Download 4-6 - Nutley Public Schools

andHL
HL
4-6
TriangleCongruence:
Congruence: ASA,
ASA, AAS,
AAS, and
4-6 Triangle
Warm Up
Lesson Presentation
Lesson Quiz
Holt
HoltGeometry
McDougal Geometry
4-6 Triangle Congruence: ASA, AAS, and HL
Warm Up
1. What are sides AC and BC called? Side
AB?
legs; hypotenuse
2. Which side is in between A and C?
AC
3. Given DEF and GHI, if D  G and
E  H, why is F  I?
Third s Thm.
Holt McDougal Geometry
4-6 Triangle Congruence: ASA, AAS, and HL
Objectives
Apply ASA, AAS, and HL to construct
triangles and to solve problems.
Prove triangles congruent by using
ASA, AAS, and HL.
Holt McDougal Geometry
4-6 Triangle Congruence: ASA, AAS, and HL
Can someone explain
what the “included”
angle means?
Holt McDougal Geometry
4-6 Triangle Congruence: ASA, AAS, and HL
An included side is the common side
of two consecutive angles in a polygon.
The following postulate uses the idea of
an included side.
Holt McDougal Geometry
4-6 Triangle Congruence: ASA, AAS, and HL
Holt McDougal Geometry
4-6 Triangle Congruence: ASA, AAS, and HL
Example 1: Applying ASA Congruence
Determine if you can use ASA to prove the
triangles congruent. Explain.
Two congruent angle pairs are given, but the
included sides are not given as congruent. So, we
cannot use ASA.
Holt McDougal Geometry
4-6 Triangle Congruence: ASA, AAS, and HL
Check It Out! Example 1
Determine if you can use ASA to
prove NKL  LMN. Explain.
By the Alternate Interior Angles Theorem. KLN  MNL.
NL  LN by the Reflexive Property. No other congruence
relationships can be determined, so ASA cannot be
applied.
Holt McDougal Geometry
4-6 Triangle Congruence: ASA, AAS, and HL
You can use the Third Angles Theorem to prove
another congruence relationship based on ASA. This
theorem is Angle-Angle-Side (AAS).
Holt McDougal Geometry
4-6 Triangle Congruence: ASA, AAS, and HL
Check It Out! Example 2
Prove the triangles congruent.
Given: JL bisects KLM, K  M
Prove: JKL  JML
A
A
S
S
A
A
Holt McDougal Geometry
4-6 Triangle Congruence: ASA, AAS, and HL
✔ ✔✔
USE: AAS
Statements
Reasons
1. ÐK @ ÐM
1. Given A
2. JL bisects ÐKLM
2. Given
3. ÐKLJ @ ÐMLJ
4. JL @ JL
3. An angle bisector cuts an
A
angle into 2 congruent angles
4. Reflexive Property
S
5. DJKL @ DJML
5. SAS
Holt McDougal Geometry
4-6 Triangle Congruence: ASA, AAS, and HL
The bad word one…
• SSA or ASS…..
• We cannot use this to prove two triangles
are congruent.
• It’s a bad word, it’s bad news, and it
doesn't’t work… so NEVER use it!
Holt McDougal Geometry
4-6 Triangle Congruence: ASA, AAS, and HL
I think of this one as
rASS
Holt McDougal Geometry
4-6 Triangle Congruence: ASA, AAS, and HL
• In a HL proof you need 3 things

The hypotenuses

The legs

Tell us that the triangles have a
right angle, thus making them right
triangles.
Holt McDougal Geometry
4-6 Triangle Congruence: ASA, AAS, and HL
Example 3A: Applying HL Congruence
Determine if you can use the HL Congruence
Theorem to prove the triangles congruent. If
not, tell what else you need to know.
According to the diagram,
the triangles are right
triangles that share one
leg.
It is given that the
hypotenuses are
congruent, therefore the
triangles are congruent by
HL.
Holt McDougal Geometry
4-6 Triangle Congruence: ASA, AAS, and HL
Example 3B: Applying HL Congruence
This conclusion cannot be proved by HL. According
to the diagram, the triangles are right triangles and
one pair of legs is congruent. You do not know that
one hypotenuse is congruent to the other.
Holt McDougal Geometry
4-6 Triangle Congruence: ASA, AAS, and HL
Check It Out! Example 3
Determine if you can use
the HL Congruence Theorem
to prove ABC  DCB. If
not, tell what else you need
to know.
Yes, so lets prove it!
Holt McDougal Geometry
4-6 Triangle Congruence: ASA, AAS, and HL
✔ ✔
✔
USE: HL-RT
Statements
Reasons
1. AC @ BD
2. CB @ CB
1. Given
3. ÐABCandÐDCB
3. Given
H
2. Reflexive Property
areright angles
4. DABCand DDCB
are right triangles
4. A right triangle
RT
has 1 right angle
5. DABC @ DDCB
5. HL
Holt McDougal Geometry
L
4-6 Triangle Congruence: ASA, AAS, and HL
Lesson Quiz: Part I
Identify the postulate or theorem that proves
the triangles congruent.
ASA
HL
SAS or SSS
Holt McDougal Geometry
4-6 Triangle Congruence: ASA, AAS, and HL
Lesson Quiz: Part II
4. Given: FAB  GED, ABC   DCE, AC  EC
Prove: ABC  EDC
Holt McDougal Geometry
4-6 Triangle Congruence: ASA, AAS, and HL
Lesson Quiz: Part II Continued
Statements
Reasons
1. FAB  GED
1. Given
2. BAC is a supp. of FAB;
DEC is a supp. of GED.
2. Def. of supp. s
3. BAC  DEC
3.  Supp. Thm.
4. ACB  DCE; AC  EC
4. Given
5. ABC  EDC
5. ASA
Holt McDougal Geometry