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Chapter 3: Q24. (page 108)

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

Expert verified

The diagonals of an equilateral quadrilateral are congruent isTrue.

Step by step solution

01

Step 1. Define equilateral quadrilateral.

An equilateral quadrilateral must have 4 equal sides.

02

Step 2. Draw the diagram.

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

03

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

In above figure consider螖AEB and螖CED

In which,

  1. Angle鈭�AEB鈮�鈭�CEDbecause vertical opposite angles.
  2. Angle 鈭�CDE鈮�鈭�ABE=45掳
  3. Angle 鈭�DCE鈮�鈭�BEA=45掳

Thus,鈭�AEB鈮�鈭�CED by AAA congruency.

Then, also side DE鈮�BE,CE鈮�AEby cpct.

If DE鈮�BE,CE鈮�AE, then AC鈮�BDbecauseAC=AE+CE and BD=DE+BE.

Hence, AC鈮�BD. Therefore, the statement that the diagonals of an equilateral quadrilateral are congruent is true.

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