Question

In the figure shown, AD - BC, DALAC, and CBLBD.Prove ACADDBCExplain your reasoning,АBCBKA

Answer 1

For this problem we will first prove that AC is congruent to BD then by the SSS criteria the triangles will are congruent.

To prove that AC is congruent to BD we use the pythagorean theorem for the right triangles CAD and DBC ( the theorem states that the sum of the square of the legs of the triangles is equal to the square of the hypotenuse):

`\begin{gathered} BC^2+BD^2=DC^2 \\ \text{AD}^2+AC^2=DC^2 \\ \Rightarrow \\ BC^2+BD^2=AD^2+AC^2 \end{gathered}`

Now, from the hypothesis we know that AD=BC ( this side is shared by both triangles) then

`\begin{gathered} AD^2=BC^2 \\ \text{Substituting in the previous equation we get} \\ AD^2+BD^2=AD^2+AC^2 \\ \text{Cancelling AD}^2\text{ on each side we get } \\ BD^2=AC^2 \\ BD=AC \end{gathered}`

Then from the SSS(side side side) criteria we have that triangle CAD is congruent to triangle DBC.