Chapter 5.5, Problem 53E

Solution

a)

To show: That there are infinitely many triangles with AAA given if the sum of the three positive angles is $180°$ .

It is shown that there are infinitely many triangles with AAA.

Given information:

The two angles are $\angle BAC=72°$ and $\angle ABC=53°$ .

Formula used:

The sine law is given by,

  $\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}$

Calculation:

Let’s draw a triangle having the given angles.

  

All triangles having proportional sides $a\text{'}$, $b\text{'}$, $c\text{'}$ with the above triangle are the problem’s solution.

  $\frac{a\text{'}}{a}=\frac{b\text{'}}{b}=\frac{c\text{'}}{c}$

Therefore, it is shown that there are infinitely many triangles with AAA.

b)

To give: The three examples of triangle where $A=30°$, $B=60°$, and $C=90°$ .

The answer is $a=1$, $b=\sqrt{3}$, and $c=2$ or r any set of three numbers proportional to these.

Calculation:

Draw a triangle having the given angles.

  

According to the Law of Sines,

  $\begin{array}{c}\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\\ \frac{a}{\frac{1}{2}}=\frac{b}{\frac{\sqrt{3}}{2}}=\frac{c}{1}\\ \frac{a}{1}=\frac{b}{\sqrt{3}}=\frac{c}{2}\end{array}$

Any $3$ numbers $a,b$, $c$ respecting above proportionality property are solutions.

Therefore, the answer is $a=1$, $b=\sqrt{3}$, and $c=2$ or r any set of three numbers proportional to these.

c)

To give: The three examples of triangle where $A=B=C=60°$, $B=60°$, and $C=90°$ .

The answer can be any set of three identical numbers.

Calculation:

Any $3$ equal numbers $a,a$, $a$ are solutions.

  $\begin{array}{c}a=b=c=1\\ a=b=c=7\\ a=b=c=10\end{array}$

Therefore, the answer can be any set of three identical numbers.