I have 7 triangles, 1 each of:I have 7 triangles, 1 each of:acute scalene, acute isosceles, equilateral, acute scalene, acute isosceles, equilateral,

right scalene, right isosceles, right scalene, right isosceles, obtuse scalene, obtuse isosceles.obtuse scalene, obtuse isosceles.

If I ask a student to draw any random triangle, If I ask a student to draw any random triangle, find:find:

(1)(1) P(exactly 2 sides congruent) =P(exactly 2 sides congruent) =

(2)(2) P(at least 2 angles congruent) =P(at least 2 angles congruent) =

(3)(3) P(2 different triangles with no sides P(2 different triangles with no sides congruent) =congruent) =

Agenda• Go over warm up.• Exploration 8.1--share answers• Review geometry concepts• Discuss attributes: Quadrilateral Hierarchy• Exploration 8.6.• More practice problems.• Assign homework.

How did you group the polygons?

• For kids… talk about attributes– Shape: # sides, special quadrilaterals– Convex or non-convex– (1 or 2) Pair of parallel sides– (1 or 2) Pair of congruent sides– (1 or 2) Pair of perpendicular sides– Nothing special about it.– Cannot do any proof or justification if kids can’t

classify and describe similarities and differences.

How do I use a protractor? I forgot!

• Line up the center and line.

0˚180˚

180˚ 0˚

135˚ 45˚

45˚ 135˚ 90˚

Can you…• Sketch a pair of angles whose

intersection is:a. exactly two points?b. exactly three points?c. exactly four points?

• If it is not possible to sketch one or more of these figures, explain why.

Use Geoboards• On your geoboard, copy the given segment.• Then, create a parallel line and a

perpendicular line if possible. Describe how you know your answer is correct.

Exploration 8.6• Do part 1 using the pattern blocks--make sure your

justifications make sense.• You may not use a protractor for part 1.• Once your group agrees on the angle measures for

each polygon, trace each onto your paper, and measure the angles with a protractor.

• List 5 or more reasons for your protractor measures to be slightly “off”.

Given m // n.• T or F: 7 and 4

are vertical.• T or F: 1 4• T or F: 2 3• T or F: m 7 + m 6 = m 1• T or F: m 7 = m 6 + m 5• If m 5 = 35˚, find all the angles you can.• If m 5 = 35˚, label each angle as acute, right, obtuse.• Describe at least one reflex angle.

7 65

43

21

mn

More practice problems• Sketch four lines such that three are

concurrent with each other and two are parallel to each other.

True or False• If 2 distinct lines do not intersect, then they are

parallel.• If 2 lines are parallel, then a single plane contains

them.• If 2 lines intersect, then a single plane contains them.• If a line is perpendicular to a plane, then it is

perpendicular to all lines in that plane.• If 3 lines are concurrent, then they are also coplanar.

Pythagorean Theorem• Remember the Pythagorean Theorem?

• a2 + b2 = c2 where c is the hypotenuse in a right triangle.

• Use your geoboard to make a right triangle whose hypotenuse is the square root of 5.

Solution…• If a2 + b2 = c2 is to be used, we want a

right triangle whose hypotenuse is square root of 5.

• So, a2 + b2 = 5.• If you do not use

a geoboard, there are lots of answers.

5

Van Hiele levels• Formal study of geometry in high school requires

that students are familiar and comfortable with many different aspects of elementary and middle school geometry.

• Visualization, analysis, informal deduction are all necessary prior to high school geometry.

• This means students need to categorize, classify, compare and contrast, and make predictions about figures based upon their attributes.

Attributes• Early childhood:

– Size: big--little– Thickness: thin--thick– Colors: red-yellow-blue-etc.– Shape: triangle, rectangle, square, circle, etc.– Texture: rough--smooth

Why do we need this??? READING!!

Talk about polygonsWhat is a polygon?

Polygon• A simple, closed, plane figure

composed of at least 3 line segments.

• Why are each of the figures below not polygons?

Convex vs. Non-convex• Both are hexagons. One is convex.

One is non-convex.

• Look at diagonals: segments connecting non-consecutive vertices.

• Boundary, interior, exterior

Names of polygons!• Triangle• Quadrilateral• Pentagon• Hexagon• Heptagon (Septagon)• Octagon• Nonagon (Ennagon)• Decagon• 11-gon• Dodecagon

Triangle Attributes• Sides: equilateral, isosceles, scalene• Angles: acute, obtuse, right.• Can you draw an acute, scalene triangle?• Can you draw an obtuse, isosceles triangle?• Can you draw an obtuse equilateral triangle?

One Attribute of Triangles• The Triangle Angle Sum is 180˚.

• This is a theorem because it can be proven.

• Exploration 8.10--do Part 1 #1 - 3 and Part 2.

Diagonals, and interior angle sum

• Triangle• Quadrilateral• Pentagon• Hexagon• Heptagon (Septagon)• Octagon• Nonagon (Ennagon)• Decagon• 11-gon• Dodecagon

Congruence vs. SimilarityTwo figures are congruent if they are exactly

the same size and shape.Think: If I can lay one on top of the other, and

it fits perfectly, then they are congruent.Question: Are these two

figures congruent?Similar: Same shape, but

maybe different size.

Quadrilateral Hierarchy

Quadrilaterals• Look at Exploration 8.13. Do 2a, 3a - f.• Use these categories for 2a:

– At least 1 right angle– 4 right angles– 1 pair parallel sides– 2 pair parallel sides– 1 pair congruent sides– 2 pair congruent sides– Non-convex

Exploration 8.13• Let’s do f together:• In the innermost region, all shapes have 4 equal

sides.• In the middle region, all shapes have 2 pairs of equal

sides. Note that if a figure has 4 equal sides, then it also has 2 pairs of equal sides. But the converse is not true.

• In the outermost region, figures have a pair of equal sides. In the universe are the figures with no equal sides.

Warm Up• Use your geoboard to make:• 1. A hexagon with exactly 2 right angles• 2. A hexagon with exactly 4 right angles.• 3. A hexagon with exactly 5 right angles.• Can you make different hexagons for each

case?

Warm-up part 2• 1. Can you make a non-convex

quadrilateral?

• 2. Can you make a non-simple closed curve?

• 3. Can you make a non-convex pentagon with 3 collinear vertices?

Warm-up Part 3• Given the diagram at

the right, name at least 6 different polygons using their vertices.

E

G

F

DC

B

A

Agenda• Go over warm up.• Complete discussion of 2-Dimensional Geometry• Polyhedra attributes• Exploration 8.15 and 8.17• Examining the Regular Polyhedra• 3 Dimensions require 3 views• Assign Homework

Quadrilateral Hierarchy• Do the worksheet.

Some formulas--know how they work.

• Number of degrees in a polygon:Take 1 point and draw all the diagonals. Triangles are formed. Each triangle has 180˚. So, (n - 2)•180˚ is the number of degrees in a polygon.

• If the polygon is regular, then each angle is (n - 2) • 180/n.

Some formulas--know how they work.

• Distance formula: This is related to the Pythagorean Theorem.

• If aa22 + b + b22 = c = c22, then c = a, then c = a22 + b + b22 . .

• Now, if a is the distance from left to right, and Now, if a is the distance from left to right, and b is the distance from top to bottom, then the b is the distance from top to bottom, then the distance formula makes sense.distance formula makes sense.

Some formulas--know how they work.

• The distance formula is • A

• B

(x1, y1)

(x2, y2)(x2 - x1)2 + (y2 - y1)2

Some formulas--know how they work.

• Midpoint formula: If the midpoint is half way between two points, then we are finding the average of the left and right, and the average of the up and down.

• Midpoint: (x2 + x1) , (y2 + y1) 2 2

Some formulas--know how they work.

• Slope of a line: change in left and right compared to the change in up and down.

• m = (y2 - y1) (x2 - x1)

Discuss answers to Explorations 8.11 and 8.13• 8.11• 1a - c

• 3a: pair 1:same area,not congruent;pair 2: different area, not congruent;

• Pair 3: congruent--entire figure is rotated 180˚.

More practice problems• Think of an analog clock.• A. How many times a day will the minute hand be

directly on top of the hour hand?• B. What times could it be when the two hands

make a 90˚ angle?• C. What angle do the hands make at 7:00?

3:30? 2:06?