# The diagonals of a trapezoid divide each other proportionall

The diagonals of a trapezoid divide each other proportionally.

Given: Trapezoid ABCD with diagonal AC, BD intersecting at O
Prove: $$\displaystyle{\frac{{{A}{O}}}{{{C}{O}}}}={\frac{{{B}{O}}}{{{D}{O}}}}$$
$\begin{array}{|c|c|}\hline \text{Statements} & \text{Reasons} \\ \hline 1. (see\ above) & \text{1. Given} \\ \hline 2. & 2.\\ \hline 3. \text{(Give 2 pairs of equal angles)} & \text{3. If two parallel lines are cut by a transveral,}\\ & \text{then the alternate interior angles are=.} \\ \hline 4.\ \triangle AOB\sim\triangle COD & 4. \\ \hline 5. \frac{AO}{CO}=\frac{BO}{DO} & 5.\\ \hline \end{array}$

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Ralph Lester

Step 1
Now,
$$\begin{array}{|c|c|}\hline \text{Statements} & \text{Reasons} \\ \hline 1. \text{Diagonal AC, BD intersecting O} & \text{1. Given} \\ \hline 2.\ AB\parallel CD & 2. \text{ABCD is Trapezoid} \\ \hline 3. \angle BDC=\angle DBA\ and\ \angle DCA=\angle CAB & \text{3. If two parallel lines are cut by a transveral,}\\ & \text{then the alternate interior angles are=.} \\ \hline 4.\ \triangle AOB\sim\triangle COD & 4. \text{If two angles of one triangle are} \\ & \text{equal respectively to two angles of} \\ & \text{another, then the triangles are similar. (A.A)} \\ \hline 5. \frac{AO}{CO}=\frac{BO}{DO} & 5. \text{C.S.S.T.P-corresponding sides of} \\ & \text{similar triangles are proportional}\\ \hline \end{array}$$