Final answer:
By employing the Vertical Angles Theorem, Isosceles Triangle Theorem, Corresponding Angles Postulate, and Converse of the Base Angles Theorem, we can prove that the line joining the angular points of two isosceles triangles with equal vertical angles is equal.
Step-by-step explanation:
The task is to prove that when two isosceles triangles with equal vertical angles are placed such that their vertices coincide, the line joining their other angular points is equal.
To start, by the Vertical Angles Theorem, we establish that the vertical angles formed at the coinciding vertices are equal. Then, applying the Isosceles Triangle Theorem, we know that the triangles have two sides that are equal in length.
Next, we use the Corresponding Angles Postulate which tells us that if the two triangles are placed such that their equal sides are parallel, then the corresponding angles are equal.
Finally, we can argue by contradiction using the Converse of the Base Angles Theorem, which states that if the base angles are equal, then the sides opposite them must be equal. If we assume the line joining the non-coinciding vertices is not equal, then the base angles would not be equal - a contradiction since we know they are equal. Therefore, the line joining the non-coinciding vertices must be equal.