**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 quadrilaterals are 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 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!

- Quadrilateral Charts and Diagrams

- 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?
- Answer: rectangle and parallelogram
- 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.