Transcript Geo_Lesson 4_6

Geometry Lesson 4.6
Isosceles, Equilateral,
and Right Triangles
Warm Up: Key Proof Concepts
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So far in Chapter 4 we have learned some
key concepts about congruent figures:
By definition: All corresponding sides and
angles are congruent
If figures , then corresp. sides and s 
If corresp. sides and s , then figures 
For triangles: It is not necessary to show ALL
sides and angles are congruent
 4 methods: SSS, SAS, ASA, AAS
1. CPC
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In this lesson we will go beyond just proving
triangles are congruent
We will prove other properties of a figure
after we know that two triangles are
congruent
We can abbreviate the definition of
congruence as follows:
CPC: Corresponding Parts are Congruent
If s , then CPC
(converse: if CPC, then s )
Example 2a: Apply CPC
Given: PQ  RQ and PS  RS
Prove: PQS  RQS
Q
Plan for proof:
Know: PQ  RQ and PS  RS
Can show: QS  QS
R
P
Logic: Show PQS  RQS using SSS,
then use CPC to show PQS  RQS
Statement
PQ  RQ and PS  RS
QS  QS
Reason
S
Given
Reflexive Property of Congruence
PQS  RQS
SSS
PQS  RQS
CPC
Example 2b: Apply CPC
Given: AB || CD and BC || DA
Prove: AB  CD
B

C
Plan for proof:
Know: parallel sides

A
D
Can show: CBD  ADB and CDB  ABD (alt int s)
and BD  BD (reflexive)
Logic: Use ASA to show ADB  CBD, then use CPC to
show AB  CD
S.
AB || CD and BC || DA
CBD  ADB
BD  BD
ADB  CBD
AB  CD
Reason
Given
Alt. Int. Angles
Reflexive Property of Congruence
SAS
CPC
Warm Up: Review of Special Triangles
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Classify the triangles below
Can you name the sides of the isosceles
and right triangles?
leg
leg
leg
base
isosceles
equilateral
right
leg
1. Base Angles Theorem
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If two sides of a triangle are congruent,
then their opposite angles are congruent
B
B and C are the
base angles of the
A
isosceles ABC
C
If two angles of a triangle are congruent,
then their opposite sides are congruent
2. Corollaries to Base Angle Theorem
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A
If a triangle is
equilateral, then it
is equiangular
If a triangle is
equiangular, then it
is equilateral
equilateral
B
(biconditional)
↔ equiangular
C
Example 1
State a reason that allows you to solve for x
(a)
(b)
Base Angles
Theorem
3x = 24
x=8
Base Angles
Theorem
(c)
if equiangular,
then equilateral
3x – 11 = 2x + 11
x = 22
2x – 1 = x + 3
x=4
Practice 1
State a reason that allows you to solve for x & y
(a)
(b)
hint: find the angle
measures first
Corollary to Base
Angle Theorem
3x= 45
X=15
y+7=45
Y= 38
Corollary to the
base angle
theorem
3x+3x+3x=180
9x=180
X=20
Example 2
Find the values of x and y: F
Finding x:
EFG is
equilateral
G
x° 120° y°
x°
H
x° y°
E
If equilateral,
then equiangular
Finding y:
GEH is isosceles (2 sides )
3x = 180
x = 60
If 2 sides , then opp. s 
(Base Angle Theorem)
mEGH = 180° - 60° = 120°
2y = 180 – 120 = 60
y = 30
3. Hypotenuse-Leg Theorem
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If the hypotenuse and one leg of a right
triangle are congruent to the hypotenuse
and corresponding leg of another right
triangle, then the triangles are congruent
A
D
B
C
E
F
If BC  EF and AC  DF, then ABC  DEF
Example 3
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Given: AB  AC and AM  BC
Prove: ABM  ACM
Plan for proof:
Show s are right, AM  AM, then
use HL to show the triangles are 
Statement
AB  AC and AM  BC
AM  AM
AMB & AMC are rt. s
Reason
Given
Reflexive Property
Definition of Perpendicular Lines
ABM & ACM are rt. s Definition of right triangles
ABM  ACM Hypotenuse-Leg Theorem
Practice 3
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Given: D is the midpoint of CE; BCD and FED
are right angles; and BD  FD
F
Prove: BCD  FED B
Statement
C
Reason
D is midpoint of CE
Given
BCD and FED are rt s
Given
BD  FD
Given
BCD and FED are rt s
CD  ED
BCD  FED
D
E
Definition of right triangle
Definition of midpoint
Hypotenuse-Leg Theorem
Assignment
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Ch 4.6 w/s