Download Geometry Chapter 4

Geometry: 5 Triangle Theorems
MathScience Innovation Center
B. Davis
What are we studying?
• 3 kinds of special triangles
– right triangles
– equilateral triangles
– isosceles triangles
• and their special properties
Congruent Triangles B. Davis
MathScience Innovation Center
Why should we study these
triangles and their properties?
• They are everywhere in the world
around us!!!
• Artists, architects, and many others use
them to create designs ,buildings, and
items that we see and use everyday.
Congruent Triangles B. Davis
MathScience Innovation Center
Like what??
National Gallery
of Art
Washington, DC
Photo by B. Davis 1995
Congruent Triangles B. Davis
MathScience Innovation Center
Like what??
National Gallery
of Art
Washington, DC
NGA post card
Congruent Triangles B. Davis
MathScience Innovation Center
Like what??
National Gallery
of Art
Washington, DC
NGA
Congruent Triangles B. Davis
MathScience Innovation Center
Like what??
What special
triangles are
these?
On the left: a
right triangle
National Gallery
of Art
On the right: an
isosceles triangle
Washington, DC
NGA
Congruent Triangles B. Davis
MathScience Innovation Center
It’s cool to hold 19 degrees in
your hands
National Gallery
of Art
Washington, DC
Photo by M. Davis 1995
Congruent Triangles B. Davis
MathScience Innovation Center
East Wing
National Gallery
of Art
Washington, DC
Photos by B. Davis 1995
Congruent Triangles B. Davis
MathScience Innovation Center
Like what??
If I hold 19
degrees at
the purple
dot, then
how many
degrees is at
the orange
dot?
National Gallery
of Art
Washington, DC
NGA
Congruent Triangles B. Davis
MathScience Innovation Center
The East Wing of the National
Gallery of Art was designed by
Ieoh Ming Pei
http://www.artcyclopedi
a.com/artists/pei_im.HT
ML
Congruent Triangles B. Davis
MathScience Innovation Center
Ieoh Ming Pei
• born in Canton, China in 1917. He left China
when he was eighteen to study
architecture at MIT and Harvard.
• Pei worked as an instructor and then as an
assistant professor at Harvard
• Pei generally designs sophisticated glass
clad buildings. He frequently works on a
large scale and is renowned for his sharp,
geometric designs.
http://www.artcyclopedi
a.com/artists/pei_im.HT
ML
Congruent Triangles B. Davis
MathScience Innovation Center
Ceiling
National Science
Foundation
Washington, DC
Photo by B. Davis 1995
Congruent Triangles B. Davis
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Wigwam
Cody,Wyoming
Photo by L. Campbell 1992
Congruent Triangles B. Davis
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US Capitol
Washington, DC
Photo by B. Davis 1995
Congruent Triangles B. Davis
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Window
Yellowstone
National Park
Congruent Triangles B. Davis
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Photo by B. Davis 1994
Interior
by Horace Pippin
Any
triangles
here?
Congruent Triangles B. Davis National Gallery of Art
MathScience Innovation Center
Interior
by Horace Pippin
Notice
the artist
used
triangles
in the
quilted
rug.
Congruent Triangles B. Davis National Gallery of Art
MathScience Innovation Center
Interior
by Horace Pippin
There are
more groups
of 3
I can see at
least 8 more
groups of 3.
Can you?
Congruent Triangles B. Davis National Gallery of Art
MathScience Innovation Center
Who was Horace Pippin
who painted Interior ?
Lived 1888 - 1946
Painted Interior
in 1944
http://artarchives.si.edu/guides/afriamer/
pippin.htm
Congruent Triangles B. Davis
MathScience Innovation Center
Horace Pippin
http://artarchives.si.edu/guides/afriamer/
pippin.htm
African-American artist Horace Pippin, was
injured by a German sniper during World War I.
Pippin was a member of the 369th Army
Regiment, the first African-American soldiers
to fight overseas for the United States.
Pippin's injury left him with a shattered right
shoulder. Doctors attached his upper arm with a
steel plate, and after healing, Horace could
never lift his right hand above shoulder level.
Congruent Triangles B. Davis
MathScience Innovation Center
...So art and architecture are
some reasons to study these
special triangles.
What do we need to learn???
Congruent Triangles B. Davis
MathScience Innovation Center
5 Key Ideas: Here is the First
• Base Angles Theorem If 2 sides of a triangle are
congruent, then the angles opposite those sides
are congruent.
Congruent Triangles B. Davis
MathScience Innovation Center
5 Key Ideas
Two are:
• Base Angles Theorem If 2 sides of a triangle are
congruent, then the angles opposite those sides
are congruent.
Converse of the Base Angles Theorem If 2
angles of a triangle are congruent, then the
sides opposite them are congruent.
x
30o
30o
Congruent Triangles B. Davis
MathScience Innovation Center
5 Key Ideas Number 3 is:
• Base Angles Theorem If 2 sides of a triangle are
congruent, then the angles opposite those sides
are congruent.
• Converse of the Base Angles Theorem If 2
angles of a triangle are congruent, then the
sides opposite them are congruent.
If a triangle is equilateral, then it is equiangular.
60o
60o
60o
Congruent Triangles B. Davis
MathScience Innovation Center
5 Key Ideas and the 4th is:
• Base Angles Theorem If 2 sides of a triangle are
congruent, then the angles opposite those sides are
congruent.
• Converse of the Base Angles Theorem If 2 angles of a
triangle are congruent, then the sides opposite them are
congruent.
• If a triangle is equilateral, then it is equiangular.
If a triangle is equiangular, then it is equilateral.
60o
60o
60o
Congruent Triangles B. Davis
MathScience Innovation Center
5 Key Ideas and the 5th and
final idea is:
• Base Angles Theorem If 2 sides of a triangle are
congruent, then the angles opposite those sides are
congruent.
• Converse of the Base Angles Theorem If 2 angles of a
triangle are congruent, then the sides opposite them are
congruent.
• If a triangle is equilateral, then it is equiangular.
• If a triangle is equiangular, then it is equilateral.
If the hypotenuse and a leg of a right triangle
are congruent to the hypotenuse and leg of
another right triangle, then the 2 triangles are
congruent.
Congruent Triangles B. Davis
MathScience Innovation Center
5 Key Ideas and the 5th and
final idea is:
If the hypotenuse and a leg of a right
triangle are congruent to the hypotenuse
and leg of another right triangle, then
the 2 triangles are congruent.
Congruent Triangles B. Davis
MathScience Innovation Center
5 Key Ideas and the 5th and
final idea is:
If the hypotenuse and a leg of a right
triangle are congruent to the hypotenuse
and leg of another right triangle, then
the 2 triangles are congruent.
Congruent Triangles B. Davis
MathScience Innovation Center
5 Key Ideas and the 5th and
final idea is:
If the hypotenuse and a leg of a right
triangle are congruent to the hypotenuse
and leg of another right triangle, then
the 2 triangles are congruent.
Congruent Triangles B. Davis
MathScience Innovation Center
5 Key Ideas and the 5th and
final idea is:
If the hypotenuse and a leg of a right
triangle are congruent to the hypotenuse
and leg of another right triangle, then
the 2 triangles are congruent.
Congruent Triangles B. Davis
MathScience Innovation Center
5 Key Ideas and the 5th and
final idea is:
If the hypotenuse and a leg of a right
triangle are congruent to the hypotenuse
and leg of another right triangle, then
the 2 triangles are congruent.
Congruent Triangles B. Davis
MathScience Innovation Center
5 Key Ideas and the 5th and
final idea is:
If the hypotenuse and a leg of a right
triangle are congruent to the hypotenuse
and leg of another right triangle, then
the 2 triangles are congruent.
Congruent Triangles B. Davis
MathScience Innovation Center