For any $△ A B C,$ let the lengths of the sides opposite angles $A,$ $B,$ and $C$ be $a,$ $b,$ and $c,$ respectively.

The Law of Sines relates the sine of each angle to the length of the opposite side.

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

Let's start by finding

m ∠ A .

Then, we will use the above law to find the values of

a

and

c .

We will find them one at a time.

Consider the triangle that satisfies the given conditions.

From the Triangle Angle Sum Theorem we know that the sum of the angles in a triangle is equal to

18 0^{\circ} .

With this information we can find

A .

m ∠ A + 1 8^{\circ} + 14 2^{\circ} = 18 0^{\circ} ⇕ m ∠ A = 2 0^{\circ}

Let's mark the measure of the angle $A$ on the graph.

We know that the length of a side is

20

and that the measure of its opposite angle is

1 8^{\circ} .

We also know that the measure of the angle that is opposite to the side we want to find is

2 0^{\circ} .

With this information and using the Law of Sines, we can write an equation in terms of

a .

\frac{sin 1 8 ^{\circ}}{2 0} = \frac{sin 2 0 ^{\circ}}{a}

Let's solve the above equation for

a

\frac{sin 1 8 ^{\circ}}{2 0} = \frac{sin 2 0 ^{\circ}}{a}

Cross multiply

sin 1 8^{\circ} \cdot a = sin 2 0^{\circ} \cdot 20

$LHS / sin 1 8^{\circ} = RHS / sin 1 8^{\circ}$

a = \frac{sin 2 0 ^{\circ} \cdot 2 0}{sin 1 8 ^{\circ}}

Use a calculator

a = 22.136008 \dots

Round to $1$ decimal place(s)

a \approx 22.1

Consider the triangle with the new information.

Exercise