Question

determine if triangle abc and triangle def are or are not similar, and, if they are, state how you know. (note that figures are not necessarily drawn to scale.)

Explanation:

Step1: Find the third angle of triangle ABC

The sum of angles in a triangle is \(180^{\circ}\). Let the third angle of \(\triangle ABC\) be \(x\). Then \(x + 82^{\circ}+53^{\circ}=180^{\circ}\). So \(x=180^{\circ}-(82^{\circ} + 53^{\circ})=45^{\circ}\).

Step2: Check the ratios of the sides

For \(\triangle ABC\) with sides \(a = 4\), \(b = 5\), \(c = 6\) and \(\triangle DEF\) with sides \(d = 21\), \(e = 25\).
Check the ratios \(\frac{4}{21}
eq\frac{5}{25}=\frac{1}{5}
eq\frac{6}{?}\) (We don't know the third - side of \(\triangle DEF\) yet, but from the two - side ratios we can see a problem).
Another way: If two triangles are similar, the ratios of their corresponding sides should be equal.
Let's assume \(\angle A=\angle D = 82^{\circ}\). If \(\triangle ABC\sim\triangle DEF\), then \(\frac{AB}{DE}=\frac{AC}{DF}=\frac{BC}{EF}\)
\(\frac{4}{21}
eq\frac{5}{25}\)

Answer:

The triangles \(ABC\) and \(DEF\) are not similar.