Notes Lesson 5.2
Congruent Triangles
Target 4.1
HELP
GEOMETRY
Congruent Figures
Lesson 5.2
Definition:
Congruent triangles are triangles that have
all corresponding sides congruent and all
corresponding angles congruent.
Write a congruence statement for
the triangles at the left.
What information is sufficient to
prove triangles congruent?
HELP
GEOMETRY
Congruent Figures
Example 2:
XYZ
KLM,
YZ= x + 10
LM= 2x
Find the value
of x and the
lengths of the
given sides.
HELP
Target 4.1
Begin marking these triangles with
corresponding angles that are congruent.
GEOMETRY
Leading to Target 4.2
HELP
GEOMETRY
HELP
GEOMETRY
HELP
GEOMETRY
Triangle Congruence by SSS and SAS
Lesson 5.3
HELP
GEOMETRY
Triangle Congruence by ASA and AAS
Example 3:
Write a two-column proof.
Given: A
Prove:
1
B, AP
APX
BP
BPY
2
Statements
1
2
Reasons
1.
A  B, AP  BP
1. Given
2.
Ð1@ Ð2
2. Vertical angles are congruent.
3.
APX  BPY
HELP
3. ASA
GEOMETRY
Triangle Congruence by ASA and AAS
Example 4:
TARGET 4.3 & 4.5
Given
2
1
Write a two-column proof that uses AAS.
Given: B D, AB || CD
Prove: ABC
CDA
Not an
included
side
Statements
Reasons
1. B
D, AB || CD
1. Given
2. 1
2
2. If lines are ||, then alternate interior angles are .
3. AC
CA
3. Reflexive Property of Congruence
4.
HELP
ABC
CDA
4. AAS Theorem
GEOMETRY
Triangle Congruence by SSS and SAS
Example 5:
Target 4.2
Given: M is the midpoint of XY, AX
Prove: AMX
AMY
AY
From the given information, can you
prove that the triangles are congruent.
Explain.
HELP
Copy the diagram.
Mark the congruent sides.
Midpoint M implies MX MY.
AM AM by the Reflexive
Property of Congruence.
AMX
AMY by the
SSS Postulate.
GEOMETRY
Triangle Congruence by SSS and SAS
Example 6:
Target 4.2
a) Draw the two congruent
triangles separately.
A
a)
D
b) AD BC. What other
information do you need
to prove ADC
BCD
by SAS?
HELP
B
C
D
C
b) DC
CD by the Reflexive Property.
You now have two pairs of corresponding
congruent sides.
Therefore if you know ADC BCD,
you can prove
ADC
BCD by SAS.
GEOMETRY
TARGET 4.3
YOU TRY #1
HELP
GEOMETRY
Review: Right triangle:
H
Hypotenuse
Leg
L
L
L
H
L
Leg
TARGET 4.4
HELP
GEOMETRY
Congruence in Right Triangles
Example 7:
What additional information
is needed to prove.
Prove: ABC
DCB
by HL.
TARGET 4.4
C
D
B
C
Since BC
CB
Reflexive Property of Congruence
You must prove that BD
To prove
HELP
A
B
ABC
CA
DCB by the (HL Theorem).
GEOMETRY
Congruence in Right Triangles
Example 8:
TARGET 4.4
One student wrote “ CPA
MPA The diagram shows the following congruent parts.
by SAS” for the diagram below.
CA MA
Is the student correct? Explain.
CPA MPA
PA
PA
The congruent angles are not
included between the
corresponding congruent sides.
The triangles are not congruent by the
SAS Postulate, but they are congruent
by the HL Theorem.
HELP
GEOMETRY
YOU TRY #2
TARGET 4.4
What additional information will allow you to prove
the triangles congruent by the HL theorem?
A. ÐS @ ÐQ
B. mÐRPQ = 90
C. SR @ QP
D. PR @ RS
HELP
GEOMETRY
YOU TRY #3
HELP
TARGET 4.4
GEOMETRY