Find the measure of the third angle of the triangle. Is it possible for sides opposite to angles of different measures to have the same length?
F
Practice makes perfect
We are given the measures of two triangle angles. Let's calculate the measure of the third one. We can use the fact that the sum of three interior angles in any triangle adds up to 180^(∘).
m∠ 1+ m∠ 2+ m∠ 3=180^(∘)
Let's substitute 45^(∘) for m∠ 1 and 92^(∘) for m∠ 2 to calculate m∠ 3.
Now that we know the measure of all the angles in the triangle, we can draw it.
As we can see, triangle △ ABC has one obtuse angle ∠ C and two acute angles ∠ A and ∠ B. According to the definition, it is an obtuse triangle. Now we need to determine if the triangle is scalene or isosceles. A triangle is isosceles if it has two sides of equal length. We can use Theorem 5.10.
If one angle of a triangle has
a greater measure than another angle,
then the side opposite to the greater angle is
longer than the side opposite to the lesser angle.
In △ ABC there are three angles of different measures. Thus, by the theorem, sides opposite to larger angles are longer and sides opposite to smaller angles are shorter. The most important is that none of them have the same length. Hence, △ ABC is an obtuse scalene triangle. The answer is F.
