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Side Angle Side Theorem By Andrew Moser

Side Angle Side Theorem

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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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Page 1: Side Angle Side Theorem

Side Angle Side TheoremSide Angle Side Theorem

By Andrew MoserBy Andrew Moser

Page 2: Side Angle Side Theorem

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.

Page 3: Side Angle Side Theorem

ExamplesExamples

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are needed to see this picture.

QuickTime™ and a decompressor

are needed to see this picture.

QuickTime™ and a decompressor

are needed to see this picture.

Page 4: Side Angle Side Theorem

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

Page 5: Side Angle Side Theorem

Side Side Side Side Side Side

Kyle SchroederKyle Schroeder

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Page 6: Side Angle Side Theorem

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

Page 7: Side Angle Side Theorem

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

Page 8: Side Angle Side Theorem
Page 9: Side Angle Side Theorem

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

Page 10: Side Angle Side Theorem

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.

Page 11: Side Angle Side Theorem

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

Page 12: Side Angle Side Theorem

ExamplesExamples

Given:

Statement :

# BWO ≅#MNA∠NAM ; ∠WOB

#BWO ≅#MNA

Reason :

HL

CPCTC

Page 13: Side Angle Side Theorem

Web LinksWeb Links

Main Concept and Some Examples CPCTC WikiPedia Examples

Main Concept and Some Examples CPCTC WikiPedia Examples

Page 14: Side Angle Side Theorem

Equilateral Triangle Equilateral Triangle

By Jake Morra By Jake Morra

Page 15: Side Angle Side Theorem

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

Page 16: Side Angle Side Theorem

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

Page 17: Side Angle Side Theorem

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

Page 18: Side Angle Side Theorem

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

Page 19: Side Angle Side Theorem

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

Page 20: Side Angle Side Theorem

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

Page 21: Side Angle Side Theorem

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

Page 22: Side Angle Side Theorem

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

Page 23: Side Angle Side Theorem

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

Page 24: Side Angle Side Theorem

Angle Side Angle Theorem Angle Side Angle Theorem

By: Daulton Moro By: Daulton Moro

Page 25: Side Angle Side Theorem

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.

Page 26: Side Angle Side Theorem

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.

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Page 28: Side Angle Side Theorem

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

Page 29: Side Angle Side 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

Page 30: Side Angle Side Theorem

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

QuickTime™ and a decompressor

are needed to see this picture.

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

Page 31: Side Angle Side Theorem

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

Page 32: Side Angle Side Theorem

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

Page 33: Side Angle Side Theorem

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

Page 34: Side Angle Side 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

Page 35: Side Angle Side Theorem

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

Page 36: Side Angle Side Theorem

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

Page 37: Side Angle Side Theorem

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