Side Angle Side Theorem
By Andrew Moser
Summary
 If two sides and the included angle of a
triangle are congruent to two sides and the
included angle of another triangle, then the
two triangles are congruent.
 If two pairs of sides of two triangles are
equal in length, and the included angles are
equal in measurement, then the triangles are
congruent.
Examples
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Web Links
 http://www.mathwarehouse.com/trigonomet
ry/area/side-angle-side-triangle.html
 http://hotmath.com/hotmath_help/topics/SA
S-postulate.html
 http://www.jimloy.com/cindy/ass.htm
Side Side Side
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Kyle Schroeder
Summary
 You can only find SSS if the three sides in
one triangle are congruent.
 We learned this when using Solving
Triangle Proofs
Rules, Properties, & Formulas
 The rule and property for SSS theorem is
that you can only determine that you have
reached SSS is that the triangle has to be
congruent to the other triangle
Web Links
 http://www.cut-theknot.org/pythagoras/SSS.shtml
 http://www.tutorvista.com/topic/proof-ofsss-theorem
 http://www.mathwarehouse.com/geometry/c
ongruent_triangles/side-side-sidepostulate.php
Proofs Involving CPCTC
by,
Nick Karach
Summary:
-CPCTC stands for:
“Corresponding Parts of Corresponding Triangles are Congruent”
 This means that once you prove two triangle congruent, you know that
corresponding sides and angles are congruent.
Rules, Properties & Formulas


First of all you must prove the Triangles congruent through a postulate such as
ASA, SAS, AAS or HL.
Second, once you state the two triangles are congruent, you can state a two
sides are congruent. Ex.
AB CD
Examples
Given:
# BWO # MNA
Statement :
Re ason :
# BWO # MNA
HL
NAM ; WOB
CPCTC
Web Links
 Main Concept and Some Examples
 CPCTC WikiPedia
 Examples
Equilateral Triangle
By Jake Morra
Equilateral Triangles
 A equilateral triangle is a triangle where all
the sides are equal in length.
 All angles opposite though sides are
congruent
Finding The Height
To find the height add an altitude from
vertexes to opposite segment
If the added
segment is a
altitude. angle
BPA and APC
are 90
degrees
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If Segment AB,
BC, and CA are all
10 then Segment
BP and PC are 5
2
2
2
a + 5 = 10
Now that you know all of this can solve
the height by the Pythagorean Theorem
Other Websites To Help You
 http://mathcentral.uregina.ca/QQ/database/
QQ.09.02/rosa2.html
 www.calcenstein.com/calc/1111_help.php
 www.ehow.com › Education › Math
Education › Triangles
Angle Bisector and Incenter
-What is an angle bisector and an
incenter?
-Example problems
-Web links
What is an angle bisector and an
incenter?
B
Incenter
An angle bisector is a segment that divided an angle in half.
When the three angle bisectors intersect they create a point
of concurrency which is called the incenter
Ex: 1- Both little angles
will be the same
measure
mBAF = 24.41 °
mCAF = 24.41 °
A
H
G
Incenter
mABG = 33.53 °
B
mGBC= 33.53 °
mHCA = 32.06 °
F
C
mBCH= 32.06 °
Ex: 2 Find x
Equation: 13x-1= 2(6x+4)
13x-1= 12x+8
-12x 12x
x-1= 8
+1 +1
X= 9
Ex:3 Incenter is ALWAYS in the middle
Acute
Right
Obtuse
A
A
Incenter
Incenter
Incenter
C
B
C
B
B
C
Helpful Links
 http://www.cliffsnotes.com/study_guide/Altitudes-Medians-and-AngleBisectors.topicArticleId-18851,articleId-18787.html
 http://jwilson.coe.uga.edu/emt725/Prob.2.35.1/Problem.2.35.1.html
 http://mathworld.wolfram.com/AngleBisector.html
Angle Side Angle Theorem
By: Daulton Moro
AAS Theorem Summary:
 The AAS theorem is one of the theorems
that is used to prove triangles congruent.
 The AAS theorem is when two angles and
one non-included side are congruent.
Sample Problems



For the first picture you would mark lines BC and CE congruent and angles A
and D would be congruent. After mark the vertical angles congruent the you
have congruence by AAS.
The second picture shows AAS because there are two angles that are
congruent and one side that is non-included.
The third picture is self explanatory and is proven by using AAS.
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Helpful Websites
 www.mathwarehouse.com
 www.library.thinkquest.org
 www.phschool.com
What exactly is an HL proof?
By Dylan Sen
 The hypotenuse leg theorem, or HL, is the
congruence theorem used to prove only right
triangles congruent.
 Also The theorem states that any two right triangles
that have a congruent hypotenuse and a
corresponding, congruent leg are congruent
triangles..
 The goal of today’s lesson is to prove right triangles
congruent using the HL theorem
Rules and Formulas
 As seen in the previous slide, if the
hypotenuse and leg of one triangle are
congruent to the hypotenuse and leg of the
other, the triangles are congruent.
 The most important formula to remember is:
uuuuuuuuur
uuuuuuuuuur
if BC  EF, and AC  DF,thenVABC VDEF
Given:
uuur suur
AC  ZY
Examples
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ACB  ZYX  90
suur suur
AB  XY
ABC VXYZ
Prove:
Reason
uStatement
uur suur
-Given
AC  ZY(leg)
suur suur (hypotenuse) - Given
AB  XY
ABC and
XYZ
-They both have a right angle.
are right triangles
ABC VXYZ
- Through the HL theorem. Since the
hypotenuse and the leg are congruent,
that means the triangles are congruent
uuur uuur
Given- AB  DE
uuur uuur
BC  EF
ACB and DFE  90
Prove:
ABC VDEF
uuur uuur
AB  DE (leg)
uuur uuur
BC  EF (hypotenuse)
ABC and DEF
are right triangles
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Given
Given
They have a right angle
ABC VDEFBecause the hypotenuse and
corresponding leg are congruent, the
triangles are congruent
Given:
uuuuuuuuur
BC
 EFr
uuuuuuuuu
AC  DF
ABC and DEF  90
Prove
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d eco mpres sor
are nee ded to s ee this picture.
ABC VDEF
Statement
uuuuuuuuur
Reason
BC
 EF r (leg)
Given
uuuuuuuuu
AC  DF (hypotenuse) Given
ABC and DEF They have a right angle
are right triangles
Because the hypotenuse and
corresponding leg are congruent, the
ABC VDEF
triangles are congruent
Useful Websites to help you
further understand HL:
 http://delta.classwell.com/ebooks/navigateBook.clg?sectionType=unit&na
vigation=1&prevNext=0&curSeq=235&curDispPage=239&xpqData=%2
Fcontent%5B%40id%3D%27mcd_ma_geo_lsn_0395937779_p236.xml%
27%5D - This is the textbook definition. It will show examples and a step
by step method of figuring out how to use HL.
 http://www.mathwarehouse.com/geometry/congruent_triangles/hypotenus
e-leg-theorem.php - Much like the textbook, this website shows great
examples and will help clarify anything you have trouble with.
 http://www.onlinemathlearning.com/hypotenuse-leg.html - this example
shows more guided examples, which will further help you understand the
HL Theorem
Medians and Centroids
Summary: A median is a segment that connects the vertex of a
triangle to the midpoint of the opposite side. The point of concurrency
(intersection) of the medians is called the centroid.
Goals: The goals of this presentation are to:
1) Review Medians and Centroids
2) Review Sample Problems
Medians and Centroids
 A median is a segment that
connects the vertex of a triangle
to the midpoint of the opposite
side
 The point of concurrency
(intersection) of the medians is
called the centroid
 The distance from the vertex to
the centroid is 2/3 of the total
distance of the median
 No matter what type of triangle
(right, acute, obtuse), the
centroid is ALWAYS inside the
triangle
Sample Problems
1) Always, Sometimes, Never: The centroid
________________ lies within the triangle.
2) Find x:
A
B
|
2x+5
|
D
3x-10
C
3) Fill In The Blank: A triangle has ____________ medians.
Helpful Links
 http://www.mathopenref.com/trianglemedians.html
 http://mathworld.wolfram.com/TriangleMedian.html
 http://www.analyzemath.com/Geometry/MediansTriangle/
MediansTriangle.html
 http://www.cut-the-knot.org/triangle/medians.shtml