the base), then the base angles (the angles adjacent to the base) are equal. Wait, no, the standard is: in an isosceles triangle, if two sides are equal, the angles opposite to them are equal.
Let's use the formula \(x+32^{\circ}+74^{\circ}=180^{\circ}\) (no, another way. Wait, the options: check each option.
Option A: If \(x = 32^{\circ}\), then sum \(32+32 + 116=180\) (no, not matching options). Option C: If \(x = 74^{\circ}\), then \(74+32+74=180\) (because \(74 + 32+74=180\)).

Answer:

C. \(x = 74^{\circ}\)