Understanding Theorems About Parallelogram and Analyzing Diagonals of a Rectangle and a Rhombus

This proof will be written in two parts.

If a Parallelogram Is a Rhombus, Then Its Diagonals Are Perpendicular

A rhombus is a parallelogram with four congruent sides. By the Parallelogram Diagonals Theorem, it can be said that its diagonals bisect each other. Let Let $A B C D$ be a rhombus with $P$ at the midpoint of both diagonals.

Note that

\overline{A B}

is congruent to

\overline{A D}

and

\overline{D P}

is congruent to

\overline{B P} .

Additionally, by the Reflexive Property of Congruence,

\overline{A P}

is congruent to itself. Therefore, by the Side-Side-Side Congruence Theorem,

△ A P B

is congruent to

△ A P D .

⎩ ⎪ ⎪ ⎨ ⎪ ⎪ ⎧ \overline{A B} ≅ \overline{A D} \overline{D P} ≅ \overline{B P} \overline{A P} ≅ \overline{A P} \Rightarrow △ A P B ≅ △ A P D

Because corresponding parts of congruent triangles are congruent,

∠ A P B

and

∠ A P D

are congruent angles. Furthermore, these angles form a linear pair, which means they are supplementary. With this information, it can be concluded that both

∠ A P B

and

∠ A P D

are right angles.

{∠ A P B ≅ ∠ A P D m ∠ A P B + m ∠ A P D = 18 0^{\circ} ⇓ m ∠ A P B = 9 0^{\circ} and m ∠ A P D = 9 0^{\circ}

This implies that

\overline{A C}

is perpendicular to

\overline{B D} .

Therefore, the diagonals of a rhombus are perpendicular.

Parallelogram $A B C D$ is a rhombus $\Rightarrow$ $\overline{A C} ⊥ \overline{B D}$

If Its Diagonals Are Perpendicular, Then a Parallelogram is a Rhombus

Conversely, let $A B C D$ be a parallelogram whose diagonals are perpendicular.

By the Parallelogram Diagonals Theorem, the diagonals of the parallelogram bisect each other. If $P$ is the midpoint of both diagonals, then $\overline{A P}$ and $\overline{C P}$ are congruent.

Since

\overline{A C}

and

\overline{B D}

are perpendicular,

∠ A P B

and

∠ C P B

measure

9 0^{\circ}

and thus are congruent angles. By the Reflexive Property of Congruence,

\overline{B P}

is congruent to itself. This means that two sides and their included angle are congruent. By the Side-Angle-Side Congruence Theorem,

△ A P B

and

△ C P B

are congruent triangles.

⎩ ⎪ ⎪ ⎨ ⎪ ⎪ ⎧ \overline{A P} ≅ \overline{C P} ∠ A P B ≅ ∠ C P B \overline{B P} ≅ \overline{B P} ⇓ △ A P B ≅ △ C P B

Because corresponding parts of congruent figures are congruent, it can be said that

\overline{A B}

is congruent to

\overline{C B} .

Furthermore, by the Parallelogram Opposite Sides Theorem, $\overline{A B}$ is congruent to $\overline{D C}$ and $\overline{A D}$ is congruent to $\overline{B C} .$ By the Transitive Property of Congruence, it follows that all sides of the parallelogram are congruent.

This means that if the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus.

$\overline{A C} ⊥ \overline{B D}$ $\Rightarrow$ parallelogram $A B C D$ is a rhombus

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Discussion

Properties of a Parallelogram's Opposite Sides

Explore

Investigating the Diagonals of a Parallelogram

Discussion

Properties of a Parallelogram's Diagonals

Discussion

Diagonals of a Rectangle

Discussion

Diagonals of a Rhombus

Rule

Rhombus Diagonals Theorem

Proof

If a Parallelogram Is a Rhombus, Then Its Diagonals Are Perpendicular

If Its Diagonals Are Perpendicular, Then a Parallelogram is a Rhombus

Example

Solving Problems Using the Diagonals of a Rhombus

Hint

Solution

Closure

Additional Applications of a Parallelogram's Properties

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Parallelogram Consecutive Angles Theorem
The consecutive angles of a parallelogram are supplementary.