Basic Theorems of Triangles

Sophomore Geometry By: Shannon Manzella

I was going to add a math pun, but I only like those to a degree.

Math puns are the first sine of madness.

Interior Angles Theorem

• In a Triangle, the Interior Angles will always add to be 180°– The Interior Angles are the inside angles between

adjacent sides of a rectilinear figure.

60°

30°70° 70°

40°

70+70+40 = 180 30+60+90 = 180

90°

Example A

• Solve for angle X.

21°

108°

X

108+21+X = 180

129+X = 180

X = 51

51°

Base Angles Theorem

• If the sides of a triangle are congruent ( like in an Isosceles triangle) then the angles opposite these sides are congruent.– Two angles are congruent if they have the value

and size.

Leg A Leg B

Base Angle B Base Angle A

Example B

• Solve for angle X.

X

76°

76+X+X = 180

76+2X = 180

2x = 104

X = 52

52°52° X

Example C

• Solve for angle X.

X54°

63+63+X = 180

126+X = 180

x = 54

63° 63°

The Midpoint Theorem

• The segment joining the midpoints of two sides of a triangle is parallel to the third side and half as long as the third side.– The Midpoint of a line segment is the point that is

halfway between the endpoints of the line segment.

X

X

Example D

Since the line segment DE is the midpoint of triangle ΔABC, line

DE is half the length of AC

2X = 24

X = 12

A

B

D E

24

X

C

12

Example E

A

B

CD

E

X3

Since the line segment DE is the midpoint of triangle ΔABC, line

DE is half the length of BC

X = 3

X = 6 6

Cites

• http://www.toxel.com/wp-content/uploads/2012/12/meltedcrayon08.jpg

• http://www.mathwarehouse.com/geometry/triangles/

• http://www.basic-mathematics.com/base-angles-theorem.html

• http://www.cliffsnotes.com/math/geometry/polygons/the-midpoint-theorem