Question

Draw several diagrams to help you decide whether each statement is true or false. If it is false, show a counterexample. It is true, draw and label a diagram you could use in a proof. List, in terms of the diagram, what is given and what is to be proved. Do not write a proof.

The diagonals of an equilateral quadrilateral are congruent.

Short answer

The diagonals of an equilateral quadrilateral are congruent isTrue.

Step-by-step solution

Step 1. Define equilateral quadrilateral.

An equilateral quadrilateral must have 4 equal sides.

Step 2. Draw the diagram.

Draw a quadrilateral $ABCD$such that all its sides are equal that is., $AB=BC=CD=AD$.

Step 3. Prove if diagonals of an equilateral quadrilateral are congruent or not.

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,$\angle AEB\cong \angle CED$ 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, the statement that the diagonals of an equilateral quadrilateral are congruent is true.