Chapter 3.6, Problem 24WE

To determine

Whether the given statement is true or false. Justify with a diagram.

Answer to Problem 24WE

The diagonal of an equilateral quadrilateral are congruent is true.

Explanation of Solution

Given information:

Given statement:

The diagonals of a quadrilateral are congruent.

Calculation:

Consider the following figure:

In above figure there is a quadrilateral $ABCD$

And equilateral means all sides must be in equal length.

This means $AB=BC=CD=AD$

To prove $AC\cong BD$

In above figure consider $\Delta AEB$ and $\Delta CED$

In which,

1. Angle $\angle AEB\cong \angle CED$because vertical opposite angles.
2. Angle $\angle CDE\cong \angle ABE=45°$
3. Angle $\angle DCE\cong \angle BEA=45°$

Thus $\Delta AEB\cong \Delta CED$ by AAA congruency

Then also side $DE\cong BE,CE\cong AE$ by cpct.

If $DE\cong BE,CE\cong AE$

Then $AC\cong BD$ because $AC=AE+CE$ and $BD=DE+BE$

Hence, $AC\cong BD$

Therefore, if the diagonal of an equilateral quadrilateral are congruent is true.