What type of a quadrilateral is one in which the opposite sides are equal? Justify your answer.

Hint: Draw a diagonal and check for congruent triangles.

Question

What type of a quadrilateral is one in which the opposite sides are equal? Justify your answer. Hint: Draw a diagonal and check for congruent triangles.

Homi · Accepted Answer

Understand the Question

A quadrilateral with opposite sides of equal length can be analyzed by drawing a diagonal.
The diagonal splits the quadrilateral into two triangles that have three pairs of equal sides.
By the SSS congruence criterion, the two triangles are congruent.
Congruent triangles have equal corresponding angles (CPCTC), which act as equal alternate interior angles, proving both pairs of opposite sides are parallel.
A quadrilateral with opposite sides parallel is defined as a parallelogram.

Step 1 · Set up the Quadrilateral and Diagonal

Let $ABCD$ be a quadrilateral where opposite sides are equal: $AB = CD \quad \text{and} \quad BC = DA$ Diagram 1

Draw diagonal $AC$ dividing the quadrilateral into $\triangle ABC$ and $\triangle CDA$ .

Step 2 · Prove Triangle Congruence

In $\triangle ABC$ and $\triangle CDA$ :

\begin{aligned} AB &= CD \quad (\text{Given}) \\ BC &= DA \quad (\text{Given}) \\ AC &= CA \quad (\text{Common side}) \end{aligned}

By SSS (Side-Side-Side) congruence criterion: $\triangle ABC \cong \triangle CDA$

Step 3 · Show Opposite Sides are Parallel

Since $\triangle ABC \cong \triangle CDA$ , by CPCTC (Corresponding Parts of Congruent Triangles are Congruent): $\angle BAC = \angle DCA$ $\angle BCA = \angle DAC$

For lines $AB$ and $DC$ with transversal $AC$ , $\angle BAC$ and $\angle DCA$ are alternate interior angles. Since they are equal: $AB \parallel DC$

For lines $AD$ and $BC$ with transversal $AC$ , $\angle DAC$ and $\angle BCA$ are alternate interior angles. Since they are equal: $AD \parallel BC$

Step 4 · Conclude the Type of Quadrilateral

Since both pairs of opposite sides are parallel ( $AB \parallel DC$ and $AD \parallel BC$ ), quadrilateral $ABCD$ is a parallelogram.

Answer

The quadrilateral is a Parallelogram.

Common Mistakes

Assuming Special Parallelograms: Assuming the quadrilateral must be a rectangle or rhombus; without right angles or four equal sides given, it is generally a parallelogram.
Alternate Angle Identification: Misidentifying which lines are parallel from the angle pairs: $\angle BAC = \angle DCA \implies AB \parallel DC$ , while $\angle BCA = \angle DAC \implies AD \parallel BC$ .
Missing SSS Condition: Forgetting to mention the shared diagonal $AC = CA$ as the third side required for SSS congruence.

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