Unit 5 Notes

Triangle Properties

Definitions

Classify Triangles by Sides

Definitions

Classify Triangles by Angles

Definitions

Interior and Exterior AnglesWhen the sides of a polygon are extended, other angles are formed. The original angles are the interior angles. The angles that form linear pairs with the interior angles are the exterior angles.

Theorem

Triangle Sum TheoremThe sum of the measures of the interior angles of a triangle is 180°.

A

B C

180m A m B m C

Theorem

Exterior Angle TheoremThe measure of an exterior angle of a triangle is equal to the sum of the measures of the two nonadjacent interior angles. A

B C D

m A m B m ACD

• The two congruent sides of an isosceles triangle are called the legs.

• The angle formed by the legs is the vertex angle.

• The third side is the base of the isosceles triangle.

• The two angles adjacent to the base are called base angles.

Parts of an isosceles triangle

Base angles theorem

• Two sides of a triangle are congruent if and only if the angles opposite them are congruent.

Copy and complete each statement

Corollary to the base angles theorem

• A triangle is equilateral if and only if it is equiangular.

Find the values of x and y in the diagram

Definition

Definition of MidsegmentA midsegment is a segment that connects the midpoints of two sides of a triangle. Every triangle has three midsegments.

Line segment BD is a midsegment of triangle AEC

ED

C

B

A

Theorem

Midsegment TheoremThe segment connecting the midsegment is parallel to the third side and is half as long as that side.

BD = ½ AEBD is parallel to AEDB

E

C

A

Definition: Median of a Triangle

•A median of a triangle is a segment from a vertex to the midpoint of the opposite side.

•The point of concurrency is called the centroid. •The centroid is the center of gravity for the triangle. • The medians must intersect inside the triangle.

Definition: Altitudes of a triangle

•An altitude of a triangle is the perpendicular segment from a vertex to the opposite side or to the line that contains the opposite side.•The point of concurrency is called the orthocenter. •It doesn’t have a special function. •The three altitudes of a triangle can intersect inside, on, or outside the triangle.

Definition: Perpendicular bisector

• The segment that is perpendicular to a side of a triangle at it’s midpoint.

• The point of concurrency is called the circumcenter. • The circumcenter is the center of the circumscribed circle

making it equidistant from all three vertices. • The three perpendicular bisectors in a triangle can intersect

inside, on, or outside the triangle.

Definition: Angle bisector:

• The segment that bisects an angle of a triangle. • The point of concurrency is called the incenter. • The incenter is the center of the inscribed circle making it

equidistant from the three sides of the triangle. • The 3 incenters can only intersect inside the triangle.

Theorem

•Perpendicular Bisector Theorem•Any point on a perpendicular bisector is equally distant from the • endpoints of the segment it is bisecting.

Theorem

•Angle Bisector Theorem•Any point on an angle bisector is equally distant from the two sides of the angle.

Theorem

•Centroid Theorem• The distance from the centroid to the vertex is 2/3 the length of the entire

median. The distance from the centroid to the midpoint is 1/3 the length of the entire median.

Side Angle Relationships in a Triangle

Theorem

Triangle Inequality TheoremThe sum of the lengths of any two sides of a triangle is greater than the length of the third side.

A

AB + BC > AC

BC + AC > AB

B C AC + AB > BC