Question

In the following figure, given the triangles at the right, which of the following can you conclude are true?

$\begin{array}{l}\mathrm{a}.\angle D\cong \angle R\mathrm{b}.\overline{DE}\cong \overline{DF}\\ \mathrm{c}.\overline{DF}\cong \overline{RT}\mathrm{d}.\angle E\cong \angle F\\ \mathrm{e}.\angle E\cong \angle S\mathrm{f}.\angle S\cong \angle T\end{array}$

Short answer

The true statements are

$\begin{array}{l}\mathrm{b}.\overline{DE}\cong \overline{DF}\\ \mathrm{c}.\overline{DF}\cong \overline{RT}\\ \mathrm{d}.\angle E\cong \angle F\\ \mathrm{f}.\angle S\cong \angle T\end{array}$

Step-by-step solution

Step 1. Check the figure.

Consider the isosceles triangles.

Step 2. Apply the concept.

In an isosceles triangle, any two sides are always congruent. And angles opposite to these equal opposite sides are also congruent.

Step 3. Step description.

From the figures, it is clear that $\overline{DE}=\overline{RT}=8$.

Thus, $\overline{DE}\cong \overline{RT}$.

In $\mathrm{\Delta }DEF$, as $\overline{DE}\cong \overline{DF}$so that $\angle E\cong \angle F$.

Similarly, in $\mathrm{\Delta }RST$, as $\overline{RS}\cong \overline{RT}$so that $\angle S\cong \angle T$.

Thus, the true statements are $\overline{DE}\cong \overline{DF},\overline{RS}\cong \overline{RT},\angle E\cong \angle F$and $\angle S\cong \angle T$.