2.6 Overlapping Triangles
When triangles overlap it is sometimes difficult to see a strategy for a proof. The reason is that sides or angles may be parts of several triangles. Consider the following example.
Example:
Given: $\overline{AB}\cong \overline{DC}$, $\overline{AB}\cong\overline{BC}$. Prove: $\triangle EBC$ is isosceles. |
Solution: The key is to note that $\angle 1$ and $\angle 2$ are parts of two different triangles. It is often helpful to pull the triangles apart as shown below.
Since $\overline{BC}$ is congruent to itself we can show $\triangle ABC\cong\triangle DCB$ by SSS, thereby making $\angle 1\cong \angle 2$ by CPCTC. But angles 1 and 2 are part of $\triangle EBC$ so $\overline{EB}\cong\overline{EC}$ by the Converse of the Isosceles Triangle Theorem, making $\triangle EBC$ isosceles.
In the following problems you may find it useful to use different colored pencils or make drawings with the triangles separated in order to see the important relationships.