1. **Problem statement:** Delma wants to estimate the distance across a river by using her walking stick AB and sighting points T (tree across the river) and R (rock on her side). We need to explain why triangle ABT is congruent to triangle ABR. 2. **Given:** - AB is vertical and common to both triangles. - Angle of sight along the top of the stick to points T and R is the same (Delma did not change the angle of her head). 3. **Key concept:** Two triangles are congruent if they satisfy criteria such as SAS (Side-Angle-Side), ASA (Angle-Side-Angle), or RHS (Right angle-Hypotenuse-Side). 4. **Explanation:** - Segment AB is common to both triangles ABT and ABR. - The angle at A between AB and the line of sight to T is equal to the angle at A between AB and the line of sight to R because Delma did not change the angle of her head. - Both triangles have a right angle at B because AB is vertical and the ground is horizontal. 5. **Conclusion:** - Triangles ABT and ABR have two angles and the included side AB equal. - By the ASA (Angle-Side-Angle) congruence rule, triangle ABT is congruent to triangle ABR. Therefore, $$\triangle ABT \cong \triangle ABR$$ because they share side AB, have equal angles at A, and right angles at B.