Download 2205 Unit 3 part A Notes

Geometry 2205
Unit 3:
Mrs. Bondi
Unit 3: Two-Dimensional Shapes
Lesson Topics:
Part A:
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)
Part B:
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)
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Geometry 2205
Unit 3:
Mrs. Bondi
Lesson 1: Congruent Figures (PH text 4.1)
Objectives:
to recognize congruent figures and their corresponding parts
Similar Polygons: same shape, but not necessarily the same size
1.
2.
Congruent polygons: exactly the same shape and size
1.
2.
When you write a congruency statement, you must list corresponding vertices in the same order.
Example 1: BIRD  CAGE Name congruent sides. Find the measures of as many angles as you can.
B
I
A
R
D
C
Example 2: ACBX  PRQY
X
E
List all of the congruent corresponding parts of ACBX and PRQY.
B
A
G
60
Y
P
R
C
Q
Theorem 4-1 Third Angles Theorem
If two angles of one triangle are congruent to two angles of another triangle, then the third
angles are congruent.
E
B
If A  D, B  E , then C  F .
D
A
2
C
F
Geometry 2205
Unit 3:
Mrs. Bondi
B
E
Two-Column Proof for Theorem 4-1
Given: A  D and B  E
Prove: C  F
A
D
C
Statements
1.) A  D and B  E
F
Reasons
1.) Given
2.) mA  mB  mC  180 and
mD  mE  mF  180
2.)
3.) mA  mB  mC  mD  mE  mF
3.)
4.)
4.) Substitution
5.) mC  mF
5.)
6.)
6.) Def. of Congruent Angles
Prove Triangles Congruent:
Given: AB ED , side measures indicated on diagram
Prove:
ABC  EDC
HW: p.222 #10-31, 33, 36-40 even, one proof from # 32, 44, 45
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Lesson 2: Triangle Congruence by SSS and SAS (PH text 4.2)
Objective:
Postulate 4-1
to prove two triangles congruent using the SSS and SAS Postulates
Side-Side-Side Postulate (SSS Postulate)
If three sides of one triangle are congruent to three sides of another triangle, then
the two triangles are congruent.
GHF  PQR
Postulate 4-2
Side-Angle-Side Postulate (SAS Postulate)
If two sides and the included angle of one triangle are congruent to two sides
and the included angle of another triangle, then the two triangles are congruent.
CBA  DFE
Given:
CA  RD
AR  DC
Prove:
CAR  RDC
Statement
C
A
D
R
Reason
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Geometry 2205
Unit 3:
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Prove Triangles Congruent:
Given: side measures indicated on diagram
Prove:
ABC  EDC
Statement
Reason
Review (frequently):
Postulates and Theorems (p.902-910)
Properties (p.900)
HW: p.230 #8, 11-18, 22, prove two from #28-33
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Lesson 3: Triangle Congruence by ASA and AAS (PH text 4.3)
Objective:
to prove two triangles congruent by the ASA Postulate and the AAS Theorem.
Postulate 4-3
Angle-Side-Angle (ASA Postulate)
If two angles and the included side of one triangle are congruent to two angles
and the included side of another triangle, then the two triangles are congruent.
GBH  KPN
B
P
G
Theorem 4-2
H
K
N
Angle-Angle-Side Theorem (AAS Theorem)
If two angles and a nonincluded side of one triangle are congruent to two angles
and the corresponding nonincluded side of another triangle, then the triangles
are congruent.
C
T
M
G
D
X
Q
Given:
XQ
TR , XR bisects QT
X
Prove:
XMQ  RMT
T
Statement
Reason
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M
R
DCM  GXT
Geometry 2205
Unit 3:
Given:
Prove:
Mrs. Bondi
N
MNP  ONP
MPN  OPN
M
MNP  ONP
O
P
Statement
Reason
P
Given:
PQ  TS
P  S
Prove:
PQR  STR
S
R
Q
Statement
T
Reason
HW: p.238 #8-13, 16-20
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Lesson 4: Using Corresponding Parts of Congruent Triangles (PH text 4.4)
Objective:
to use triangle congruence and CPCTC to prove that parts of two triangles are
congruent
Accepted Fact based on definition of congruent triangles:
CPCTC – Corresponding Parts of Congruent Triangles are Congruent
We can use this fact in proving congruence of sides or angles.
M
Given:
M  P
X is midpoint of MP
Prove:
MN  PO
X
O
Statement
N
Reason
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P
Geometry 2205
Unit 3:
Mrs. Bondi
Given:
D  B and DAC  BCA
Prove:
AB  CD
Statement
A
D
Reason
B
C
Legs of an ironing board bisect each other. Prove that the board is parallel to the floor.
(You make the drawing and the given/prove statements.)
Given:
Prove:
Statement
Reason
HW: p.246 #5-16
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Lesson 5: Isosceles and Equilateral Triangles (PH text 4.5)
Objective: Use and apply properties of isosceles and equilateral triangles.
Isosceles Triangle Legs –
Base –
Vertex Angle –
Base Angles –
Theorem 4-3 Isosceles Triangle Theorem
If two sides of a triangle are congruent, then the angles opposite those sides are congruent.
C
A
B
Theorem 4-4 Converse of the Isosceles Triangle Theorem
If two angles of a triangle are congruent, then the sides opposite those angles are congruent.
C
A
B
Write the Isosceles Triangle Theorem and its converse as a biconditional statement.
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Geometry 2205
Unit 3:
Example 1)
Mrs. Bondi
Explain all angle and side relationships you can. Justify.
A
B
C
D
E
Theorem 4-5
The bisector of the vertex angle of an isosceles triangle is the perpendicular bisector of the base.
C
A
B
D
Examples:
2)
If mL  52 , find the values of x and y.
x = ______
M
y = ______
x
52 
L
y
O
N
3)
AB  ____
B
FD  ____
D
E
CE  ____
A
F
C
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Corollary to Isosceles Triangle Theorem - If a triangle is equilateral, then it is equiangular.
Y
X
Z
Corollary to Converse of Isosceles Triangle Theorem - If a triangle is equiangular, then it is equilateral.
Y
X
Z
Write a biconditional statement for these two corollaries.
Examples:
4)
Find the measures of A, B , and ADC .
5)
What is the measure of  A ?
6)
What is the value of x?
HW: p.254 # 6-19, 22, 30-32, 37-40
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Lesson 6: Congruence in Right Triangles (PH text 4.6)
Objective:
Theorem 4-6
to prove triangles congruent using the Hypotenuse-Leg Theorem
Hypotenuse-Leg Theorem (HL Theorem)
If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse
and leg of another right triangle, then the triangles are congruent.
PQR  XYZ
To use the HL Theorem in proofs you must show that these three conditions are met.



There are two right triangles
There is one pair of congruent hypotenuses
There is one pair of congruent legs
Proof
Given:
KJWZ is a rectangle
Prove:
WJZ  KZJ
Statement
Reason
23
K
Z
X
J
W
Geometry 2205
Unit 3:
Mrs. Bondi
Are the two triangles congruent? If so, write the congruence statement. If not, tell why.
Practice using the HL Theorem.
Given:
BD  AC
AB  CB
Prove:
ADB  CDB
Statement
B
A
Reason
HW: p.261 #1-5, 8-13 (two system of equations problems)
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D
C
Geometry 2205
Unit 3:
Mrs. Bondi
Proof Practice:
Given:
CD  EA
..
Prove:
CBD  EBA
A
C
E
B
D
Statement
Given:
LP bisects MLN ,
PM  LM , PN  LN
Prove:
PM  PN
Reason
L
N
M
P
Statement
Reason
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Lesson 7: Congruence in Overlapping Triangles (PH text 4.7)
Objective:
to identify congruent overlapping triangles
to prove two triangles congruent using other congruent triangles
Sometimes triangles overlap one another and share a side or an angle. We can separate and redraw the
triangles in order to better see the corresponding parts.
What common side does BAE and DEA share?
B
D
C
E
A
What common angle does BAE and DAF share?
B
D
C
F
E
A
Given:
Prove:
BA  DE
BE  DA
BEA  DAE
B
D
C
A
E
Statement
Given:
Prove:
Reason
B
1  2
BD  BE
3  4
DAF  ECF
1 2
D
3
A
E
F
4
C
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HW: p.268
#8-15, 19-21
Geometry 2205
Unit 3:
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Review 4.1-7
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Practice for Ch.4Test
After ch.4 test HW: p.281 #3-15
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Lesson 8: Midsegments of Triangles (PH text 5.1)
Objective:
To use properties of midsegments to solve problems
Midsegment of a triangle:
Theorem 5-1 Triangle Midsegment Theorem
If D is the midpoint of CA and E is the midpoint of CB , then DE AB and DE = ½ AB.
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Examples:
4.
What all can be determined, given that OP is a midsegment of MLN ?
M
65o
89o
O
P
N
L
HW: p.288 #6-26 even, 32-44 even, 48
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Lesson 9: Perpendicular and Angle Bisectors (PH text 5.2)
Objective:
Use properties of perpendicular bisectors and angle bisectors
Equidistant -
Theorem 5-2
Perpendicular Bisector Theorem
If a point is on the perpendicular bisector of a segment,
then it is equidistant from the endpoints of the segment.
P
N
A
B
M
Theorem 5-3
Converse of the Perpendicular Bisector Theorem
If a point is equidistant from the endpoints of a segment,
then it is on the perpendicular bisector of the segment.
R
U
I
Biconditional:
Examples:
1) What is the length of AB ?
2)
37
What is the length of QR ?
Geometry 2205
Unit 3:
Mrs. Bondi
Distance from a point to a line:
Length of the perpendicular segment from the point to the line (the shortest distance)
C
A
Theorem 5-4
B
Angle Bisector Theorem
If a point is on the bisector of an angle, then it is equidistant from the sides of the angle.
Theorem 5-5
Converse of the Angle Bisector Theorem
If a point in the interior of an angle is equidistant from the sides of the angle, then it is on the
angle bisector.
Example:
3)
What is the length of FB ?
HW: p.296 #6-8, 12-22, 29-35 (complete at least two proofs)
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Lesson 10: Bisectors in Triangles (PH text 5.3)
Objective:
to identify properties of perpendicular bisectors and angle bisectors, altitudes and
medians of a triangle.
Concurrent –
Point of concurrency –
Theorem 5-6
Concurrency of Perpendicular Bisectors Theorem
The perpendicular bisectors of the sides of a triangle are
concurrent at a point equidistant from the vertices.
Circumcenter of the triangle
– the point of concurrency of
the perpendicular bisectors
It can be inside, on, or outside
the triangle.
The circle that has its center as the circumcenter of the triangle is circumscribed about the triangle.
The circle contains each of the triangle’s vertices.
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Example: A local park commission wants to build
a snack shack at a central point from the tennis
courts, ball fields, and playgrounds. How should they
determine where the snack shack should be built?
BF
TC
PG
Theorem 5-7
Concurrency of Angle Bisectors Theorem
The bisectors of the angles of a triangle are concurrent at a point
equidistant from the sides of the triangle.
Incenter of the triangle – the point of concurrency of angle bisectors - It is always inside the triangle.
The incenter of a triangle is the center of the circle that is inscribed in the triangle (when points on the
circle and points on the triangle’s sides are shared).
How would you solve this? (think algebraically)
HW: p.304 #5, 7-8, 14-18, 23-29, 33-35
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Lesson 11: Medians and Altitudes (PH text 5.4)
Objective:
to identify properties of perpendicular bisectors and angle bisectors, altitudes and
medians of a triangle.
Median of a triangle –
Theorem 5-8
Concurrency of Medians Theorem
The medians of a triangle are concurrent at a point that is
2/3 the distance from each vertex to the midpoint of the
opposite side.
Centroid of the triangle – the point of concurrency of the medians – also called the center of gravity of
a triangle because it is the point where a triangular shape will balance. It is always inside the triangle.
Example 1:
Assume ZA = 9. What is the length of FC ?
What is the ratio of ZA to AC?
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Altitude of a triangle –
Theorem 5-9
Concurrency of Altitudes Theorem
The lines that contain the altitudes of a triangle are concurrent.
Orthocenter of the triangle – the point of concurrency of the lines that contain the altitudes
It can be inside, on, or outside the triangle.
Practice:
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Concept Summary: Special Segments and Lines in Triangles
Name and label the point of concurrency for each example.
Perpendicular
Bisectors
Angle Bisectors
Medians
HW: p.312 #5, 8-14, 17-21, 24-31 (Bring the paper triangles to class.)
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Altitudes
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Lesson 12: Indirect Proof (PH text 5.5)
Objective:
To write convincing proofs by using indirect reasoning
Puzzle using indirect reasoning:
Fill in the empty squares with numbers. The numbers one
through four must appear exactly once in each row and column.
1
4
3
2
1
1
2
4
1
4
2
Indirect Reasoning – all possibilities are considered, then all but one are proven false. The remaining
possibility must then be true.
Indirect Proof – a proof using indirect reasoning – sometimes called a proof by contradiction
Often a statement and its negation are the only possibilities. When one of these leads to a conclusion
that contradicts a fact you know to be true, you can eliminate that possibility.
Explain why the two statements are contradictory.
1)
2)
3)
4)
A is acute
mA  90
__________________________________________
M is the midpoint of CD
CM  5, MD  10
__________________________________________
m forms right angles with n
m n
__________________________________________
1 is a complement of 2
m1  m2  180
__________________________________________
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Writing an Indirect Proof:
1 – State as a temporary assumption the negation of what you want to prove.
2 – Show that this assumption leads to a contradiction.
3 – Conclude that the assumption must be false, so that what you want to prove must be true.
Write an indirect proof.
P
S
1)
Given: QS does not bisect PQR
Prove: mPQS  mSQR
Q
R
Assume
______________________________________________________
Then
______________________________________________________
This contradicts the fact that __________________________________________
Therefore
2)
______________________________________________________
1 2
3 4
Given: m is not to n
Prove: m3  m6
5 6
7 8
m
n
Assume
______________________________________________________
Then
______________________________________________________
This contradicts the fact that __________________________________________
Therefore
______________________________________________________
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T
3)
Given: TU is not  to UW
Prove: m1  m2  90
V
1
U
2
W
Assume
______________________________________________________
Then
______________________________________________________
This contradicts the fact that __________________________________________
Therefore
4)
______________________________________________________
Given: ABC is scalene.
Prove: A, B, and C all have different measures.
HW: p.319 #3-21, do at least two proofs from 22-23 & 29-30
p.323 – review and/or practice – you are expected to be able to use and solve linear inequalities
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Lesson 13: Inequalities in One Triangle (PH text 5.6)
Objective:
Use inequalities involving triangle side lengths and angle measures to solve problems.
Comparison Property of Inequality
If a = b + c and c > 0, then a > b.
Corollary to the Triangle Exterior Angle Theorem
The measure of an exterior angle of a triangle is greater than the measure of each of its remote
interior angles
If  1 is an exterior angle, then m1  m2 and m1  m3 .
Use It:
Given the information in the figure, explain why m2  m3 .
Theorem 5-10
If two sides of a triangle are not congruent, then the larger angle lies opposite the longer side.
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Theorem 5-11
If two angles of a triangle are not congruent, then the longer side lies opposite the larger angle.
B
40 
92 
48 
C
A
Practice:
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Theorem 5-12 Triangle Inequality Theorem
The sum of the lengths of any two sides of a triangle is greater than the length of the third side.
Is it possible for a triangle to have sides of the following lengths?
1)
2 in, 5 in, 8 in
2)
4 cm, 6 cm, 9 cm
Application:
Given the lengths if two sides of a triangle, write an inequality to represent the range of values for the
length of the third side, s.
a) 15cm, 5 cm
b) 3 cm, 10 cm
c) a, b (assume a ≥ b)
HW: p.328 # 4-38 even
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Lesson 14: Inequalities in Two Triangles (PH text 5.7)
Objective:
to apply inequalities in two triangles
Think about a set of triangles that has two pairs of congruent sides. What do you know about the
relationship of the included angle and the third side? (If it helps, imagine two sets of rulers taped
together to form a hinge.)
Theorem 5-13
The Hinge Theorem (SAS Inequality Theorem)
If two sides of one triangle are congruent to two sides of another triangle, and the included
angles are not congruent, then the longer third side is opposite the larger included angle.
Examples:
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Unit 3:
Theorem 5-14
Mrs. Bondi
Converse of the Hinge Theorem (SSS Inequality Theorem)
If two sides of one triangle are congruent to two sides of another triangle, and the third sides
are not congruent, then the larger included angle is opposite the longer side.
Use an indirect proof to prove the Converse of the Hinge Theorem
Examples:
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HW: p.336 #4-18, 23 or 25
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Review:
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Lessons 5-5 through 5-7
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