Chapter 1.3, Problem 7PSA

a.

To determine

To Find: How the sum of two acute angles could be acute angle?

Answer to Problem 7PSA

  $0°<{\theta }_1<45°\text{ and }0°<{\theta }_2<45°$

  $\Rightarrow {\theta }_1+{\theta }_2\text{is an acute angle}$

Explanation of Solution

Given:

Two acute angles.

Concept Used:

An angle $\theta$ is said to be acute angle if $0°<\theta <90°$

Calculation:

Let ${\theta }_1\text{ and }{\theta }_2$ be two acute angle, such that $0°<{\theta }_1<45°\text{ and }0°<{\theta }_2<45°$

Now,

  $\begin{array}{l}0°<{\theta }_1+{\theta }_2<90°\\ \Rightarrow {\theta }_1+{\theta }_2\text{is an acute angle}\end{array}$

Conclusion:

  $0°<{\theta }_1<45°\text{ and }0°<{\theta }_2<45°$

b.

To determine

To Find: How the sum of two acute angles could be obtuse angle?

Answer to Problem 7PSA

  $45°<{\theta }_1<90°\text{ and }45°<{\theta }_2<90°$

  $\Rightarrow {\theta }_1+{\theta }_2\text{is an obtuse angle}$

Explanation of Solution

Given:

Two acute angles.

Concept Used:

An angle $\theta$ is said to be obtuse angle if $90°<\theta <180°$

Calculation:

Let ${\theta }_1\text{ and }{\theta }_2$ be two acute angle, such that $45°<{\theta }_1<90°\text{ and }45°<{\theta }_2<90°$

Now, $90°<{\theta }_1+{\theta }_2<180°$

  $\Rightarrow {\theta }_1+{\theta }_2\text{is an obtuse angle}$

Conclusion:

  $45°<{\theta }_1<90°\text{ and }45°<{\theta }_2<90°$

  $\Rightarrow {\theta }_1+{\theta }_2\text{is an obtuse angle}$

c.

To determine

To Find: How the sum of two acute angles could be right angle?

Answer to Problem 7PSA

When both acute angles are complement of each other.

Explanation of Solution

Given:

Two acute angles.

Concept Used:

An angle $\theta$ is said to be right angle if $\theta =90°$

Calculation:

Let ${\theta }_1\text{ and }{\theta }_2$ be two acute angle, such that ${\theta }_1+{\theta }_2=90°$ .

This is possible only when both angles are complement of each other.

Conclusion:

When both acute angles are complement of each other.