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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. - PowerPoint PPT Presentation
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Side Angle Side TheoremSide Angle Side Theorem
By Andrew MoserBy Andrew Moser
SummarySummary
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.
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.
ExamplesExamples
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are needed to see this picture.
QuickTime™ and a decompressor
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QuickTime™ and a decompressor
are needed to see this picture.
Web LinksWeb Links
http://www.mathwarehouse.com/trigonometry/area/side-angle-side-triangle.html
http://hotmath.com/hotmath_help/topics/SAS-postulate.html
http://www.jimloy.com/cindy/ass.htm
http://www.mathwarehouse.com/trigonometry/area/side-angle-side-triangle.html
http://hotmath.com/hotmath_help/topics/SAS-postulate.html
http://www.jimloy.com/cindy/ass.htm
Side Side Side Side Side Side
Kyle SchroederKyle Schroeder
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SummarySummary
You can only find SSS if the three sides in one triangle are congruent.
We learned this when using Solving Triangle Proofs
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 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
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 LinksWeb Links
http://www.cut-the-knot.org/pythagoras/SSS.shtml
http://www.tutorvista.com/topic/proof-of-sss-theorem
http://www.mathwarehouse.com/geometry/congruent_triangles/side-side-side-postulate.php
http://www.cut-the-knot.org/pythagoras/SSS.shtml
http://www.tutorvista.com/topic/proof-of-sss-theorem
http://www.mathwarehouse.com/geometry/congruent_triangles/side-side-side-postulate.php
Proofs Involving CPCTCby,
Nick Karach
Proofs Involving CPCTCby,
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.
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 & FormulasRules, 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.
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
ExamplesExamples
Given:
Statement :
# BWO ≅#MNA∠NAM ; ∠WOB
#BWO ≅#MNA
Reason :
HL
CPCTC
Web LinksWeb Links
Main Concept and Some Examples CPCTC WikiPedia Examples
Main Concept and Some Examples CPCTC WikiPedia Examples
Equilateral Triangle Equilateral Triangle
By Jake Morra By Jake Morra
Equilateral TrianglesEquilateral Triangles
A equilateral triangle is a triangle where all the sides are equal in length.
All angles opposite though sides are congruent
A equilateral triangle is a triangle where all the sides are equal in length.
All angles opposite though sides are congruent
Finding The HeightFinding The Height
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To find the height add an altitude from vertexes to opposite segment
If Segment AB, BC, and CA are all 10 then Segment
BP and PC are 5
If the added segment is a altitude. angle BPA and APC are 90 degrees
Now that you know all of this can solve the height by the Pythagorean Theorem
a + 5 = 102 2 2
Other Websites To Help YouOther 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
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 Angle Bisector and Incenter
-What is an angle bisector and an incenter?
-Example problems
-Web links
-What is an angle bisector and an incenter?
-Example problems
-Web links
What is an angle bisector and an incenter?
What is an angle bisector and an 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
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
Incenter
B
Ex: 1- Both little angles will be the same measure
Ex: 1- Both little angles will be the same measure
m∠BCH = 32.06 °
m∠HCA = 32.06°
m∠GBC = 33.53°
m∠ABG = 33.53°
m∠CAF = 24.41°m∠BAF = 24.41°
H G
F
Incenter
A
B C
Ex: 2 Find xEx: 2 Find x
Equation: 13x-1= 2(6x+4)13x-1= 12x+8-12x 12xx-1= 8 +1 +1 X= 9
Equation: 13x-1= 2(6x+4)13x-1= 12x+8-12x 12xx-1= 8 +1 +1 X= 9
Ex:3 Incenter is ALWAYS in the middle
Ex:3 Incenter is ALWAYS in the middle
Incenter
A
B
C
Acute
Incenter
B C
Right
Incenter
A
B C
Obtuse
Helpful Links Helpful Links
http://www.cliffsnotes.com/study_guide/Altitudes-Medians-and-Angle-Bisectors.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
http://www.cliffsnotes.com/study_guide/Altitudes-Medians-and-Angle-Bisectors.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 Angle Side Angle Theorem
By: Daulton Moro By: Daulton Moro
AAS Theorem Summary: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.
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 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.
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.
QuickTime™ and a decompressor
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are needed to see this picture.
Helpful WebsitesHelpful Websites
www.mathwarehouse.com www.library.thinkquest.org www.phschool.com
www.mathwarehouse.com www.library.thinkquest.org www.phschool.com
What exactly is an HL proof? By Dylan Sen
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
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 FormulasRules 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:
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:
if BC ≅EFu ruuuuuuuu
,andAC ≅DF,u ruuuuuuuuu
thenVABC ≅VDEF
ExamplesExamples
Given:
Prove: Statement Reason
(leg) -Given (hypotenuse) - Given
and -They both have a right angle.are right triangles
- Through the HL theorem. Since the hypotenuse and the leg are congruent, that means the triangles are congruent
Given:
Prove: Statement Reason
(leg) -Given (hypotenuse) - Given
and -They both have a right angle.are right triangles
- Through the HL theorem. Since the hypotenuse and the leg are congruent, that means the triangles are congruent
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ABs ruu
≅XYs ruu
ACu ruu
≅ZYs ruu
∠ACB = ∠ZYX = 90°
VABC ≅VXYZ
ACu ruu
≅ZYs ruu
ABs ruu
≅XYs ruu
VABC VXYZ
VABC ≅VXYZ
Given- and
Prove:
(leg) Given(hypotenuse) Givenand They have a right angle
are right triangles
Because the hypotenuse and corresponding leg are congruent, the triangles are congruent
Given- and
Prove:
(leg) Given(hypotenuse) Givenand They have a right angle
are right triangles
Because the hypotenuse and corresponding leg are congruent, the triangles are congruent
QuickTime™ and a decompressor
are needed to see this picture.
ABu ruu
≅DEu ruu
BCu ruu
≅EFu ruu
∠ACB ∠DFE =90°
BCu ruu
≅EFu ruu
ABu ruu
≅DEu ruu
VABC VDEF
VABC ≅VDEF
VABC ≅VDEF
Given:
and
Prove
Statement Reason
(leg) Given
(hypotenuse) Given
and They have a right angle
are right triangles
Because the hypotenuse and corresponding leg are congruent, the
triangles are congruent
Given:
and
Prove
Statement Reason
(leg) Given
(hypotenuse) Given
and They have a right angle
are right triangles
Because the hypotenuse and corresponding leg are congruent, the
triangles are congruent
QuickTime™ and a decompressor
are needed to see this picture.
BC ≅EFu ruuuuuuuu
AC ≅DFu ruuuuuuuu
VABC ≅VDEF
∠ABC ∠DEF =90°
BC ≅EFu ruuuuuuuu
AC ≅DFu ruuuuuuuu
VABC VDEF
VABC ≅VDEF
Useful Websites to help you further understand HL:
Useful Websites to help you further understand HL:
http://delta.classwell.com/ebooks/navigateBook.clg?sectionType=unit&navigation=1&prevNext=0&curSeq=235&curDispPage=239&xpqData=%2Fcontent%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/hypotenuse-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
http://delta.classwell.com/ebooks/navigateBook.clg?sectionType=unit&navigation=1&prevNext=0&curSeq=235&curDispPage=239&xpqData=%2Fcontent%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/hypotenuse-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 CentroidsSummary: 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 CentroidsSummary: 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 CentroidsMedians 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
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 ProblemsSample Problems1) Always, Sometimes, Never: The centroid ________________
lies within the triangle.
2) Find x:
3) Fill In The Blank: A triangle has ____________ medians.
1) Always, Sometimes, Never: The centroid ________________ lies within the triangle.
2) Find x:
3) Fill In The Blank: A triangle has ____________ medians. ||
3x-102x+5D
A
B C
Helpful LinksHelpful 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
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
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