# 5.5 – Rhombi and Squares

Key Terms

• Adjacent – Next to each other.
• Two sides or two angles of a figure are adjacent if they are next to each other (side-by-side).
• Rhombus – A quadrilateral with four congruent sides.
• The plural of rhombus is rhombi.
• All rhombi are parallelograms because their opposite sides are parallel and congruent.
• Square – A quadrilateral with four right angles and four congruent sides.
• All squares are parallelograms.
• All squares are rhombi.
• All squares are rectangles.

Review

• Polygons
• All parallelograms are quadrilaterals, and therefore polygons.
• All rectangles are parallelograms, and therefore quadrilaterals and polygons.
• A square is a regular quadrilateral because all its sides and angles are congruent to one another.

• Alternate Interior Angles
• When two parallel lines are cut by a transveral, their alternate interior angles are congruent.

Notes

• Rhombi
• Properties & Rules

• A rhombus is a parallelogram with four congruent sides.
• So, if a parallelogram has two adjacent sides that are congruent, then the parallelogram is a rhombus because the opposite sides of a parallelogram are congruent.

• Diagonals
• In a rhombus, the angles where the diagonals intersect always measure 90° (perpendicular).

• The diagonals of a rhombus divide the interior angle at each vertex into two congruent angles.

• Proof: To prove a parallelogram Is a rhombus:
• Show that all the sides are congruent.
• Show that one pair of adjacent sides is congruent.
• Show that its diagonals are perpendicular to each other.
• Show that its diagonals bisect each interior angle.

• Prove: The diagonals of a rhombus intersect at perpendicular angles (right angles).

•  Squares
• A square is a rhombus with four right angles.
• Squares have all of the properties of quadrilaterals, parallelograms, rhombi, and rectangles.
• If a parallelogram has four right angles and four congruent sides, then it is a square.
• The diagonals of a square are always $\sqrt{2}$ times the length of its side.
• This value (the square root of 2) is known as Pythagoras’s constant. It was the first number proven to be irrational!

• Summary of Rules
• All squares are rhombi, but not all rhombi are squares.
• All squares are rectangles, but not all rectangles are squares.
• All rhombi and squares are parallelograms.
• All rhombi and squares are quadrilaterals.
• Squares and rectangles are quadrilaterals that have congruent consecutive angles (each angle is 90°).
• Rhombi and squares are the only two quadrilaterals that have diagonals that intersect at 90° angles (their diagonals are perpendicular).
• The diagonals of a rhombus are perpendicular and bisect each other.

• Examples
• If ABCD is a rhombus, then it might be a square.
• If ABCD is a rhombus, then it MUST be a parallelogram.
• If PQRS is a parallelogram, then it might be a rhombus.
• If PQRS is a parallelogram with two adjacent congruent sides, then is MUST be a rhombus.
• If ZXYW is a square, then it MUST be a rhombus.
• If MNOP is a rhombus with NO right angles, then it cannot be a square.

More Examples

• Ex 1. In a given quadrilateral, each side is parallel to its opposite side and the diagonals are not perpendicular. What could it be?
• Reason: rectangles and parallelograms are not always squares.
• Sometimes they are long and wide, which means the diagonals will not cross at 90° angles.

• Ex 2. Given that ABCD is a rhombus, what is the value of x?
• Since we know it is a rhombus, we know the diagonals are perpendicular (90°).
• We also know that the three angles of a triangle add up to 180°.
• Setup: (x) + (3x + 7) + (90) = 180
• Combine like terms: 4x + 97 = 180
• Subtract 97 on both sides: 4x = 83
• Divide by 4 on both sides to solve for x
• x = 20.75°

• Ex 3. When is a rhombus a square?
• Answer: When its angles are right angles.