5. Proof 4
• Two triangles are said to be similar if their corresponding angles are of
equal measures and their corresponding sides are in the same ratio.
Also, if the angles are of the same measure, then we can say by using
the sine law, that the corresponding sides will also be in the same
ratio. Hence, corresponding angles in similar triangles will lead us to
equal ratios of side lengths.In triangle ABD and triangle ACB:
• ∠A = ∠A (common)
• ∠ADB = ∠ABC (both are right angles)
• Thus, triangle ABD and triangle ACB are equiangular, which means that
they are similar by AA similarity criterion. Similarly, we can prove
triangle BCD similar to triangle ACB. Since triangles ABD and ACB are
similar, we have AD/AB = AB/AC. Thus, we can say that AD × AC = AB2.
Similarly, triangles BCD and ACB are similar. That gives us CD/BC =
BC/AC. Thus, we can also say that CD × AC = BC2. Now, using both of
these similarity equations, we can say that AC2 = AB2 + BC2. Hence
Proved.