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Geometry: 5 Triangle Theorems
MathScience Innovation Center
B. Davis
Congruent Triangles B. Davis MathScience Innovation Center
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??
National Gallery of Art
Washington, DC
NGA
What special triangles are these?
On the left: a right triangle
On the right: an isosceles triangle
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??
National Gallery of Art
Washington, DC
NGA
If I hold 19 degrees at the purple dot, then how many degrees is at the orange dot?
Congruent Triangles B. Davis MathScience Innovation Center
The East Wing of the National Gallery of Art was designed by Ieoh Ming Pei
http://www.artcyclopedia.com/artists/pei_im.HTML
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.artcyclopedia.com/artists/pei_im.HTML
Congruent Triangles B. Davis MathScience Innovation Center
Ceiling
National Science Foundation
Washington, DC
Photo by B. Davis 1995
Congruent Triangles B. Davis MathScience Innovation Center
Wigwam
Cody,Wyoming
Photo by L. Campbell 1992
Congruent Triangles B. Davis MathScience Innovation Center
US Capitol
Washington, DC
Photo by B. Davis 1995
Congruent Triangles B. Davis MathScience Innovation Center
Window
Yellowstone National Park
Photo by B. Davis 1994
Congruent Triangles B. Davis MathScience Innovation Center
Interior by Horace Pippin
National Gallery of Art
Any triangles here?
Congruent Triangles B. Davis MathScience Innovation Center
Interior by Horace Pippin
National Gallery of Art
Notice the artist used triangles in the quilted rug.
Congruent Triangles B. Davis MathScience Innovation Center
Interior by Horace Pippin
National Gallery of Art
There are more groups
of 3
I can see at least 8 more groups of 3.
Can you?
Congruent Triangles B. Davis MathScience Innovation Center
Who was Horace Pippinwho painted Interior ?
http://artarchives.si.edu/guides/afriamer/pippin.htm
Lived 1888 - 1946
Painted Interior
in 1944
Congruent Triangles B. Davis MathScience Innovation Center
Horace Pippin
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.
http://artarchives.si.edu/guides/afriamer/pippin.htm
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.
2020
40o x40 o
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.
30o 30o
100 x100
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.