Special Parallelograms: Rectangles, Rhombuses, and Squares

Comprehensive practice flashcards covering properties, theorems, and algebraic problems related to rectangles, rhombuses, and squares based on the lecture notes.

What is Theorem 1 regarding rectangles?

If a parallelogram has a right angle, then it has four right angles and the parallelogram is a rectangle.

What is Theorem 2 regarding rectangles?

The diagonals of a rectangle are congruent.

How is a rhombus defined?

A rhombus is a parallelogram with four congruent sides.

How is a rectangle defined?

A rectangle is a parallelogram with four right angles.

How is a square defined?

A square is a parallelogram with four congruent sides and four right angles.

In rectangle HOPE, if $HE = 7\,cm$, what is the length of $OP$ and why?

$OP$ measures $7\,cm$ because opposite sides of a rectangle are congruent.

In rectangle HOPE, if $HP = 12\,cm$, what is the length of $OE$ and why?

$OE$ measures $12\,cm$ because diagonals of a rectangle are congruent.

What is Theorem 3 regarding a rhombus?

The diagonals of a rhombus are perpendicular.

What is Theorem 4 regarding a rhombus?

Each diagonal of a rhombus bisects a pair of opposite angles.

What are the main properties of a rhombus regarding its sides, angles, and diagonals?

All sides are congruent; opposite sides are parallel and congruent; opposite angles are congruent; diagonals are perpendicular and bisect each other; consecutive angles are supplementary; and each diagonal divides the rhombus into two congruent triangles.

Why is a square described as the most special parallelogram?

Because all the properties of a parallelogram and all the theorems on rectangles and rhombuses are true to all squares.

In square FAITH, if diagonal $FI = 14\text{ inches}$, what is the measure of diagonal $AT$?

The measure of $AT$ is $14\text{ inches}$ because the diagonals of a square are equal.

In square FAITH with diagonals intersecting at point $L$, if $HF = 7.5\text{ inches}$, what is the measure of $HL$?

The measure of $HL$ is $7.5\text{ inches}$ because the diagonals of a square bisect each other.

In square FAITH with diagonals intersecting at point $L$, if $AT = 18\text{ inches}$, what is the measure of $FH$?

The measure of $FH$ is $9\text{ inches}$ because the diagonals of a square bisect each other.

In square FAITH with diagonals intersecting at point $L$, what is the measure of angle $\angle 5$ formed by the intersecting diagonals?

The measure of $\angle 5$ is $90^\circ$ because the diagonals of a square are perpendicular.

In rectangle QRST, the diagonals are $QS = 5x - 31$ and $RT = 2x + 11$. What is the value of $x$ and the length of each diagonal?

Solving $5x - 31 = 2x + 11$ gives $3x = 42$, so $x = 14$. Substituting $x = 14$ gives each diagonal a length of $39\text{ units}$.

In rectangle BOWL, the diagonals are given as $BW = 4x + 7$ and $LO = 13 + x$. What is the value of $x$ and the measure of $BW$ and $LO$?

Setting $4x + 7 = 13 + x$ yields $3x = 6$, so $x = 2$. Thus, $BW = LO = 15\text{ units}$.

In rhombus EFGH, side $EH = 2y + 3$ and side $HG = 5y - 6$. What is the value of $y$ and the length of each side?

Setting $2y + 3 = 5y - 6$ gives $3y = 9$, so $y = 3$. Substituting $y = 3$ into either expression gives each side a length of $9\text{ units}$.

In square MATH, two sides are given as $HT = 8x - 13$ and $AT = 7x + 11$. How do you set up the equation to find $x$?

Set $8x - 13 = 7x + 11$ because a square is a parallelogram with four congruent sides.