Geometry 21 January 2014

1) WARM UP– on pink sheet- back side

2) Do your Khan Quiz

ObjectiveStudents will explore and apply properties of midsegments by doing an investigation and working problems.

HOMEWORK due today: Khan video notesUpcoming HOMEWORK DUE FRIDAY- Quadrilaterals Packet Give brief reason/ evidence of thinking for each problemDUE next Monday- Khan assignment

Summarizing Properties of Quadrilaterals

Quadrilateral

Kite Parallelogram Trapezoid

Rhombus Rectangle

Square

Isosceles Trapezoid

Unit Question

How do the properties of quadrilaterals

determine the classification of quadrilaterals?

A midsegment of a triangle is a segment connecting the midpoints of two sides of a triangle

T

RX

Z

Y

Today we will explore midsegments.ADD sketches to your properties of polygons

handoutmidsegment- the segment that connects the

midpoints of two sides of a triangle

midsegment

Midsegment of a trapezoid

• The midsegment of a trapezoid is the segment that connects the midpoints of its legs.

midsegment

B C

DA

Triangle Midsegments

Continue with investigation 1…. - compare angles what can you conclude about the mid-segment and the third side?

- compare length of mid-segment with the length of the 3rd side what can you conclude?

Investigations, page 275Investigation 11) draw a large SCALENE triangle on patty paper2) PINCH midpoint of each side. Draw 3 midsegments.3) with a 2nd piece of patty paper- trace ONE of the four triangles. WHAT DO YOU NOTICE?

Using the corresponding angles conjecture, what do you notice about angles? what does this tell you about the midsegment and the 3rd side?

Triangle Midsegment

The segment connecting the midpoints of two sides of a triangle is parallel to the 3rd side and is half as long.

T R

X

Z Y

YZRTandRTYZ21

||

Trapezoid midsegments

Do investigation 2, page 276

are corresponding angle congruent? What does that tell you about the midsegment and how it relates to the bases?

How long is the midsegment compared to the bases?

Midsegment of a trapezoid

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

• MN║AD, MN║BC• MN = ½ (AD + BC)

NM

A D

CB

Ex. 3: Finding Midsegment lengths of trapezoids

• LAYER CAKE A baker is making a cake like the one at the right. The top layer has a diameter of 8 inches and the bottom layer has a diameter of 20 inches. How big should the middle layer be?

Ex. 3: Finding Midsegment lengths of trapezoids

• Use the midsegment theorem for trapezoids.• DG = ½(EF + CH)=

½ (8 + 20) = 14”C

D

E

D

G

F

Examples

The midsegment of the trapezoid is ST. Find the value of x.

8

S T11

x

11 = ½ (8 + x)22 = 8 + x14 = x

Classwork- complete Geometry Worksheet- Kites and Trapezoids• Be ready to share your answers with the class.• Due end of class

• SUBMIT for a CLASSWORK GRADE

Debrief

What are the special properties of a triangle midsegement? a trapezoid midsegment?

Geometry 22/23 January 2014Warm UpPlease have your Properties of Polygons Handout on your desk for reference

Continue to work on 1) BASIC- do problems 2) ADVANCED: Kites and Trapezoids (to be submitted for classwork)

Be ready to present your work to the class in 15 minutes.

ObjectiveStudents will explore and apply properties of polygons by doing an investigation, taking notes and working problems.

Upcoming HOMEWORK DUE FRIDAY- Quadrilaterals Packet Give brief reason/ evidence of thinking for each problemDUE next Monday- Khan assignment

Summarizing Properties of Quadrilaterals

Quadrilateral

Kite Parallelogram Trapezoid

Rhombus Rectangle

Square

Isosceles Trapezoid

Unit Question

How do the properties of quadrilaterals

determine the classification of quadrilaterals?

Definition of kite

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

Properties of Kites

Kite theorems

• If a quadrilateral is a kite, then its diagonals are perpendicular.

• bisects

so AF = FC• bisects B and D⦟ ⦟

B

C

A

DF

BD AC

AC BD

BD

Kite theorems

• If a quadrilateral is a kite, then non-vertex angles are congruent.

• A ≅ C, B ≅ DB

C

A

D

Definition of Trapezoid

Properties of Trapezoids

Consecutive angles between the bases are supplementary.

Base angles of isosceles trapezoids are congruent. ⦟P S and Q ⦟ ⦟ R⦟

Isosceles Trapezoids

Trapezoid Theorems

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

• ABCD is isosceles if and only if AC = BD.

A B

D C

Triangle Midsegment

The segment connecting the midpoints of two sides of a triangle is parallel to the 3rd side and is half as long.

T R

X

Z Y

YZRTandRTYZ21

||

Trapezoid Midsegment is parallel to the bases

Mid-segments of a trapezoid

Watch video: http://www.youtube.com/watch?v=O2bf1mZQWXI

http://math.kendallhunt.com/x19429.html

Definitions…..review

Rectangle- an equiangular parallelogram Rhombus- an equilateral parallelogram

Square- an equilateral, equiangular parallelogram

Parallelogram Propertiespg. 281- 282; pg. 291- 294

See the book for ideas. Basically, construct each figure and explore the properties of each.

Are any angles congruent? supplementary?What about sides? parallel? congruent?

Check diagonals? Bisected? 90⁰? any sections of the diagonal congruent?TOOLS? protractor, ruler, patty paper, compass

BE READY TO SHARE YOUR RESULTS in 15 minutes.

Parallelogram Propertieshttp://math.kendallhunt.com/x19430.html

http://www.mathwarehouse.com/geometry/quadrilaterals/parallelograms/#anglesParallelogram

http://www.youtube.com/watch?v=ioTBllxJ91s

if a quadrilateral is a parallelogram

opposite sides are parallel (DEFINITION)opposite sides are congruentopposite angles are congruentconsecutive angles are supplementarydiagonals bisect each other

Using Properties of Parallelograms

• PQRS is a parallelogram. Find the angle measure. m< R m< Q

Q R

SP70°

=70 °

70 ° + m < Q = 180 °

m< Q = 110 °

Using Algebra with Parallelograms

• PQRS is a parallelogram. Find the value of h.

P Q

RS3h 120°

3h + 120 = 180 - 120 -120

3h = 60divide both sides by 3

h = 20

Rhombus

Rhombus PropertiesRhombuses share all properties of a parallelogram AND:(1) Consecutive sides of a rhombus are congruent.(2) The diagonals of a rhombus bisect pairs of opposite angles.(3) The diagonals of a rhombus are perpendicular.

Example

• PQRS is a rhombus. What is the value of b?

P Q

RS

2b + 3

5b – 6

2b + 3 = 5b – 6 9 = 3b 3 = b

Rectangle Properties

All properties of a parallelogram AND:• Diagonals are congruent• All angles are congruent (90⁰)

Squares share all properties of a parallelogram AND rectangles AND rhombus CAN YOU NAME all the properties?

Square Properties

Debrief

• What was easy?• What is still difficult?• what do you need to study so you can apply

polygon properties to solve problems?