Geometry: 5 Triangle Theorems MathScience Innovation Center B. Davis.

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Presentation transcript:

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 a.com/artists/pei_im.HT ML

Congruent Triangles B. Davis MathScience Innovation Center Ieoh Ming Pei born in Canton, China in 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. 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 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 Pippin who painted Interior ? pippin.htm Lived 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. 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 o x

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. 30 o 100 x

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. 60 o

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. 60 o

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.