Question

two angle measures in a triangle are 37° and 37° and one side is 6 cm long. how many triangles can be made with these measurements? triangle(s) can be made. no one two three

Explanation:

Step1: Find the third angle

The sum of angles in a triangle is \(180^\circ\). Given two angles are \(37^\circ\) and \(37^\circ\), the third angle is \(180 - 37 - 37 = 106^\circ\).

Step2: Analyze triangle uniqueness

We have two equal angles (\(37^\circ\)), so the triangle is isosceles with the two sides opposite the \(37^\circ\) angles equal. But we are given one side length (6 cm). However, in a triangle with two angles fixed (\(37^\circ, 37^\circ, 106^\circ\)) and one side length, by the Angle - Angle - Side (AAS) or Angle - Side - Angle (ASA) congruence criteria, there is only one unique triangle (up to congruence) that can be formed. Wait, no—wait, actually, if the given side is either the side between the two \(37^\circ\) angles (included side) or a side opposite one of the \(37^\circ\) angles, but in terms of triangle construction with two angles and a side, when two angles are known, the third is fixed, and with a side, the triangle is uniquely determined in shape and size (up to congruence). Wait, but the options have "Two" as a choice? Wait, no, maybe I made a mistake. Wait, the triangle has angles \(37^\circ, 37^\circ, 106^\circ\). If the given side is 6 cm, if it's the base (opposite the \(106^\circ\) angle) or one of the equal sides (opposite the \(37^\circ\) angles). Wait, no—actually, in the case of two equal angles (isosceles), if we are given a side, depending on which side it is, but the problem says "one side is 6 cm long". Wait, no, the key is that when two angles are fixed, the triangle is similar, but with a given side, it's congruent. Wait, no, the triangle angle - angle - side: if two angles and a side (either included or not) are given, for a triangle, if the two angles are fixed, the triangle is determined up to congruence. Wait, but the options have "Two" as a selected option? Wait, no, maybe I messed up. Wait, let's re - think. The triangle has angles \(37^\circ, 37^\circ, 106^\circ\). If the given side is 6 cm, if it's the side between the two \(37^\circ\) angles (the included side), then the triangle is unique. If it's a side opposite a \(37^\circ\) angle, then also, since the angles are fixed, the triangle is unique. Wait, but the answer here—wait, the problem is about how many triangles can be made with these measurements. Wait, maybe the confusion is between ambiguous cases (like SSA), but here we have two angles and a side, which is AAS or ASA, which is a congruence condition, meaning only one triangle. But the option "Two" is highlighted. Wait, no, maybe I made a mistake. Wait, no—wait, the triangle with angles \(37^\circ, 37^\circ, 106^\circ\) and a side of 6 cm: if the 6 cm is the length of the equal sides (the legs) or the base (the unequal side). But in reality, for a triangle with two angles \(37^\circ\) and a side, the triangle is uniquely determined. Wait, but the options have "Two" as a choice. Wait, maybe the problem is considering that the 6 cm side could be either the side between the two \(37^\circ\) angles or a side opposite a \(37^\circ\) angle, leading to two different triangles? Wait, no, because the angles are fixed. Let's calculate the sides. Let's denote the triangle as \(ABC\) with \(\angle A=\angle B = 37^\circ\), \(\angle C=106^\circ\). If \(AB = 6\) cm (the base, opposite \(\angle C\)), then using the Law of Sines: \(\frac{AB}{\sin C}=\frac{AC}{\sin B}\), so \(AC=\frac{AB\sin B}{\sin C}=\frac{6\sin37^\circ}{\sin106^\circ}\). If \(AC = 6\) cm (one of the equal sides), then \(AB=\frac{AC\sin C}{\sin B}=\frac{6\sin106^\circ}{\sin37^\circ}\). But the problem says "one side is 6 cm long"—it doesn't specify which side. So there are two possible triangles: one where the 6 cm side is the base (opposite \(106^\circ\)) and one where the 6 cm side is one of the equal sides (opposite \(37^\circ\)). Ah, that's the key! So depending on which side the 6 cm is (the included side or the side opposite), but in this case, since the triangle is isosceles, the 6 cm can be either the equal side or the unequal side, leading to two different triangles (in terms of side lengths, but same angles). So that's why the answer is two.

Answer:

Two (The option "Two" is correct, so the number of triangles that can be made is two.)