Name: ______________________________
Honors Geometry
Congruence in Overlapping Triangles
Example 1:
Given: 𝐵𝐶 ≅ 𝐴𝐷
𝐴𝐶 ≅ 𝐷𝐵
Prove: ∠𝐴𝐶𝐵 ≅ ∠𝐵𝐷𝐴
The first instinct is to try to prove ∆𝐶𝐸𝐴 ≅ ∆𝐵𝐸𝐷, however, the given information yields one
pair of equal sides (𝐶𝐴 ≅ 𝐷𝐵) and a pair of vertical angles (∠𝐶𝐸𝐴 ≅ ∠𝐷𝐸𝐵). The last piece of given
information (𝐴𝐶 ≅ 𝐷𝐵) won’t help to prove those two triangles congruent. It becomes necessary to
look at the image again.
Look for a side or an angle which is an angle of both of the triangles (reflexive) will help prove
the congruence. If the triangles are “pulled apart” into ∆𝐶𝐴𝐵 and ∆𝐷𝐵𝐴, and the given information is
remarked on the sketch, it becomes obvious that these triangles can be proven congruent. Examine the
sketch below with the triangles separated. Complete the following proof below.
Example 2:
Given: 𝐴𝐵 ≅ 𝐴𝐶
𝐴𝐷 ≅ 𝐴𝐸
Prove: 𝐵𝐸 ≅ 𝐶𝐷
In this proof, it is necessary to “pull apart” the triangles. If the first attempt doesn’t yield
enough information for a congruence, you may have to try to use two other triangles.
First Attempt: Try to prove ∆𝐷𝐵𝐶 ≅ ∆𝐸𝐶𝐵. You have 𝐵𝐶 ≅ 𝐵𝐶, by reflexive, and you could use
SAP, substitution, and subtraction to get 𝐵𝐷 ≅ 𝐸𝐶. However, there is no other information to help
prove these triangles congruent.
Second Attempt: Try to prove ∆𝐵𝐴𝐸 ≅ ∆𝐶𝐴𝐷. You have ∠𝐴 ≅ ∠𝐴 (reflexive) , 𝐴𝐷 ≅ 𝐴𝐸 and
𝐴𝐵 ≅ 𝐴𝐶 (from the given). Using the sketches below, remark the pictures and complete the proof.
Example 3: Try the following proof:
Given: 𝐴𝐷 ≅ 𝐵𝐶
E is midpoint of AD, F is midpoint of BC
∠𝐸𝐷𝐶 ≅ ∠𝐹𝐶𝐷
Prove: 𝐶𝐸 ≅ 𝐷𝐹