These angles are not labeled as congruent and are not any of the congruent angle pairs we recognize such as vertical angles, right angles, or corresponding angles. The only way we can justify their congruence is if the triangles themselves are congruent. First, justify the congruence of the triangles. Then justify the congruence of the angles by using the corresponding parts of congruent triangles theorem.

From the labels in the diagram, we can see that \overline{AB}\cong \overline{EF}, \angle{A}\cong \angle{E}, and \overline{AC}\cong \overline{ED}. We can conclude that \triangle{ABC} \cong \triangle{EFD} by the side-angle-side congruence theorem. Since \angle{ACB} corresponds to \angle{EDF}, we can justify that \angle{ACB}\cong \angle{EDF} because corresponding parts of congruent triangles are congruent.

Make sure the diagram is labelled clearly with the information that can be deduced by the information already given. This will help us determine which triangle congruency theorem we are using. Sometimes it helps to work backwards with what we're trying to show, so we know we need to show \triangle{RMP} is an isosceles, therefore we need to show that \overline{MR}\cong \overline{MP}. Since we weren't given any information about the length of these sides, the only way to show they're equal is to justify why \triangle RMQ \cong \triangle PMQ.

Here is an updated diagram, using the information already given on the diagram previously:

From this labeling we can see that these triangles are congruent by side-angle-side. We're now ready to complete our explanation.

It is given in the diagram that \overline{RQ}\cong \overline{QP} and that \overline{RP}\perp \overline{MQ}. So \angle{MQR} and \angle{MQP} are right angles. Since \overline{MQ} is a shared side, we can see that \triangle RMQ \cong \triangle PMQ by Side-Angle-Side (SAS) congruence. So we know \overline{MR}\cong \overline{MP} since corresponding parts of congruent triangles are congruent (CPCTC). Therefore, \triangle{RMP} is an isosceles.

In order to find the length of the arch we need to justify the congruence of the triangles. Then we can find the length because it will be congruent to the side it corresponds to in the other triangle where all three lengths are known.

The sides of lengths 26\text{ ft} and 30\text{ ft} are congruent pairs because they have the same lengths, and the angles included by these sides are congruent because vertical angles are congruent. This means that the two triangles are congruent by side-angle-side congruence. The width of the arch corresponds to the side of the other triangle that is 24\text{ ft} long, and so the width of the arch is 24\text{ ft} as well since corresponding parts of congruent triangles are congruent.