This exercise consists of two parts. First, we describe the method and second, we explain why it works.
The method
Let's examine the situation as seen from above.
To measure the distance across the canyon, we first place a stake at some point C, below D, so that DC⊥ DE.
Next, we have to find the midpoint of DC which we mark as B.
Finally, we have to locate a third point, A, so that AB⊥ DC and A, B and E are collinear.
Why it works
If we examine the diagram, we see that our triangles are both right triangles with one congruent corresponding leg. We also notice that the angles at B are vertical angles. According to the Vertical Angles Congruence Theorem, vertical angles are congruent. Let's mark this in our diagram.
Since two angles and the included side of â–ł BED are congruent with two angles and the included side of â–ł BAC, we can prove using the ASA Congruence Theorem that they are congruent. Additionally DE and AC are congruent corresponding sides. Therefore, we can measure the distance across the canyon be measuring AC.
