Download Lesson 4.3 and 4.4 Proving Triangles are Congruent

Lesson 4.3 and 4.4 Proving
Triangles are Congruent
p. 212
Learning Target
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I can list the conditions (SAS, SSS) to prove
triangles are congruent.
I can identify and use reflexive, symmetric and
transitive property in my proof.
How To Find if Triangles are
Congruent
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Two triangles are congruent if they have:
 exactly the same three sides and
 exactly the same three angles.
But we don't have to know all three sides and all
three angles ...usually three out of the six is
enough.
There are five ways to find if two triangles are
congruent: SSS, SAS, ASA, AAS and HL.
1. SSS (side, side, side)
SSS stands for "side, side, side“
and means that we have two triangles
with all three sides equal.
 For example:
is congruent to:

If three sides of one triangle are equal to three sides of
another triangle, the triangles are congruent.
2. SAS (side, angle, side)


SAS stands for "side, angle, side"
and means that we have two triangles
where we know two sides and the included angle
are equal.
For example:
is congruent to:
If two sides and the included angle of one triangle are equal
to the corresponding sides and angle of another triangle, the
triangles are congruent.
3. ASA (angle, side, angle)


ASA stands for "angle, side, angle“
and means that we have two triangles
where we know two angles and the
included side are equal.
For example:
is congruent to:
If two angles and the included side of one triangle are equal
to the corresponding angles and side of another triangle, the
triangles are congruent.
4. AAS (angle, angle, side)

AAS stands for "angle, angle, side“
and means that we have two triangles
where we know two angles and the
non-included side are equal.

For example:
is congruent to:
If two angles and the non-included side of one triangle are
equal to the corresponding angles and side of another
triangle, the triangles are congruent.
5. HL (hypotenuse, leg)

HL stands for "Hypotenuse, Leg" (the longest
side of the triangle is called the "hypotenuse",
the other two sides are called "legs")
and
HL applies only to right angled-triangles!
5. HL (hypotenuse, leg)

It means we have two right-angled triangles with
the same length of hypotenuse and
 the same length for one of the other two legs.
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It doesn't matter which leg since the triangles
could be rotated.
For example:
is congruent to
If the hypotenuse and one leg of one right-angled triangle
are equal to the corresponding hypotenuse and leg of
another right-angled triangle, the two triangles are congruent.
Caution ! Don't Use "AAA" !
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AAA means we are given all three
angles of a triangle, but no sides.
This is not enough information to decide if
two triangles are congruent!
Because the triangles can have the same angles
but be different sizes:
For example:
is congruent to
Without knowing at least one side, we can't be sure if two
triangles are congruent..
Proving Triangles are Congruent
You have learned to prove that two triangles are congruent by the
definition of congruence – that is, by showing that all pairs of
corresponding angles and corresponding sides are congruent.
THEOREM
B
Theorem 4.4 Properties of Congruent Triangles
A
Reflexive Property of Congruent Triangles
Every triangle is congruent to
itself.
Symmetric Property of Congruent Triangles
D
If ABC  DEF , then

.
DEF ABC
Transitive Property of Congruent Triangles
If ABC  DEF and

, ABC
then JKL J
DEF JKL
C
E
F
L
.
K
Goal 2
Using the SAS Congruence Postulate
Prove that
 AEB DEC.
1
2
3
Stateme
nts
AE  DE, BE  CE
1 2
 AEB   DEC
1
2
Reasons
Given
Vertical Angles Theorem
SAS Congruence Postulate
MODELING A REAL-LIFE SITUATION
Proving Triangles Congruent
ARCHITECTURE You are designing the window shown in the drawing. You
want to make  DRA congruent to  DRG. You design the window so that
DR AG and RA  RG.
Can you conclude that  DRA   DRG ?
D
SOLUTION
GIVEN
PROVE
DR
AG
RA
RG
 DRA
A
 DRG
R
G
Proving Triangles Congruent
GIVEN
PROVE
DR
AG
RA
RG
 DRA
D
 DRG
A
Statements
R
G
Reasons
Given
1
DR
AG
2
DRA and DRG
are right angles.
If 2 lines are , then they form
4 right angles.
3
DRA 
4
RA  RG
Given
5
DR  DR
Reflexive Property of Congruence
6
 DRA   DRG
SAS Congruence Postulate
DRG
Right Angle Congruence Theorem
S
Given: SP  QR; QP  PR
Prove  SPQ  SPR
Statements
1. SP  QR; QP  PR
2. QPS and RPS are right
’s.
Q
Reasons
1. Given
P
2. Def. of 
3. QPS  PRS
3. Rt.   Thm.
4. SP  SP
4. Reflexive POC
5.  SPQ  SPR
5. SAS  Post.
R
Pair-share
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Work on classwork on “Congruence Triangle”
Sage and Scribe on #21 to #24