Question

Use diagram shown.

1. For $\Delta ABC$ and $\Delta DEC$, find the measure of each angle and the length of each side.
2. How are $\Delta ABC$ and$\Delta DEC$ related? Explain your reasoning.

Short answer

(a) The angle of each side is$90°$, $60°$ and$30°$.

(b) The $\Delta ABC$ and $\Delta DEC$ are related to each other by angle-angle theorem.

Step-by-step solution

Step 1. Given Information.

The figure is shown below.

Step 2. Concept used.

If the sum of two angles is $90°$ then the angles are called complementary angles and if the sum is $180°$ then the angles are called supplementary angles.

Step 3. Calculation.

From the diagram,

$\begin{array}{c}\angle ACB+\angle DCE=180°\\ 150°+\angle DCE=180°\\ \angle DCE=180°-150°\\ \angle DCE=30°\end{array}$

This implies that,

$\angle C=30°$

So, in $\Delta ABC$,

$\begin{array}{c}\angle CAB+\angle ABC+\angle BCA=180°\\ \angle A+\angle B+\angle C=180°\\ \angle A+90°+30°=180°\\ \angle A=180°-120°\end{array}$

This implies that,

$\angle A=60°$

And similarly, in $\Delta DEC$.

$\begin{array}{c}\angle DCE+\angle CED+\angle EDC=180°\\ \angle C+\angle E+\angle D=180°\\ 30°+90°+\angle D=180°\\ \angle D=180°-120°\end{array}$

This implies that,

$\angle D=60°$

Thus, the angles are $\angle A=60°$, $\angle C=30°$, and $\angle D=60°$.

Since, $\angle A=60°$ , $\angle D=60°$ and $\angle E=90°$, $\angle B=90°$ therefore, $\Delta ABC$ and $\Delta DEC$ are similar, by angle-angle theorem.

$\begin{array}{c}\frac{AB}{DE}=\frac{BC}{CE}\\ \frac{4}{2}=\frac{BC}{CE}\\ \frac{2}{1}=\frac{BC}{CE}\end{array}$

Thus, $BC=2\text{and}CE=1$.

Since, it is aright angled triangle therefore,

$\begin{array}{c}AC=\sqrt{A{B}^2+B{C}^2}\\ AC=\sqrt{4^2+2^2}\\ AC=2\sqrt{5}\end{array}$

Also,

$\begin{array}{l}CD=\sqrt{C{E}^2+E{D}^2}\\ CD=\sqrt{1^2+2^2}\\ CD=\sqrt{5}\end{array}$

Step 4. Conclusion.

The angle of each side are $\angle A=60°$, $\angle C=30°$, and $\angle D=60°$, and the length of each side are $BC=2\text{and}CE=1$ , $AC=2\sqrt{5}$ and $CD=\sqrt{5}$.

b.Step 1. Given Information.

The figure is shown below.

Step 2. Concept used.

When two angles of a triangle are same then by angle-angle theorem they are called as similar triangles.

Step 3. Calculation.

Since in $\Delta ABC$ and $\Delta DEC$, $\angle A=60°$, $\angle D=60°$ and $\angle E=90°$, $\angle B=90°$. Therefore, by angle-angle theorem $\Delta ABC$ and $\Delta DEC$ are similar.

Step 4. Conclusion.

The $\Delta ABC$ and $\Delta DEC$ are related to each other.