altitudes and orthocenters

Altitudes and OrthocentersAltitudes and OrthocentersAn altitude is a height in a triangle from

the vertex. The segment is also perpendicular to the opposite side of

the vertex it comes from. An orthocenter is the intersection point of

all three altitudes on a triangle.

Goals1. To learn what an altitude is

2. To learn what an orthocenter is

An altitude is a height in a triangle from the vertex. The segment is also

perpendicular to the opposite side of the vertex it comes from. An

orthocenter is the intersection point of all three altitudes on a triangle.

Goals1. To learn what an altitude is

2. To learn what an orthocenter is

FactsFacts

Altitudes can also be medians and angle bisectors when it is in an isosceles triangle.

Orthocenters can be outside,Inside, or on a vertex dependingOn the type of triangle

Altitudes can also be medians and angle bisectors when it is in an isosceles triangle.

Orthocenters can be outside,Inside, or on a vertex dependingOn the type of triangle

QuickTime™ and a decompressor

are needed to see this picture.

QuickTime™ and a decompressorare needed to see this picture.

Sample ProblemsSample Problems

Write sometimes, always, or never.1. Altitudes cut angles in half.(Sometimes)2. What is the angle measure a of a? (90 degrees) 3. Where does the orthocenter lie in an obtuse

triangle? (Outside)

Write sometimes, always, or never.1. Altitudes cut angles in half.(Sometimes)2. What is the angle measure a of a? (90 degrees) 3. Where does the orthocenter lie in an obtuse

triangle? (Outside)

Web Links

http://www.cliffsnotes.com/study_guide/Altitudes-Medians-and-Angle-Bisectors.topicArticleId-18851,articleId-18787.html

http://www.homeschoolmath.net/teaching/g/altitude.php

http://www.pinkmonkey.com/studyguides/subjects/geometry/chap2/g0202401.asp

Web Links

http://www.cliffsnotes.com/study_guide/Altitudes-Medians-and-Angle-Bisectors.topicArticleId-18851,articleId-18787.html

http://www.homeschoolmath.net/teaching/g/altitude.php

http://www.pinkmonkey.com/studyguides/subjects/geometry/chap2/g0202401.asp

Isosceles TriangleSummary

Isosceles TriangleSummary

An Isosceles Triangle is a triangle with two equal sides.

The Isosceles Triangle have legs and a base. They also have a vertex angle and base angles.

When solving an isosceles triangle problem, there are many theorems or postulates used; some are the Base Angle Theorem and the Triangle Sum Theorem.

There are examples on slide 3 of isosceles triangle problems.

An Isosceles Triangle is a triangle with two equal sides.

The Isosceles Triangle have legs and a base. They also have a vertex angle and base angles.

When solving an isosceles triangle problem, there are many theorems or postulates used; some are the Base Angle Theorem and the Triangle Sum Theorem.

There are examples on slide 3 of isosceles triangle problems.

Meg BerlengiPeriod 4





Rules, Properties, and Formulas

Rules, Properties, and Formulas

Isosceles Triangle Theorem- the angles opposite the two equal sides are equal.

Base Angle Theorem- If two sides of a triangle are congruent, then the angles opposite them are congruent.

Converse Base Angle Theorem- If two angles of a triangle are congruent, then the sides opposite them are congruent.

Triangle Sum Theorem- the sum of the measures of the interior angles of a triangle is 180°

The vertex angle is the angle opposite the base. The base angles are the two angles adjacent to the base.

Isosceles Triangle Theorem- the angles opposite the two equal sides are equal.

Base Angle Theorem- If two sides of a triangle are congruent, then the angles opposite them are congruent.

Converse Base Angle Theorem- If two angles of a triangle are congruent, then the sides opposite them are congruent.

Triangle Sum Theorem- the sum of the measures of the interior angles of a triangle is 180°

The vertex angle is the angle opposite the base. The base angles are the two angles adjacent to the base.



ExamplesExamples



Solution:If two angles of a triangle are congruent, the sides opposite them are congruent.Set: 6x - 8 = 4x + 2 2x = 10 x = 5

Note: The side labeled 2x + 2 is a distracter and is not used in finding x

1. 2.



Solution:If two sides of a triangle are congruent, the angles opposite them are congruent.So m<1 = m<2 and m<3 = 40 degrees.180 - 50 = 130 180 - (40 + 40) = 100m<1 = 65 degrees m <4 = 100 degreesm<2 = 65 degrees

3. Solve for x.

Solve for the missing angles.

Solve for x.

Solution:68 + 68 =136180 −136 =444x−2 + 44 =1804x+ 42 =1804x=48x=12

Web LinksWeb Links http://www.mathwarehouse.com/geometry/congruent_triangles/

isosceles-triangle-theorems-proofs.php

http://hotmath.com/hotmath_help/topics/converse-of-isosceles-triangle-theorem.html

http://www.regentsprep.org/regents/math/geometry/GP6/PracISOS.htm

http://www.mathwarehouse.com/geometry/congruent_triangles/isosceles-triangle-theorems-proofs.php

http://hotmath.com/hotmath_help/topics/converse-of-isosceles-triangle-theorem.html

http://www.regentsprep.org/regents/math/geometry/GP6/PracISOS.htm

Converse, Inverse, Contrapositive

Converse, Inverse, Contrapositive

Kylie HelfenbeinPeriod 4

Kylie HelfenbeinPeriod 4

Summary Summary

The converse, inverse, and contrapositive are based off the basic understanding of a conditional statement. A converse is formed when the hypothesis and conclusion

of a conditional statement are exchanged. An inverse is formed by negating the hypothesis and

conclusion of a conditional statement. A contrapositive is formed by negating the hypothesis

and conclusion of a converse statement.

The converse, inverse, and contrapositive are based off the basic understanding of a conditional statement. A converse is formed when the hypothesis and conclusion

of a conditional statement are exchanged. An inverse is formed by negating the hypothesis and

conclusion of a conditional statement. A contrapositive is formed by negating the hypothesis

and conclusion of a converse statement.

Rules, Formulas, Properties

Rules, Formulas, Properties

To express all three statements, work off the basis of a conditional, where p-->q. The converse can be expressed by saying

q --> p The inverse can be expressed by saying

~p --> ~q, where the “~” negates the statement.

The contrapositive can be expressed by saying ~q --> ~p

To express all three statements, work off the basis of a conditional, where p-->q. The converse can be expressed by saying

q --> p The inverse can be expressed by saying

~p --> ~q, where the “~” negates the statement.

The contrapositive can be expressed by saying ~q --> ~p

Truth Value Truth Value

The conditional and the contrapositive have the same truth value.

The converse and the inverse have the same truth value.

The conditional and the contrapositive have the same truth value.

The converse and the inverse have the same truth value.

Sample problems Sample problems

1. If you are in North America, then, you are in the United States. Form a converse, inverse, and contrapositive from

these statements. Answer:

Converse: If you are in the United States, then, you are in North America.

Inverse: If you are not in North America, then you are not in the United States.

Contrapositive: If you are not in the United States, then, you are not in North America.

1. If you are in North America, then, you are in the United States. Form a converse, inverse, and contrapositive from

these statements. Answer:

Converse: If you are in the United States, then, you are in North America.

Inverse: If you are not in North America, then you are not in the United States.

Contrapositive: If you are not in the United States, then, you are not in North America.

Sample ProblemSample Problem

Using the same conditional, determine the truth value. “If you are in North America, then you are in

the United States.”

Answer: The conditional and contrapositive are false, you

do not have to be in the U.S. to be in North America.

The inverse and converse are true, because if you have to be in North America to be in the U.S.

Using the same conditional, determine the truth value. “If you are in North America, then you are in

the United States.”

Answer: The conditional and contrapositive are false, you

do not have to be in the U.S. to be in North America.

The inverse and converse are true, because if you have to be in North America to be in the U.S.

Sample ProblemSample Problem

Which of the following represents a converse to the statement, k--> h? A. ~k --> ~h B. ~h --> ~k C. h --> k

Answer: C. h --> k, to find a converse, you must exchange the

wording of the original conditional statement.

Which of the following represents a converse to the statement, k--> h? A. ~k --> ~h B. ~h --> ~k C. h --> k

Answer: C. h --> k, to find a converse, you must exchange the

wording of the original conditional statement.

WeblinksWeblinks

How to form statements http://hotmath.com/hotmath_help/topics/converse-in

verse-contrapositive.html http://mathforum.org/library/drmath/view/55349.htm

l

definitions of terms http://library.thinkquest.org/2647/geometry/

glossary.htm

How to form statements http://hotmath.com/hotmath_help/topics/converse-in

verse-contrapositive.html http://mathforum.org/library/drmath/view/55349.htm

l

definitions of terms http://library.thinkquest.org/2647/geometry/

glossary.htm

http://hotmath.com/hotmath_help/topics/converse-inverse-contrapositive.html


http://mathforum.org/library/drmath/view/55349.html






Triangle Exterior Angle Theorem/Exterior Angle Sum Theorem

By: Aubrey Postolakis

SUMMARY

Triangle Exterior Angle Theorem/Exterior Angle Sum Theorem

By: Aubrey Postolakis

SUMMARY Triangle Exterior Angle Theorem- is when the measure of

an exterior angle of a triangle is equal to the sum of the two angle measures of the two nonadjacent interior angles.

Exterior Angle Sum Theorem- is when the sum of all three angle measures of the interior angles of the triangle equals 180 degrees.

Triangle Exterior Angle Theorem- is when the measure of an exterior angle of a triangle is equal to the sum of the two angle measures of the two nonadjacent interior angles.

Exterior Angle Sum Theorem- is when the sum of all three angle measures of the interior angles of the triangle equals 180 degrees.

FORMULASFORMULAS

Triangle Exterior Angle Theorem- this is used to find the measure of the angle of the exterior angle and you find this by finding the sum of the two nonadjacent interior angles of the triangle

m ∠4 = m ∠1 + m ∠2

Exterior Angle Sum Theorem- this theorem proves that when you add up all of the three interior angles of a triangle it will equal 180 degrees.

m ∠1 + m ∠2 + m ∠3 = 180

Triangle Exterior Angle Theorem- this is used to find the measure of the angle of the exterior angle and you find this by finding the sum of the two nonadjacent interior angles of the triangle

m ∠4 = m ∠1 + m ∠2

Exterior Angle Sum Theorem- this theorem proves that when you add up all of the three interior angles of a triangle it will equal 180 degrees.

m ∠1 + m ∠2 + m ∠3 = 180

EXAMPLESEXAMPLES

Triangle Exterior Angle Theorem

Triangle Exterior Angle Theorem 3.) Example Problem

1.) Example Problem

2.) Solution

Exterior Angle Sum Theorem

2.) Example Problem

1.)

3.)

HELPFUL LINKSHELPFUL LINKS Triangle Exterior Angle Theorem http://www.kwiznet.com/p/takeQuiz.php?ChapterID=10730&Curriculu

mID=42&Num=6.3 http://www.cliffsnotes.com/study_guide/Exterior-Angle-of-a-Triangle.to

picArticleId-18851,articleId-18784.html http://www.mathopenref.com/triangleextangle.html

Exterior Angle Sum Theorem http://www.winpossible.com/lessons/Geometry_Triangle_Angle-Sum_Th

eorem.html http://www.brightstorm.com/math/geometry/triangles/triangle-angle-s

um http://www.cliffsnotes.com/study_guide/Angle-Sum-of-a-Triangle.topicA

rticleId-18851,articleId-18783.html

Math Text Book- Chapter 4 http://www.classzone.com/eservices/index.cfm?

Triangle Exterior Angle Theorem http://www.kwiznet.com/p/takeQuiz.php?ChapterID=10730&Curriculu

mID=42&Num=6.3 http://www.cliffsnotes.com/study_guide/Exterior-Angle-of-a-Triangle.to

picArticleId-18851,articleId-18784.html http://www.mathopenref.com/triangleextangle.html

Exterior Angle Sum Theorem http://www.winpossible.com/lessons/Geometry_Triangle_Angle-Sum_Th

eorem.html http://www.brightstorm.com/math/geometry/triangles/triangle-angle-s

um http://www.cliffsnotes.com/study_guide/Angle-Sum-of-a-Triangle.topicA

rticleId-18851,articleId-18783.html

Math Text Book- Chapter 4 http://www.classzone.com/eservices/index.cfm?

http://www.kwiznet.com/p/takeQuiz.php?ChapterID=10730&CurriculumID=42&Num=6.3


http://www.cliffsnotes.com/study_guide/Exterior-Angle-of-a-Triangle.topicArticleId-18851,articleId-18784.html


http://www.mathopenref.com/triangleextangle.html

http://www.winpossible.com/lessons/Geometry_Triangle_Angle-Sum_Theorem.html


http://www.brightstorm.com/math/geometry/triangles/triangle-angle-sum


http://www.cliffsnotes.com/study_guide/Angle-Sum-of-a-Triangle.topicArticleId-18851,articleId-18783.html


http://www.classzone.com/eservices/index.cfm





http://www.mathopenref.com/triangleextangle.html







http://www.classzone.com/eservices/index.cfm

Crook ProblemsCrook Problems

By Lauren SnieckusBy Lauren Snieckus

What are crook problems?What are crook problems? A crook problem is a type of set of parallel lines

and transversals. In a crook problem, the transversal is an angle

that forms a “crook”, rather than a straight line. They can be solved by drawing a horizontal line

through the angle inside of the two parallel lines, forming another parallel line.

Or, they can be solved by drawing a line perpendicular to the two parallel lines, intersecting the point of the angle inside of them.

A third method used is continuing the segments forming the angles of the “crook” until they intersect with the two parallel lines. With this type of problem, there is now multiple transversals.

A crook problem is a type of set of parallel lines and transversals.

In a crook problem, the transversal is an angle that forms a “crook”, rather than a straight line.

They can be solved by drawing a horizontal line through the angle inside of the two parallel lines, forming another parallel line.

Or, they can be solved by drawing a line perpendicular to the two parallel lines, intersecting the point of the angle inside of them.

A third method used is continuing the segments forming the angles of the “crook” until they intersect with the two parallel lines. With this type of problem, there is now multiple transversals.

Example 1Example 1 Find x if l||m. Find x if l||m.

X=110

Example 2Example 2 Find x and y if s||t Find x and y if s||t

X=48

Y=108

altitudes and orthocenters

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