Question

For each exercise place the statements in an appropriate order for a proof. (There may be more than one correct order.)

Given:$\overline{\mathrm{RS}}\cong \overline{\mathrm{RT}}$

Prove:$\angle \mathbf{3}\cong \angle \mathbf{4}$

(a)$\angle \mathbf{3}\cong \angle \mathbf{4}$

(b)$\angle \mathbf{3}\cong \angle \mathbf{1}\mathbf{;}\angle \mathbf{2}\cong \angle \mathbf{4}$

(c)$\overline{\mathrm{RS}}\cong \overline{\mathrm{RT}}$

(d)$\angle \mathbf{1}\cong \angle \mathbf{2}$

Short answer

Order of the proof is$\left(c\right)\mathbf{,}\mathbf{}\left(d\right)\mathbf{,}\mathbf{}\left(b\right)\mathbf{}\mathbf{and}\mathbf{}\left(a\right)$.

Step-by-step solution

Step 1. Apply isosceles triangle theorem.

If the two sides of a triangle are congruent, then the angles opposite to those sides are congruent.

From the figure it can be observed that,$\overline{RS}\cong \overline{RT}$.

Therefore, $\angle 1\cong \angle 2$.

Step 2. Vertically opposite angles.

From the figure it can be observed that, $\angle 1\text{and}\angle 3$, $\angle 2\text{and}\angle 4$are vertically opposite angles. Therefore,

$\begin{array}{c}\angle 1\cong \angle 3\\ \angle 2\cong \angle 4\end{array}$

Step 3. Transitive property of congruence.

If$\angle P\cong \angle Q$ and$\angle Q\cong \angle S$ then $\angle P\cong \angle S$.

Therefore, $\angle 3\cong \angle 4$.

From the above explanation, the order of proof is $\left(c\right),\left(d\right),\left(b\right)\text{and}\left(a\right)$.