Question

Explain how each corollary of Theorem 3-11 follows from the theorem

Corollary 3

Short answer

It is concluded that in a triangle there can be at most one right angle or obtuse angle.

Step-by-step solution

Step 1. State Theorem 3-11

Theorem 3-11 states that the sum of the angles is ${180}^0$.

Step 2. Statement of corollary 3

Corollary 3 states that in a triangle there can be at most one right angle or obtuse angle.

Step 3. Consider a triangle with obtuse angle

Consider an obtuse triangle $\Delta ABC$such that:.

$$\begin{gathered} \begin{array}{c}\angle A=25°\\ \angle B=120°\\ \angle C=35°\end{array} \\ \angle A+\angle B+\angle C=180° \end{gathered}$$

Here, if two angles of a triangle would have been obtuse, the sum of the angles will be more than $180°$.

Step 4. Consider a triangle with right angle

Consider a triangle $\Delta PQR$such that it has two right angles:

$$\begin{gathered} \begin{array}{c}\angle P=90°\\ \angle Q=90°\\ \angle R=0°\end{array} \\ \angle P+\angle Q+\angle R=180° \end{gathered}$$

Here, it is not possible in a triangle because the sum of three angles of a triangle is $180°$.

Thus, it can be concluded that in a triangle there can be at most one right angle or obtuse angle.