Given two angles that measure 50 degrees and 80 degrees and side that measures 4 feet, how many triangles, if any, can be constructed?
The Correct Answer and Explanation is:
The problem is asking about the number of triangles that can be formed with two given angles, 50 degrees and 80 degrees, and a side length of 4 feet. To solve this, we must apply the basic principles of geometry, particularly those related to the Law of Sines and the properties of triangle angle sums.
Step 1: Find the Third Angle
In any triangle, the sum of the three interior angles must always equal 180 degrees. We are given two angles, 50 degrees and 80 degrees. To find the third angle, we subtract the sum of the two given angles from 180 degrees:
[
\text{Third angle} = 180^\circ – (50^\circ + 80^\circ) = 180^\circ – 130^\circ = 50^\circ.
]
So, the third angle of the triangle is 50 degrees.
Step 2: Determine the Type of Triangle
The triangle in question has two angles of 50 degrees and one angle of 80 degrees. This is a valid triangle because the sum of the angles equals 180 degrees, and none of the angles is 0 degrees or negative. The triangle is classified as an acute triangle, since all its angles are less than 90 degrees.
Step 3: Apply the Law of Sines
The Law of Sines can be used to determine the relationships between the sides and angles of the triangle. According to this law:
[
\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}
]
where (a), (b), and (c) are the sides opposite angles (A), (B), and (C), respectively.
Given that we know two angles (50° and 80°) and one side length (4 feet), we can use the Law of Sines to calculate the unknown sides. However, since the problem only gives one side (without specifying which side it is), we can say that there is exactly one triangle that can be formed. There is no ambiguity, because the angle measurements fully determine the shape of the triangle, with no possibility for multiple configurations.
Conclusion:
Only one triangle can be constructed with the given conditions. The side length does not introduce any ambiguity in the triangle’s construction since the angles are fixed.