Question

1. triangle abc and def are similar. answer the following questions:

a. what relationship can you write for the angles of these two triangles?
b. what is the length of the missing side?

Explanation:

Part a:

Step1: Property of similar triangles

For similar triangles, corresponding angles are equal.
So, \(\angle A=\angle D\), \(\angle B = \angle E\), \(\angle C=\angle F\)

Part b:

Step1: Find the scale factor

The ratio of corresponding sides of similar triangles is called the scale factor.
Let's take the sides \(BC = 8m\) and \(EF=24m\). The scale factor \(k=\frac{BC}{EF}=\frac{8}{24}=\frac{1}{3}\)

Step2: Set up proportion for missing side

Let the missing side in \(\triangle ABC\) (corresponding to \(DF = 21m\) in \(\triangle DEF\)) be \(x\)
Using the proportion \(\frac{x}{DF}=\frac{BC}{EF}\)
Substitute \(DF = 21m\) and \(\frac{BC}{EF}=\frac{1}{3}\)
\(\frac{x}{21}=\frac{8}{24}\)

Step3: Solve for \(x\)

Cross - multiply: \(24x=8\times21\)
\(24x = 168\)
\(x=\frac{168}{24}=7\)

Answer:

a. \(\angle A=\angle D\), \(\angle B = \angle E\), \(\angle C=\angle F\)
b. \(7m\)