Rhombuses Or Rhombi

What makes a quadrilateral a rhombus?

Rhombuses Or Rhombi

A rhombus is an equilateral parallelogram.– All sides are

congruent

Rhombus Corollary

A quadrilateral is a rhombus if and only if it has four congruent sides.

Rectangles

What makes a quadrilateral a rectangle?

Rectangles

A rectangle is an equiangular parallelogram.

• All angles are congruent

Example 1

What must each angle of a rectangle measure?

Rectangle Corollary

A quadrilateral is a rectangle if and only if it has four right angles.

Squares

What makes a quadrilateral a square?

Squares

A square is a regular parallelogram.

• All angles are congruent

• All sides are congruent

Square Corollary

A quadrilateral is a square if and only if it is a rhombus and a rectangle.

Properties of Rhombuses, Rectangles, and Squares

Objectives:

1. To discover and use properties of rhombuses, rectangles, and squares

2. To find the area of rhombuses, rectangles, and squares

Example 2

Below is a concept map showing the relationships between some members of the parallelogram family. This type of concept map is known as a Venn Diagram. Fill in the missing names.

Example 2

Below is a concept map showing the relationships between some members of the parallelogram family. This type of concept map is known as a Venn Diagram.

Example 3

For any rhombus QRST, decide whether the statement is always or sometimes true. Draw a sketch and explain your reasoning.

1. Q S2. Q R

Example 4

For any rectangle ABCD, decide whether the statement is always or sometimes true. Draw a sketch and explain your reasoning.

1. AB CD2. AB BC

Example 5

Classify the special quadrilateral. Explain your reasoning.

Diagonal Theorem 1

A parallelogram is a rectangle if and only if its diagonals are congruent.

Example 6

The previous theorem is a biconditional. Write the two conditional statements that must be proved separately to prove the entire theorem.

Example 7

You’ve just had a new door installed, but it doesn’t seem to fit into the door jamb properly. What could you do to determine if your new door is rectangular?

Diagonal Theorem 2

A parallelogram is a rhombus if and only if its diagonals are perpendicular.

Diagonal Theorem 3

A parallelogram is a rhombus if and only if each diagonal bisects a pair of opposite angles.

Example 8

Prove that if a parallelogram has perpendicular diagonals, then it is a rhombus.

Given: ABCD is a

parallelogram;

AC BDProve: ABCD is a

rhombus

Use Properties of Trapezoids and Kites

Objectives:

1. To discover and use properties of trapezoids and kites

2. To find the area of trapezoids and kites

Trapezoids

What makes a quadrilateral a trapezoid?

Trapezoids

A trapezoid is a quadrilateral with exactly one pair of parallel opposite sides.

Trapezoid Parts

• The parallel sides are called bases

• The non-parallel sides are called legs

• A trapezoid has two pairs of base angles

Example 1

Find the value of x.

100

xA D

B C

Trapezoid Theorem 1

If a quadrilateral is a trapezoid, then the consecutive angles between the bases are supplementary.

r

ty

xA D

B C

If ABCD is a trapezoid, then x + y = 180° and r + t = 180°.

Midsegment

A midsegment of a trapezoid is a segment that connects the midpoints of the legs of a trapezoids.

Isosceles Trapezoid

An isosceles trapezoid is a trapezoid with congruent legs.

Trapezoid Theorem 2

If a trapezoid is isosceles, then each pair of base angles is congruent.

Trapezoid Theorem 3

A trapezoid is isosceles if and only if its diagonals are congruent.

Ti

Trapezoid Theorem 4

The midsegment of a trapezoid is parallel to each base and its length is one half the sum of the lengths of the bases.

If

Example 2

Find the measure of each missing angle.

Example 3

For a project, you must cut an 11” by 14” rectangular piece of poster board. Knowing how poorly you usually wield a pair of scissors, you decide to do some measuring to make sure your board is truly rectangular. Thus, you measure the diagonals and determine that they are in fact congruent. Is your board rectangular?

Example 4

Find the value of x.

Kites

What makes a quadrilateral a kite?

Kites

A kite is a quadrilateral that has two pairs of consecutive congruent sides, but opposite sides are not congruent.

Angles of a Kite

You can construct a kite by joining two different isosceles triangles with a common base and then by removing that common base.

Two isosceles triangles can form one kite.

Angles of a Kite

Just as in an isosceles triangle, the angles between each pair of congruent sides are vertex angles. The other pair of angles are nonvertex angles.

Kite Theorem 1

If a quadrilateral is a kite, then the nonvertex angles are congruent.

Kite Theorem 2

If a quadrilateral is a kite, then the diagonal connecting the vertex angles is the perpendicular bisector of the other diagonal.

E

A

B

C

D

and CE AE.

Kite Theorem 3

If a quadrilateral is a kite, then a diagonal bisects the opposite non-congruent vertex angles.

A

B

C

D

If ABCD is a kite, then BD bisects B and D.

Example 5

Quadrilateral DEFG is a kite. Find mD.

Example 6

Find the measures of each side of kite PQRS. Write your answers in simplest radical form.

Example 7

Example 8

Example 9