Question

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

Corollary 4

Short answer

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Step-by-step solution

Step 1. State Theorem 3-11

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

Step 2. Statement of corollary 4

Corollary 4- Acute angles of a right triangles are complementary.

Step 3. Illustrate for Triangle ABC

In order to show that acute angles of a right triangles are complementary that is., sum of their measures is $90°$consider a right-angle triangle $\Delta ABC$such that $\angle A=90°$. Since, sum of three angles of a triangle is $180°$, therefore,

$$\begin{gathered} \begin{array}{c}\angle A+\angle B+\angle C=180°\\ 90°+\angle B+\angle C=180°\end{array} \\ 90°+\angle B+\angle C-90°=180°-90° \\ \angle B+\angle C=90° \end{gathered}$$

As angles$\angle B$ and$\angle B$ are acute and complementary angles therefore, it can be concluded that acute angles of right triangle are complementary.