GeometryGeometry

5.4 Special Parallelograms5.4 Special Parallelograms

Set Up a Flow Chart to Fill in as We GoSet Up a Flow Chart to Fill in as We Go

RectangleRectangle

A quadrilateral with four right angles.A quadrilateral with four right angles.

Why is a rectangle a parallelogram?

Both Pairs of Opp. Angles are Congruent

RhombusRhombus

A quadrilateral with four congruent sides.A quadrilateral with four congruent sides.

Why is a rhombus a parallelogram?

Both Pairs of Opp. Sides are Congruent

Square Square (Rhom-tangle Ha! Ha!)(Rhom-tangle Ha! Ha!)

A quadrilateral with four congruent sides A quadrilateral with four congruent sides and four right angles.and four right angles.

Why is a rhombus a parallelogram?

Both Pairs of Opp. Sides are Congruent

Both Pairs of Opp. Angles are Congruent

Review: Rectangles, Rhombuses, and Review: Rectangles, Rhombuses, and Squares all Share these Properties of a Squares all Share these Properties of a

Parallelogram…Parallelogram… 1)1) Opp. Sides are //Opp. Sides are //2)2) Opp. Angles are congruentOpp. Angles are congruent3)3) Opp. Sides are congruentOpp. Sides are congruent4)4) Diagonals Bisect Each OtherDiagonals Bisect Each Other

In addition, rectangles, rhombuses, and In addition, rectangles, rhombuses, and squares all have their own special squares all have their own special properties. These are the focus of this properties. These are the focus of this lesson.lesson.

Theorem: The diagonals of a Theorem: The diagonals of a Rectangle are CongruentRectangle are Congruent

Draw two congruent intersecting lines that bisect each Draw two congruent intersecting lines that bisect each other.other.

Connect the corners. You drew a rectangle.Connect the corners. You drew a rectangle.

Theorem: The diagonals of a Theorem: The diagonals of a rhombus are perpendicular.rhombus are perpendicular.

Draw two lines that bisect each other & are perpendicular.Draw two lines that bisect each other & are perpendicular.

Connect the corners. You have drawn a rhombus.Connect the corners. You have drawn a rhombus.

Theorem: Each diagonal of a rhombus Theorem: Each diagonal of a rhombus bisects two angles of the rhombus.bisects two angles of the rhombus.

Draw a rhombus and its diagonals.Draw a rhombus and its diagonals.

You bisected all four angles.You bisected all four angles.

Theorem: The midpoint of the hypotenuse of a Theorem: The midpoint of the hypotenuse of a right triangle is equidistant from all three vertices.right triangle is equidistant from all three vertices.

Draw a right triangle and put a point at the midpoint of the Draw a right triangle and put a point at the midpoint of the hypotenuse. hypotenuse.

Draw a line from that point to the vertex of the right angle.Draw a line from that point to the vertex of the right angle.

All three distances are equal.All three distances are equal.

.

Theorem: If an angle of a parallelogram is a right Theorem: If an angle of a parallelogram is a right angle, then the parallelogram is a rectangle.angle, then the parallelogram is a rectangle.

Draw one right angle.Draw one right angle.

Draw the two other sides parallel to the opposite side.Draw the two other sides parallel to the opposite side.

You have drawn a rectangle.You have drawn a rectangle.

Why is it a rectangle?

Opp. Angles of a ParallelogramAre congruent

Parallel lines imply SS Int. anglesare supplementary.

Theorem: If two consecutive sides of a Theorem: If two consecutive sides of a parallelogram are congruent, then the parallelogram are congruent, then the

parallelogram is a rhombus.parallelogram is a rhombus.Draw two congruent sides of an angle.Draw two congruent sides of an angle.

Draw the two other sides parallelDraw the two other sides parallel

to the opposite sides.to the opposite sides.

You have drawn a rhombus.You have drawn a rhombus.

Why is it a rhombus?

Opp. Sides of a Parallelogram are congruent.

Given Quad. WXYZ is a rectangle. Complete the statements with numbers. Make sure your + and – are clear!

3. If TX = 4.5, then WY = _____.

4. If WY = 3a + 16 and ZX = 5a – 18, then a = _____, WY = _____ and ZX = _____.

5. If m<TWZ = 70, then m<TZW = _____ and

m<WTZ = _____.

Z Y

XW

T

7. If m<4 = 25, then m<5 = _____.

8. If m<DAB = 130, then m<ADC = _____.

9. If m<4 = 3x – 2 and m<5 = 2x + 7,

then x = ____, m<4 = ____, and m<5 =____.

11. If m<2 = 3y + 9 and m<4 = 2y – 4,

then y = _____, m<2 = _____, and m<4 = ____.

Given Quad. ABCD is a rhombus. Complete the statements with numbers.

D C

B A

5

4

3

2 1

Given Quad. JKLM is a square. Complete the statements with numbers.

14. If JL =18, then MK = _____, JX = _____, and XK = _____.

15. m<MJK = _____, m<MXJ = _____ and m<KLJ = _____.

M

L

K

J

x

M L

J K

X

HWHW

P. 186 (1-11)P. 186 (1-11) P. 187 (1-10) (11-27 Odd)P. 187 (1-10) (11-27 Odd)If you forget the theorems, it helps to draw a picture…i.e. draw a rhombus and If you forget the theorems, it helps to draw a picture…i.e. draw a rhombus and

then its diagonals and see if they are congruent or pependicular.then its diagonals and see if they are congruent or pependicular.

A HW Jumpstart P. 187 # 5-8A HW Jumpstart P. 187 # 5-8

PropertyProperty ParallelogramParallelogram RectangleRectangle RhombusRhombus SquareSquare

5) Diags. Bisect each other

X X X X

6) Diags. Areconguent X X

X

XX

X7) Diags. ArePerpendicular

8) A diagonal Bisects 2 angles