Review of Lesson 5

1
Review of Lesson 5

• Midpoint Theorem
• Statement
• Proof
• Generalization
• Some Applications
• Intercept Theorem
• Converse
• Proof
• Some Applications

2
Lesson 8 Polygons
What do a triangle, a rectangle, a hexagon, and a
circle have in common? What are their
differences?

3

• Definitions
• A curve is a one-dimensional continuum??? I.e.,
  a drawing with no breaks.
• A simple curve does not cross itself.
• A closed curve starting and stopping at the
  same point.
• A polygon is a simple, closed curve whose sides
  which are segments.
• A point where two sides of a polygon meet is a
  vertex.
• Convex curves are simple, closed and have no
  indentations.
• Question Is a circle a polygon? Why?

4

• Group Discussion
• For each of the following, draw a curves that
  satisfies the property. Every curve must be
  different.
• Simple
• Simple and closed
• Closed but not simple
• Polygon
• Convex polygon
• Convex, simple and closed.

5

• Definitions
• Any two sides of a polygon having a common vertex
  determine an interior angle.
• An exterior angle of a convex polygon is
  determined by a side of the polygon and the
  extension of a contiguous side of a polygon.
• A line segment connecting 2 non-adjacent
  vertices of a polygon is a diagonal.

6

• Polygons

What is a 3-sided polygon called? What is a
4-sided polygon called?
7

• Angle Properties of Polygons
• Sum of the measures of interior angles

8

• Angle Properties of Polygons
• Sum of the measures of interior angles for a
  quadrilateral
• Consider a quadrilateral ABCD.
• Draw a diagonal BD.
• Two triangles are formed.
• Sum of the measures of interior angles for a
  quadrilateral
• Sum of the measures of interior angles for
  triangle ?ABD
• Sum of the measures of interior angles for
  triangle ?BCD
• (Why?)
• ?

9

• Angle Properties of Polygons
• Sum of the measures of interior angles for a
  pentagon

Group Discussion Use similar methods as in the
case of quadrilaterals, calculate the Sum of the
measures of interior angles for a pentagon.
10

• Activity I Sum of the measures of interior
  angles of n-gons
• Draw a convex polygon with n 7 sides.
• Pick a vertex of the polygon, call it O.
• Connect O to all other vertices of the polygon,
  how many triangles are there?
• Label these triangles as ?1, ?2,
• The sum of the measures of interior angles of the
  polygon x 180o.
• If you repeat steps (1) to (3) above for a
  general n-gon, you will get ____ triangles.
• Therefore Sum of interior angles of a n-gon
  _______ ? 180o.

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Conclusion Theorem 1 Sum of the measures of
interior angles of an n-gon (n-2)x180o .
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• Angle Properties of Polygons
• Sum of the measures of exterior angles

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Activity II Sum of the measures of exterior
angles A. Walk the triangle Mark 3 points A, B,
C on the floor, at least 3 meters apart from each
other. Start at vertex A facing B, walk along the
edges of the triangle ABC, from A to B, then to
C, and stop at A. Questions 1. What is the
measure of the change in direction right after
youd passed point B? 2. How many complete
rounds did you turn when you finish the walk? How
is this related to the sum of exterior angles of
the triangle?
B. Walk the Quadrilateral Repeat the activity
with a quadrilateral ABCD. C. Walk the
Polygon Repeat the activity with a hexagon. What
can you conclude?
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Conclusion Theorem 2 Sum of the measures of
exterior angles of an n-gon 360o .
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• Group Discussion
• What is the relation between an interior angle
  and its corresponding exterior angle?
• Prove Theorem 1 by
• assuming Theorem 2.

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Quadrilaterals
It is generally difficult to classify polygons.
We will only consider the simpler
ones. Triangles are the simplest type of
polygons. What is the next simplest type?

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Definitions of Different Quadrilaterals 1. A
trapezoid is a quadrilateral with at least one
pair of parallel sides. 2. A parallelogram is a
quadrilateral in which each pair of opposite
sides is parallel. 3. A rectangle is a
parallelogram with a right angle. 4. A rhombus is
a quadrilateral with all sides congruent. 5. A
square is a rectangle with all sides
congruent. 6. A kite is a quadrilateral with two
distinct pairs of congruent adjacent sides.

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• Group Discussion
• For each of the following, draw a figure that
  satisfies the definition. Every figure must be
  different.
• Trapezoid
• Parallelogram
• Rectangle
• Rhombus
• Square
• Kite

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Properties of Quadrilaterals Sides Example Show
that opposite sides of a parallelogram have equal
lengths. Solution Main Ideas (1) Recall the
definition of parallelogram. (2) Introduce a
diagonal to obtain two triangles. (3) Use a
congruency test to show that the two triangles
are congruent. (4) Conclude that opposite sides
are of equal length.

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Properties of Quadrilaterals Sides Group
Discussion Recall the previous example opposite
sides of a parallelogram have equal
lengths. Write down the complete proof.
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Properties of Quadrilaterals Angles Example
Show that opposite sides of a parallelogram have
equal lengths. Solution Main Ideas (1) Recall
the definition of parallelogram. (2) Introduce a
diagonal to obtain two triangles. (3) Use a
congruency test to show that the two triangles
are congruent. (4) Conclude that opposite sides
are of equal length.

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Properties of Quadrilaterals Angles Example
Show that a rectangle has four right
angles. Solution Main Ideas (1) Recall the
definition of rectangle. (2) Use properties of
parallel lines to conclude that corresponding
angles are congruent.

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Properties of Quadrilaterals Angles Example
Show that a rhombus has equal opposite interior
angles. Solution Main Ideas Recall the
definition of a rhombus. Introduce a diagonal to
obtain two triangles. Use test of triangle
congruency to show that the two triangles are
congruent. Conclude that the two respective
angles are congruent.

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• Properties of Quadrilaterals Angles
• Group Discussion
• Recall the previous example opposite angles of a
  rhombus are congruent.
• Write down the complete proof.
• Deduce that a rhombus is a parallelogram.

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26

• Properties of Quadrilaterals Diagonals
• Group Discussion
• Recall the previous example two diagonals of a
  rectangle have the same length.
• Write down the complete proof.
• Deduce from the proof that ?ODC and ?OCD are
  congruent.
• Deduce that OD OC and hence conclude that
  the two diagonals bisect each other.

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• Group Discussion
• Examine the definitions and properties carefully
  and answer the following questions.
• Is a parallelogram a trapezoid?
• Is a square a rhombus?
• Is a square a rectangle?
• Is a rhombus a parallelogram?
• Is a rhombus a rectangle?
• Is a rhombus a kite?

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Group Discussion Draw a Venn Diagram for the
classes of quadrilaterals.

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• Regular Polygons
• What is so special about equilateral triangles
  when compared to other triangles?
• What is so special about squares when compared to
  other quadrilaterals?

30

• Regular Polygons
• Definition
• A regular polygon is a polygon with
• all sides congruent (equilateral)
• all interior angles congruent (equiangular)

31

• Regular Polygons
• Discussion
• If a polygon has all sides congruent, must it be
  regular?
• If a polygon has all interior angles congruent,
  must it be regular?
• What is the measure of an interior angle of a
  regular n-gon?
• Must a regular polygon be convex?

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Regular Polygons A regular polygon is a convex
polygon with measure of each its interior angles