# Thread: Need help with CONGRUENCY!! (trapezium) @___@

1. ## Need help with CONGRUENCY!! (trapezium) @___@

A trapezium is divided into 4 triangles by its diagonals. Prove the ares of the coloured triangles are equal. Thank you

2. Hello, kapitanvicki!

A trapezium is divided into 4 triangles by its diagonals.
Prove the areas of the coloured triangles are equal.
Code:
           B                   C
*  *  *  *  *  *  *
*|:::*         *:::|:*
*:|:::::::::*:::::::|:::::*
*::|:::::*   O     *:|:::::::*
*:::|:*               |   *:::::::*
*::* |                 |         *:::*
*::*  *  *  *  *  *  *  *  *  *  *  *  *  *
A      E                 F                  D

We have trapezoid $\displaystyle ABCD$ with diagonals $\displaystyle AC$ and $\displaystyle BD$,
. . intersecting at $\displaystyle O.$

$\displaystyle \Delta ABD$ and $\displaystyle \Delta ACD$ have a common base, $\displaystyle AD$, and equal altitudes, $\displaystyle BE = CF.$
. . Hence: .$\displaystyle \text{area }\Delta ABD \:=\: \text{area }\Delta ACD$

They share $\displaystyle \Delta AOD.$

$\displaystyle \begin{array}{cccc} \text{We have:} & \Delta ABD & = & \Delta ACD \\ \text{Subtract }\Delta AOD\!: & \text{-}\Delta AOD & = & \text{-}\Delta AOD \\ \text{Therefore:}& \Delta AOB & = & \Delta COD \end{array}$