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Lesson 5.1
Congruence and Triangles
Lesson 5.1 Objectives
Identify congruent figures and their
corresponding parts.
Prove two triangles are congruent.
Apply the properties of congruence to
triangles.
Congruent Triangles
When two triangles are congruent, then


Corresponding angles are congruent.
Corresponding sides are congruent.
Corresponding, remember, means that
objects are in the same location.

So you must verify that when the triangles are
drawn in the same way, what pieces match up?
Naming Congruent Parts
Be sure to pay attention to the proper
notation when naming parts.

 ABC   DEF
 This is called a congruence statement.
B
D
F
AD
BE
C F
and
A
C
E
AB  DE
BC  EF
AC  DF
Third Angles Theorem
If two angles of one triangle are
congruent to two angles of another
triangle, then the third angles are
congruent.
Prove Triangles are Congruent
In order to prove that two triangles are
congruent, we must


Show that ALL corresponding angles are
congruent, and
Show that ALL corresponding sides are
congruent.
We must show all 6 are congruent!
Side-Side-Side Congruence Postulate
If three sides of one triangle are
congruent to three sides of a second
triangle, then the two triangles are
congruent.
Side-Angle-Side Congruence
Postulate
If two sides and the included angle of
one triangle are congruent to two sides
and the included angle of a second
triangle, then the two triangles are
congruent.
Angle-Side-Angle Congruence
If two angles and the included side of one
triangle are congruent to two angles and
the included side of a second triangle, then
the two triangles are congruent.
Angle-Angle-Side Congruence
If two angles and a nonincluded side of
one triangle are congruent to two angles and
the corresponding nonincluded side of the
second triangle, then the two triangles are
congruent.
Hypotenuse-Leg Congruence
Theorem
If the hypotenuse and a leg of a right triangle are
congruent to the hypotenuse and a leg of a second
right triangle, then the two triangles are congruent.

Abbreviate using
 HL
Tuesday’s Schedule
Collect signed syllabi
Correct lesson 5.1 day 1 assignment
Review lesson 5.1 assignment
Lesson 5.1 day 2
Lesson 5.1 day 2 assignment
Lesson 5.1 Day 2
Which postulate or theorem to use??
Which postulates/theorems can be
used to prove triangle congruence?
SSS
SAS
ASA
AAS
HL

(side-side-side)
(side-angle-side)
(angle-side-angle)
(angle-angle-side)
(hypotenuse-leg)
HL can only be used in right triangles!!
Decide whether or not the congruence
statement is true. Explain your reasoning!
Reflexive Property of
Congruence
Because the segment is shared
between two triangles, and yet it
is the same segment
The statement is true
because of
SSS Congruence
The statement is not
true because the
vertices are
out of order.
Decide whether or not there is enough
information to conclude triangle congruence. If
so, state the postulate or theorem.
Reflexive Property of
Congruence
SAS Congruence
No
Decide whether or not there is enough
information to conclude triangle congruence. If
so, state the postulate or theorem.
Reflexive Property of
Congruence
Yes they are congruent!
HL
Reflexive Property of
Congruence
Not congruent
Decide whether or not there is enough
information to conclude triangle congruence. If
so, state the postulate or theorem.
Reflexive Property of Congruence
Reflexive Property of Congruence
Yes they are congruent!
Yes they are congruent!
ASA
AAS
Wednesday
Collect signed syllabi
Correct/review 5.1 day 2
Notes over lesson 5.2
Assignment 5.2
Lesson 5.2
Proving Triangles are Congruent
Review
What does congruent mean?
Draw two triangles that appear to be
congruent.
Label your drawings to make the two
triangles congruent.
Complete the proof
If 2(x+12)=90, then x=33
1. 2(x+12)=90
1. Given
2. 2x+24=90
2. Distributive Property
3. 2x=66
3. SPOE
4. X=33
4. DPOE
Complete the proof
Given
Given
Reflexive POC
SSS Congruence
Complete the proof
Given: AD
BC , AD  BC
Prove:DAB  BCD
1. AD
BC
2. AD  BC
1.Given
2. Given
3. ADB  CBD 3. AIA
4. BD  BD
4.
Reflexive
5. DAB  BCD 4. SAS
Construct a proof
1. AB  BE
1. Definition of midpoint
2. DB  BC
2. Definition of midpoint
3. ABD  EBC
3. Vertical angles
4. ABD  EBC
4. SAS
Surveying
MNP  MKL

Given
•
Segment NM  Segment KM
– Definition of a midpoint
•
LMK  PMN
– Vertical Angles Theorem
•
KLM  NPM
– ASA Congruence
•
Segment LK  Segment PN
– Corresponding Parts of Congruent Triangles
Lesson 5.3
Similar Triangles
Ratio
If a and b are two quantities measured
in the same units, then the
ratio of a to b is a/b.

It can also be written as a:b.
 A ratio is a fraction, so the denominator cannot
be zero.
Ratios should always be written in
simplified form.
/  1/2
 5 10
Proportional
If two ratios are equal after they are
simplified, then they are said to be
proportional.
6
3

10
5
12
3

20
5
These two ratios are
proportional.
Similarity of Trianlges
Two Triangles are similar when the following
two conditions exist


Corresponding angles are congruent
Correspondng sides are proportional
 Means that all side fit the same ratio.
The symbol for similarity is
 ~
 ABC ~ FGH

This is called a similarity statement.
Scale Factor
Since all the ratios should be equivalent to each
other, they form what is called the scale factor.

We represent scale factor with the letter k.
This is most easily found by find the ratio of one pair
of corresponding side lengths.

Be sure you know the polygons are similar.
k = 20/5
20
5
k=4
Angle-Angle Similarity Postulate
If two angles of one triangle are
congruent to two angles of another
triangle, then the two triangles are
similar.
Theorem 8.2:
Side-Side-Side Similarity
If the corresponding sides of two triangles
are proportional, then the triangles are
similar.

Your job is to verify that all corresponding sides fit
the same exact ratio!
10
10
6
5
5
3
Theorem 8.3:
Side-Angle-Side Similarity
If an angle of one triangle is congruent to an
angle of a second triangle and the lengths of the
sides including these angles are proportional, then
the triangles are similar.

Your task is to verify that two sides fit the same exact ratio
and the angles between those two sides are congruent!
10
6
5
3
Using Theorems…which one do I
use?
These theorems share the abbreviations with those
from proving triangles congruent.


SSS
SAS
So you now must be more specific




SSS Congruence
SSS Similarity
SAS Congruence
SAS Similarity
You chose based on what are you trying to show?


Congruence
Similarity