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2016
Chapter 4 Geometry
FST
4-1 Classifying Triangles
HW: p.219 #12-19, 24-30 even, 32-34, 41-43, 44, 45, 47, 48
New Terms:
Acute:
Obtuse:
Acute triangle:
Equiangular triangle:
Right triangle:
Obtuse triangle:
Equilateral triangle:
Isosceles triangle:
Scalene triangle:
You can classify triangles by their _____________ or their _____________.
Label the vertices of the following triangle:
List the sides:
List the angles:
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Chapter 4 Geometry
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Example 1:
Classify triangle BDC by its angle measures.
B is a ______ angle.
So BDC is a _______ triangle.
ADC is a ________ triangle.
Example 2:
Classify triangle ABD by its angle measures.
ABD and CBD form a linear pair,
so they are _______________.
Therefore mABD + mCBD =
____°.
By substitution, mABD + 100° = 180°. So mABD = ____°. ABD is a
___________ triangle by definition.
Important Note: You cannot assume sides or angles are congruent just by
looking at the picture. They need the proper notation or you need to prove
it!
Example 3:
Classify EHF by its side lengths.
From the figure,
EHF is ________.
2
. So HF = ____, and
2016
Chapter 4 Geometry
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Example 4:
Classify EHG by its side lengths.
By the Segment Addition Postulate, EG = ___
+ ___ = 10 + 4 = 14.
Since no sides are congruent, EHG is
________.
Example 5:
Find the side lengths of
JKL.
Example 6:
A steel mill produces roof supports by welding pieces of steel beams into
equilateral triangles. Each side of the triangle is 18 feet long. How many
triangles can be formed from 420 feet of steel beam?
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Chapter 4 Geometry
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4-2 Angle Relationships in Triangles
HW: 9-16, 19, 22-24, 26-32
Complete the Lab Activity on p.222.
Answers to questions 1-4:
1.
2.
3.
4.
A _______________ is a line that is added
to a figure to aid in a proof.
Example 1:
After an accident, the positions of cars are
measured by law enforcement to investigate the
collision. Use the diagram drawn from the
information collected to find mXYZ.
A ____________ is a theorem whose proof follows directly from another
theorem. Here are two corollaries to the Triangle Sum Theorem.
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Chapter 4 Geometry
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Example 2:
One of the acute angles in a right triangle measures 2x°. What is the measure
of the other acute angle?
Example 3:
The measure of one of the acute angles in a right triangle is 63.7°. What is
the measure of the other acute angle?
Additional Vocabulary:
- Interior Angle:
Exterior
- Exterior Angle:
- Remote Interior Angles:
- Exterior Angle Theorem:
Example 4:
Find mB.
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Chapter 4 Geometry
4-3 Angle Relationships in Triangles
HW: 9, 11, 17, 19, 23, 27, 29, 31, 33, 35
ABC  DEF (triangle ABC is congruent to triangle DEF )
List the congruent sides:
_____  _____
_____  _____
_____  _____
List the congruent angles:
_____  _____
_____  _____
_____  _____
When you write a statement such as ABC  DEF, you are also stating
which parts are congruent!
In a congruence statement, the order of the vertices indicates the
corresponding parts.
Example 1:
Given: ∆PQR  ∆STW
Identify all pairs of corresponding congruent parts.
Angles: P  
Sides: PQ 
, Q  
, QR 
, R  
, PR 
Example 2:
Given: ∆ABC  ∆DBC
Find the value of x.
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Chapter 4 Geometry
Example 3:
Given: ∆ABC  ∆DEF
Find mF.
Example 4:
Given:
YWX and YWZ are right angles. YW bisects XYZ.
W is the midpoint of XZ. XY  YZ .
Prove: XYW  ZYW
Statements
Reasons
Example 5:
Given:
AD bisects BE. BE bisects AD.
AB  DE, A  D
Prove: ABC  DEC
Statements
Reasons
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Chapter 4 Geometry
Example 6:
The diagonal bars across a gate give it support.
Since the angle measures and the lengths of the
corresponding sides are the same, the triangles are
congruent.
Given:
PR and QT bisect each other.
PQS  RTS, QP  RT.
Prove: ΔQPS  ΔTRS
Statements
Reasons
Example 7:
Use the diagram to prove the following.
Given:
MK bisects JL. JL bisects MK.
JK  ML. JK || ML.
Prove: JKN  LMN
Statements
Reasons
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2016
Chapter 4 Geometry
4-4 Triangle Congruence: SSS and SAS
HW: p.245 #9,13,19-35 odd
Triangle Rigidity:
Example 1:
Given: Diagram
Prove: ∆ABC  ∆DBC
Statements
Reasons
Example 2:
Given: Diagram
Prove: ∆ABC  ∆CDA
Statements
Reasons
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Chapter 4 Geometry
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An __________ __________ is an angle
formed by two adjacent sides of a polygon.
B is the __________ __________ between
sides _____ and _____.
Caution!!! The letters SAS are written in that order because the congruent
angles must be between pairs of congruent corresponding sides.
Example 3:
Given: Diagram
Prove: ∆XYZ  ∆VWZ
Statements
Reasons
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Chapter 4 Geometry
Example 4:
Given: BC || AD, BC  AD
Prove: ABD  CDB
Statements
Reasons
Example 5:
Given: QP bisects RQS. QR  QS.
Prove: RQP  SQP
Statements
Reasons
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2016
Chapter 4 Geometry
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4-5 Triangle Congruence: ASA, AAS, and HL
HW:
An __________ __________ is the common side of two consecutive angles
in a polygon. The following postulate uses the idea of an included side.
Example 1:
Given: See Picture
Prove: NKL  LMN
Statements
Reasons
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Chapter 4 Geometry
Example 2:
Given: X  V, YZW  YWZ, XY  VY
Prove: XYZ  VYW
Statements
Reasons
Example 3:
Given: JL bisects KLM, K  M
Prove: JKL  JML
Statements
Reasons
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Chapter 4 Geometry
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Example 4:
Given: See picture
Prove: ABC  DCB
Statements
Proving Triangles are Congruent
Def.  
SSS
Words
Reasons
SAS
Picture
14
ASA
AAS
HL
2016
Chapter 4 Geometry
FST
4-6 Triangle Congruence: CPCTC
HW: p.262 #7, 9, 11, 15, 17, 19, 31
CPCTC is an abbreviation for the phrase “Corresponding Parts of
Congruent Triangles are Congruent.” It can be used as a justification in a
proof after you have proven two triangles congruent.
Example 1:
A and B are edges of the ravine. What is AB?
Example 2:
Given: YW bisects XZ, XY  YZ .
Prove: XYW  ZYW
Statements
Reasons
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Example 3:
Given: PR bisects QPS and QRS.
Prove: PQ  PS
Statements
Reasons
Example 4:
Given: NO || MP, N  P
Prove: MN || OP
Statements
Reasons
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Chapter 4 Geometry
Example 5:
Given: J is the midpoint of KM and NL.
Prove: KL || MN
Statements
Reasons
Example 6:
Given: D(–5, –5), E(–3, –1), F(–2, –3), G(–2, 1),
H(0, 5), and I(1, 3)
Prove: DEF  GHI
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