Altitudes and Orthocenters
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.
Goals
1. To learn what an altitude is
2. To learn what an orthocenter is
Facts
 Altitudes can also be medians
and angle bisectors when it is
in an isosceles triangle.
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Orthocenters can be outside,
Inside, or on a vertex depending
On the type of triangle
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Sample 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)
Web Links
http://www.cliffsnotes.com/study_guide/AltitudesMedians-and-Angle-Bisectors.topicArticleId18851,articleId-18787.html
http://www.homeschoolmath.net/teaching/g/altitude.
php
http://www.pinkmonkey.com/studyguides/subjects/g
eometry/chap2/g0202401.asp
Isosceles Triangle
Summary
Meg Berlengi
Period 4

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.
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Rules, Properties, and
Formulas
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
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.
Examples
Solve for x.
3.
2.
1.
Solve for x.
Solve for the
missing angles.
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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
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) = 100
m<1 = 65 degrees
m <4 = 100 degrees
m<2 = 65 degrees
Solution:
68  68  136
180  136  44
4x  2  44  180
4x  42  180
4x  48
x  12
Web Links

http://www.mathwarehouse.com/geometry/congruent_triangles/isoscelestriangle-theorems-proofs.php

http://hotmath.com/hotmath_help/topics/converse-of-isosceles-triangletheorem.html

http://www.regentsprep.org/regents/math/geometry/GP6/PracISOS.ht
m
Converse, Inverse,
Contrapositive
Kylie Helfenbein
Period 4
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.
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
Truth Value
 The conditional and the contrapositive
have the same truth value.
 The converse and the inverse have the
same truth value.
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.
Sample 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.
Sample 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.
Weblinks
 How to form statements
 http://hotmath.com/hotmath_help/topics/converse-inversecontrapositive.html
 http://mathforum.org/library/drmath/view/55349.html
 definitions of terms
 http://library.thinkquest.org/2647/geometry/glossary.htm
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.
FORMULAS




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
Exterior Angle Sum Theorem
EXAMPLES
1.)
1.) Example Problem

Triangle Exterior Angle
2.)
Example
Theorem
3.) Example Problem
Problem
2.) Solution
3.)
HELPFUL LINKS










Triangle Exterior Angle Theorem
http://www.kwiznet.com/p/takeQuiz.php?ChapterID=10730&CurriculumID=42&
Num=6.3
http://www.cliffsnotes.com/study_guide/Exterior-Angle-of-aTriangle.topicArticleId-18851,articleId-18784.html
http://www.mathopenref.com/triangleextangle.html
Exterior Angle Sum Theorem
http://www.winpossible.com/lessons/Geometry_Triangle_AngleSum_Theorem.html
http://www.brightstorm.com/math/geometry/triangles/triangle-angle-sum
http://www.cliffsnotes.com/study_guide/Angle-Sum-of-a-Triangle.topicArticleId18851,articleId-18783.html
Math Text Book- Chapter 4
http://www.classzone.com/eservices/index.cfm?
Crook Problems
By Lauren Snieckus
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.
Example 1
 Find x if l||m.
X=110
Example 2
 Find x and y if s||t
X=48
Y=108