Geometry: 5 Triangle Theorems MathScience Innovation Center B. Davis

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

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.

Like what??

National Gallery of Art

Washington, DC

Photo by B. Davis 1995

Like what??

Washington, DC

NGA post card

Like what??

Washington, DC

Like what??

Washington, DC

What special triangles are these?

On the left: a right triangle

On the right: an isosceles triangle

It’s cool to hold 19 degrees in your hands

Washington, DC

Photo by M. Davis 1995

East Wing

Washington, DC

Photos by B. Davis 1995

Like what??

Washington, DC

If I hold 19 degrees at the purple dot, then how many degrees is at the orange dot?

The East Wing of the National Gallery of Art was designed by Ieoh Ming Pei

http://www.artcyclopedia.com/artists/pei_im.HTML

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

Ceiling

National Science Foundation

Washington, DC

Wigwam

Cody,Wyoming

Photo by L. Campbell 1992

US Capitol

Washington, DC

Window

Yellowstone National Park

Interior by Horace Pippin

Any triangles here?

Notice the artist used triangles in the quilted rug.

There are more groups

I can see at least 8 more groups of 3.

Can you?

Who was Horace Pippinwho painted Interior ?

http://artarchives.si.edu/guides/afriamer/pippin.htm

Lived 1888 - 1946

Painted Interior

in 1944

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

...So art and architecture are some reasons to study these special triangles.

What do we need to learn???

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.

40o x40 o

5 Key Ideas Two are:• Base Angles Theorem If 2 sides of a triangle

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

5 Key Ideas Number 3 is:• Base Angles Theorem If 2 sides of a triangle

• 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

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.

• If a triangle is equilateral, then it is equiangular.

If a triangle is equiangular, then it is equilateral.

60o 60o

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.

• 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.

Geometry: 5 Triangle Theorems MathScience Innovation Center B. Davis

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