LESSON TWENTY-THREE:

RHOMBI APOCALYPSE!!!

RHOMBI AND SQUARES

• So right now we have discussed the definition and properties of rectangles and parallelograms.

• Two new figures we will be discussing today are rhombi and squares.

RHOMBI AND SQUARES

• A rhombus is a parallelogram with four congruent sides.

• Since a rhombus is a parallelogram, it has all the properties that a parallelogram does.

RHOMBI AND SQUARES• Those properties are…– Opposite sides are parallel and congruent.– Opposite angles are congruent.– Consecutive angles are supplementary.– The diagonals bisect each other.

RHOMBI AND SQUARES

• Rhombi have properties also that are all their own.

• The first says that if a parallelogram is a rhombus, his its diagonals are perpendicular.

RHOMBI AND SQUARES

• The second property states that if a parallelogram is a rhombus, then each diagonal bisects a pair of opposite angles.

RHOMBI AND SQUARES

• The other figure, we will discuss today is called a square.

• A square is the most specific of the figures because we can think of it as a combination of a rhombus and rectangle.

RHOMBI AND SQUARES

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

• Because a square is both a rhombus and rectangle, the special properties for rhombi and rectangles, both apply to squares.

RHOMBI AND SQUARES

• Therefore if a parallelogram is a square, then…– Its diagonals are congruent.– Its diagonals are perpendicular.– Its diagonals bisect a pair of opposite angles.

RHOMBI AND SQUARES

• Using what we know about the distance formula, we can now identify what figure is represented on a plane.

• What type of figure would have 4 congruent sides and 4 congruent diagonals?

RHOMBI AND SQUARES• For the figure below our points are (-3, 0), (-1, 3),

(4, 3), (2, 0).• Is this a rhombus, rectangle, square or none of

these?

A

D C

B

RHOMBI AND SQUARES• In solving these distances we find that AB • This equals = = • By this process, we find that DC is also• We find that AD and BC equal

RHOMBI AND SQUARES• Finally, let’s find the diagonals.• Using the distance formula, we find that AC =

and BD = • So we have opposite sides that are congruent,

but non-congruent diagonals.• Is this a rhombus, rectangle, square or none of

these?

RHOMBI AND SQUARES

• What type of figure would have all four sides congruent, but non-congruent diagonals?

• This is the type of problem you’ll be practicing today and the logic you’ll need for it.