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Geometry Lesson 1 Triangles

Grade 9 Honors Geometry

- By Dusan Vidovic

Purpose of the Lesson

- Addressing 4th Indiana Academic Standard for

Geometry. - Identify and describe triangles.
- Define, identify, and construct altitudes,

medians, and angle bisectors. - Identify congruent triangles.

But First, What Is a Triangle?

- A polygon with 3 sides which are straight

line segments. - Base
- 2 legs
- Any 3 non-collinear points determine a

triangle. - Vertices
- 3 angles summing up to 180 degrees.

Triangle Classification

- According to the lengths of their sides.
- Isosceles
- Scalene
- Equilateral
- Equiangular

- According to the size of their largest internal

angle. - Right
- Obtuse
- Acute

Isosceles Triangle

- At least two sides are of equal length.
- It also has two congruent angles.

Scalene Triangle

- All sides have different lengths.
- The internal angles are all different.

Equilateral/Equiangular Triangle

- All sides are of equal length.
- All its internal angles are equal.
- An equilateral triangle is also equiangular.

Right Triangle

- It has one 90 internal angle (a right angle).
- The side opposite to the right angle is the

hypotenuse (the longest side in the right

triangle). - The other two sides are the legs.
- Opposite
- Adjacent

Obtuse Triangle

- It has one internal angle larger than 90.

Acute Triangle

- It has internal angles that are all smaller than

90 (three acute angles).

Triangle Properties

- All of the previous triangles that we have seen

have the following properties - Medians
- Altitudes
- Angle Bisectors
- The hyperlink on each term will lead you to more

information including formulas which are not

covered in today's lecture.

Median

- It is a line joining a vertex on one angle to the

midpoint of the opposite side. - It divides the triangle into two parts of equal

area. - Every triangle has three medians which intersect

in the triangle's center of mass.

Altitude

- It is a straight line through a vertex and

perpendicular to (i.e. forming a right angle

with) the opposite side. - The intersection between the (extended) side and

the altitude is called the foot of the altitude.

This opposite side is called the base of the

altitude. - The length of the altitude is the distance

between the base and the vertex.

Angle Bisector

- The (interior) bisector of an angle is the line

segment that divides the angle into two equal

parts.

Triangle Congruence

- Two triangles are congruent if their

corresponding sides and angles are equal. - Usually it is sufficient to establish the

equality of three corresponding parts and use one

of the following results to conclude the

congruence of the two triangles. - SAS - SSS ASA AAS

Side Angle Side

- Two triangles are congruent if a pair of

corresponding sides and the included angle are

equal.

Side Side - Side

- Two triangles are congruent if their

corresponding sides are equal.

Angle Side - Angle

- Two triangles are congruent if a pair of

corresponding angles and the included side are

equal.

Angle Angle - Side

- Two triangles are congruent if two angles and a

non - included side - of one triangle are congruent to two angles and

the corresponding - non-included side of another triangle.

Important Remarks

- An equilateral triangle is also an isosceles

triangle, but not all isosceles triangles are

equilateral triangles. - An equilateral triangle is an acute triangle, but

not all acute triangles are equilateral

triangles. - AAA (Angle-Angle-Angle) is not a method of

proving triangle congruence because it says

nothing about the size of the two triangles and

hence shows only similarity and not congruence. - While the AAS (Angle-Angle-Side) condition

guarantees congruence, SSA (Side-Side-Angle) does

not. WHY????????? - More information on triangles can be found at

Math World.

Sources

- http//mathworld.wolfram.com/Triangle.html
- http//www.indianastandards.org/standard.asp?Subje

ctmathGradeGStandard4 - http//library.thinkquest.org/20991/geo/ctri.html

SAS - Pictures
- http//ostermiller.org/calc/triangle.png
- http//academics.sru.edu/ModLang/French/Activities

/french20photo20gallery/paris20-20rose/glass2

0pyramid20-20louvre.jpg - http//www.pbs.org/wgbh/nova/pyramid/geometry/imag

es/scaleintro.jpeg - http//en.wikipedia.org/wiki/ImageTriangle.Equila

teral.svg - http//en.wikipedia.org/wiki/ImageTriangle.Isosce

les.svg - http//en.wikipedia.org/wiki/ImageTriangle.Scalen

e.svg - http//en.wikipedia.org/wiki/ImageTriangle.Right.

svg - http//en.wikipedia.org/wiki/ImageTriangle.Obtuse

.svg - http//en.wikipedia.org/wiki/ImageTriangle.Acute.

svg - http//en.wikipedia.org/wiki/ImageTriangle.Centro

id.svg - http//en.wikipedia.org/wiki/ImageTriangle.Orthoc

enter.svg - http//library.thinkquest.org/20991/geo/ctri.html

SAS

THE END

- Homework
- Study the material presented today.
- Do Pg. 49, numbers 3,5,7,9,11,12, and 14 for

Monday. - Have a great weekend!!!

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