Download 2205 Unit 3 part B NOTES

Geometry 2205
Unit 3:
Mrs. Bondi
Unit 3: Two-Dimensional Shapes
Lesson Topics:
Lesson 1: Congruent Figures (PH text 4.1)
Lesson 2: Triangle Congruence by SSS and SAS (PH text 4.2)
Lesson 3: Triangle Congruence by ASA and AAS (PH text 4.3)
Lesson 4: Using Corresponding Parts of Congruent Triangles (PH text 4.4)
Lesson 5: Isosceles and Equilateral Triangles (PH text 4.5)
Lesson 6: Congruence in Right Triangles (PH text 4.6)
Lesson 7: Congruence in Overlapping Triangles (PH text 4.7)
Lesson 8: Midsegments of Triangles (PH text 5.1)
Lesson 9: Perpendicular and Angle Bisectors (PH text 5.2)
Lesson 10: Bisectors in Triangles (PH text 5.3)
Lesson 11: Medians and Altitudes (PH text 5.4)
Lesson 12: Indirect Proof (PH text 5.5)
Lesson 13: Inequalities in One Triangle (PH text 5.6)
Lesson 14: Inequalities in Two Triangles (PH text 5.7)
Lesson 15: The Polygon-Angle Sum Theorems (PH text 6.1)
Lesson 16: Properties of Parallelograms (PH text 6.2)
Lesson 17: Proving that a Quadrilateral is a Parallelogram (PH text 6.3)
Lesson 18: Properties of Rhombuses, Rectangles, and Squares (PH text 6.4)
Lesson 19: Conditions for Rhombuses, Rectangles, and Squares (PH text 6.5)
Lesson 20: Trapezoids and Kites (PH text 6.6)
Lesson 21: Polygons in the Coordinate Plane (PH text 6.7)
Lesson 22: Applying Coordinate Geometry (PH text 6.8)
Lesson 23: Proofs Using Coordinate Geometry (PH 6.9)
Lesson 24: Proportions in Triangles (PH text 7.5)
Lesson 25: Areas of Parallelograms and Triangles (PH text 10.1)
Lesson 26: Areas of Trapezoids, Rhombuses and Kites (PH text 10.2)
Lesson 27: Areas of Regular Polygons (PH text 10.3)
Lesson 28: Perimeters and Areas of Similar Figures (PH text 10.4)
Lesson 29: Trigonometry and Area (PH text 10.5)
Lesson 30: Geometric Probability
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Geometry 2205
Unit 3:
Mrs. Bondi
Get Ready for Ch.6
Define:
Polygon:
Complete:
Polygon Chart:
(Create a chart to help you remember the names and number of sides of polygons with up to 12 sides.)
1
2
3
4
5
6
7
8
9
10
11
12
Do p.349 #1-15.
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Geometry 2205
Unit 3:
Mrs. Bondi
Lesson 15: The Polygon-Angle Sum (PH text 6.1)
Objective:
to find the sum of the both interior and exterior angles of a polygon
Theorem 6-1
Polygon Angle-Sum Theorem
The sum of the measures of the interior angles of an n-gon is __________________
Example 1: If m∠B = m∠D, find m∠B in ABCD.
B
A
100°
40°
D
Equiangular Polygon
C
Equilateral Polygon
Regular Polygon
Corollary to Polygon Interior Angle-Sum Theorem
The measure each interior angle of a regular n-gon is __________________
Example 2: What is the measure each interior angle of a regular nonagon?
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Geometry 2205
Unit 3:
Theorem 6-2
Mrs. Bondi
Polygon Exterior Angle-Sum Theorem
The sum of the measures of the exterior angles of a polygon, one at each vertex, is ________
Example 3: Find the measure of each interior angle. What is the sum of the interior angles?
Extend one side at each vertex and find the exterior angle that was formed. What is the sum of
the exterior angles?
Check out this explanation. http://www.mathsisfun.com/geometry/exterior-angles-polygons.html
Example 4: Find the measure of an interior and an exterior angle of a regular pentagon.
Practice:
HW: p.356 #6-24 even, 32
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Geometry 2205
Unit 3:
Mrs. Bondi
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Unit 3:
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Geometry 2205
Unit 3:
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Lesson 16: Properties of Parallelograms (PH text 6.2)
Objective:
to find relationships among angles, sides, and diagonals of parallelograms
Parallelogram:
Theorem 6-3
Opposite sides of a parallelogram are congruent.
Prove the theorem.
Given:
ABCD
Prove:
AB  CD
BC  DA
B
A
Statement
C
D
Reason
Consecutive Angles:
Theorem 6-4
Theorem 6-5
Consecutive angles of a parallelogram are supplementary.
Opposite angles of a parallelogram are congruent.
A
7
B
C
A
D
B
C
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Geometry 2205
Unit 3:
Theorem 6-6
Mrs. Bondi
The diagonals of a parallelogram bisect each other.
B
C
A
Theorem 6-7
D
If three (or more) parallel lines cut off congruent segments on one transversal,
then they cut off congruent segments on every transversal.
If AB CD EF and AC  CE , then BD  DF .
A
C
E
(Try dividing an index card into four congruent vertical sections using notebook paper.)
Practice:
HW: p.364 #10-30 even, 45-48
8
B
D
F
Geometry 2205
Unit 3:
Mrs. Bondi
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Geometry 2205
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Geometry 2205
Unit 3:
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Lesson 17: Proving that a Quadrilateral is a Parallelogram (PH text 6.3)
Objective:
Theorem 6-8
Prove it:
Given:
Prove:
to determine whether a quadrilateral is a parallelogram
If both pairs of opposite sides of a
quadrilateral are congruent, then the
quadrilateral is a parallelogram.
B
AB  CD
BC  DA
ABCD is a
C
A
Statement
D
Reason
Theorem 6-9
If an angle of a quadrilateral is supplementary
to both of its consecutive angles, then the
quadrilateral is a parallelogram.
Theorem 6-10
If both pairs of opposite angles of a quadrilateral
are congruent, then the quadrilateral is a
parallelogram.
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Unit 3:
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Example 1: Find the value of each variable to make the figure a parallelogram.
Theorem 6-11
If the diagonals of a quadrilateral
bisect each other, then the
quadrilateral is a parallelogram.
Theorem 6-12
If one pair of opposite sides of a
quadrilateral is both congruent and
parallel, then the quadrilateral is a
parallelogram.
Is it possible to prove that the quadrilateral is a parallelogram based on the given information? Explain.
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Unit 3:
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Practice:
HW: p.372 # 6-16 even, 22, 24, 29, 32-34
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Geometry 2205
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Lesson 18: Properties of Rhombuses, Rectangles, and Squares (PH text 6.4, 6.6)
Objectives:
to define and classify special types of parallelograms.
to use properties of diagonals of rhombuses and rectangles
Special Quadrilaterals:
Parallelogram:
Example:
Rhombus:
Rectangle:
Example:
Example:
Square:
Kite:
Example:
Example:
Trapezoid:
Isosceles Trapezoid:
Example:
Example:
Example 1: Judging by appearance, name the quadrilateral in as many ways as possible.
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Example 2: Given C(1, 2), D(5, 2), E(7,-2), F(-1, -2), determine the
most precise name for quadrilateral CDEF.
Theorem 6-13
Theorem 6-14
The diagonals of a rhombus are perpendicular.
Each diagonal of a rhombus bisects
the two opposite angles.
Example 3: If mABC = 120, find the measures of the numbered angles.
B
Rhombus ABCD
C
5
1
A
17
2
3
4
D
Geometry 2205
Unit 3:
Mrs. Bondi
Theorem 6-15
The diagonals of a rectangle are congruent.
Prove it:
Given:
Prove:
Rectangle ABCD
AC  DB
Statement
B
C
A
D
Reason
Practice:
HW: p.379 # 6-38 even, 42-46 even, 59-61
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Geometry 2205
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Lesson 19: Conditions for Rhombuses, Rectangles, and Squares (PH text 6.5)
Objectives:
to determine whether a parallelogram is a rhombus or a rectangle.
Theorem 6-16
If the diagonals of a parallelogram are perpendicular, then the parallelogram is a
rhombus.
Theorem 6-17
If one diagonal of a parallelogram bisects a pair of opposite angles, then the
parallelogram is a rhombus.
Theorem 6-18
If the diagonals of a parallelogram are congruent, then the parallelogram is a
rectangle.
Examples: Use the given information to identify each quadrilateral with any names that work
for it. Also identify its most specific name. (It is often helpful to draw a sketch.)
1. a diagonal bisects two angles
2. the diagonals are congruent
3. the diagonals bisect each other
4. the angle measures are 35, 145, 35, 145
5. the diagonals are perpendicular
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6. Example 6: For what values of x is
B
3x
2x + 12
A
ABCD a rhombus?
C
D
Example 7: For what values of x is
B
ABCD a rectangle?
C
7x + 2
A
5x + 8
D
Practice:
HW: p.386 #6-18 even, 32-34, 36-38
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(The expressions are labeling the pieces of the diagonals.)
Geometry 2205
Unit 3:
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Lesson 20: Trapezoids and Kites (PH text 6.6)
Objective:
bases
to verify and use properties of trapezoids and kites
base angles
trapezoid –
isosceles trapezoid –
Theorem 6-19
Example 1:
legs
Base angles of an isosceles trapezoid are congruent.
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Find the height of the trapezoid.
13
13
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Example 2: A circular dart board has 20 sections meeting in the middle.
What are the base angles of the trapezoids formed in the second ring?
Theorem 6-20
The diagonals of an isosceles trapezoid
are congruent.
Midsegment of a trapezoid –
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Geometry 2205
Unit 3:
Theorem 6-21
Mrs. Bondi
Trapezoid Midsegment Theorem
The midsegment of a trapezoid is parallel to its bases AND the length of the
midsegment is half the sum of the lengths of the bases
Example 2:
x+3
For what values of x can we call this a midsegment?.
2x - 4
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Kite –
Theorem 6-22
The diagonals of a kite are perpendicular.
R
Problem: Given Kite RSTW, find the value of RT.
RS = 13 cm
ST = 15 cm
SW = 24 cm
RT = ?
W
S
T
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Geometry 2205
Unit 3:
Mrs. Bondi
Example 3: Refer to the kite diagram below. If KT  x  4 , KB  2x  4 , JB  y  4 , and JT  2 x ,
find the values of x and y.
T
J
K
B
Quadrilateral Summary:
(include 8 types of figures)
Practice:
HW: p.394 #8-36 even, 57-62, 67-70
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Geometry 2205
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Mid-Chapter Quiz – p.398
Prepare for a quiz!
Algebra Review – Simplifying Radicals – p.399
Be sure you remember these skills!
More Radical Practice:
1. A pool is shaped like a rectangle with a length 4 times its width w. What is an expression for the
distance between opposite corners of the pool?
2. Evelyn rode her horse along a triangular path. The distance she traveled south was five times the
distance she traveled east. Then she rode directly back to her starting point. What is an expression for
the total distance she rode?
3. You are making a mosaic design on a square table top. You have already covered half of the
table top with 150 1-inch square tile pieces.
a. What are the dimensions of the table top?
b. What is the measure of the diagonal from one corner to the opposite corner of the table top?
4. The equation r 
SA
gives the radius r of a sphere with surface area SA. What is the radius of a
4
sphere with the given surface area? Use 3.14 for π.
a. 1256 in2.
b. 200.96 cm 2
c. 379.94 ft2
5. Open-Ended What are three radical expressions that simplify to 2 x 3 ?
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Algebra Review:
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Geometry 2205
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Lesson 21: Polygons in the Coordinate Plane (PH text 6.7)
Objective:
to classify polygons in the coordinate plane
Reminder:
In Coordinate Geometry, we combine our knowledge of geometry and algebra to
identify shapes, find missing vertices, or prove shape characteristics. This is useful for
many real-world applications.
Examples:
1)
Name the type of triangle is formed by
A (2, 0), B (3, 4), C (-1 , 1)
Use the distance formula to find length of:
AB
BC
CA
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2) Prove
EFGH is a parallelogram.
E (-5, -3), F (-2, 6), G (2 , 7), H (-1 , -2)
Use the slope formula to find slope of:
EF
FG
GH
HE
3)
Give the most precise name for JKLM .
J (-2, 4), K (2, 2), L (4 , -2), M (-4 , -4)
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Geometry 2205
Unit 3:
Mrs. Bondi
A quadrilateral formed by joining the midpoints of the sides of any quadrilateral will be
a parallelogram.
4)
Plot the vertices of kite ABCD. What is the shape of the
quadrilateral formed by the midpoints?
A (-2, 4), B (2, 2), C (4 , -2), D (-4 , -4)
Use the midpoint formula to find midpoints of:
AB
BC
CD
DA
(Is it a special parallelogram? Use slope to check for ║, and distance formula to check side lengths.)
Practice:
HW: p.403 #6-24 even, 33, 45-50 (need graph paper)
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Mrs. Bondi
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Lesson 22: Applying Coordinate Geometry (PH text 6.8)
Objective:
to name coordinates of special figures by using their properties
We can use Coordinate Geometry to supply missing information.
1)
Square ABCD
Find the coordinates of the other vertices.
A (-3, 3), B (-3, -3), C (
,
), D (
, )
Hints:
If this is a square, …
What do you know about side lengths?
What do you know about the sides’ relationship to one another?
What do you know about the slope of the sides?
2)
Rhombus EFGH
Find the coordinates of the other vertex.
E (0, 0), F (3, 4), G (8 , 4 ), H (
,
)
Hints:
If this is a rhombus, …
What do you know about side lengths?
What do you know about the diagonals’ relationship to one
another?
What do you know about the slope of the diagonals?
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Geometry 2205
Unit 3:
3)
Mrs. Bondi
Find the coordinates of the missing vertices of isosceles trapezoid EFGH. FG  EH
E (-1, -2), F (-1, 5), G (4 , 8), H (
,
)
In Coordinate Geometry, we can use variables to name coordinates of a figure. This is
useful to demonstrate a general case for a given situation.
Use variables to name the coordinates of the vertices of each figure.
Square
Isosceles Triangle
Rectangle
Rhombus
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Geometry 2205
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What do you know about the diagonals of a parallelogram?
How can we find the coordinates of D?
We can use coordinates with variables to prove a general geometric relationships. This is called a
coordinate proof. Sometimes using a coordinate proof is the easiest way to prove a statement.
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Geometry 2205
Unit 3:
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Practice:
HW: p.410 #8-13, 17-19, 38-40
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Lesson 23: Proofs Using Coordinate Geometry (PH 6.9)
Objective:
to prove theorems using figures in the coordinate plane
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Chapter 6 Review
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Unit 3:
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Chapter 6 Review (continued)
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