How do we analyze the special segments that exist inside triangles? Agenda: Warmup Go over homework Segments inside triangles notes/practice QUIZ Thursday

Warmup Complete each statement: 1) A(n) ________________  has no congruent sides. 2) A(n) ______________  has two congruent sides. 3) A(n) _____________  has three congruent sides. 4) A(n) ____________  has three congruent angles. 5) A(n) ____________  has three acute angles. 6) A(n) __________  has 1 obtuse  and 2 acute  s. 7) A(n) ___________  has 1 right  and 2 acute  s.

The congruent sides are called the legs of the triangle. The third side is called the base. The two angles across from the legs are called the base angles. The angle across from the base is called the vertex angle. ***If 2 sides of a triangle are congruent, then the angles opposite them are congruent and vice versa. Base Angles Base Leg Vertex Angle MORE ABOUT ISOSCELES TRIANGLES *The base of a triangle does not have to be on the bottom

Example 52 ° 5x + 7 What kind of triangle is this? Solve for x What is the measure of the remaining angle?

EXAMPLE 2 5x – 10 3x 15 WHAT KIND OF TRIANGLE IS THIS? SOLVE FOR X

EXAMPLE 3 WHAT KIND OF TRIANGLE IS THIS? SOLVE FOR X 13 X + 8

Recall the following definitions: Segment Bisector – a line (or segment, or ray) that intersects a given segment at its midpoint. Angle Bisector – a ray that begins at the vertex of an angle and divides an angle into two , adjacent angles. Midpoint – the point on a segment that divides the segment into two congruent segments. ABC If A is the midpoint of BC, then BA  AC. A B C D E If DA bisects BC, then E is the midpoint of BC. (BE  EC). If AB bisects  CAD, then  CAB   BAD. C A D B

A median of a triangle is a segment from a vertex of the triangle to the midpoint of the opposite side. A D C B If BD is a median of  ABC, then AD  DC.

B A C Z X Y AX, BY, and CZ. Every triangle has 3 medians, one from each vertex. The intersection of the medians is known as the centroid. (Point Q) Q AZ = BX = AY = Medians: ZB; Z is the midpoint of AB. XC; X is the midpoint of BC. YC; Y is the midpoint of AC.

An altitude of a triangle is a segment in a triangle that begins at one vertex and is perpendicular to the opposite side. Altitude

Every triangle also has 3 altitudes, one from each vertex. The intersection point of the three altitudes is called the orthocenter.

A perpendicular bisector of a segment/triangle is a line, ray or segment that meets the given segment at a 90° angle and also intersects the segment at its midpoint. A D CB E AD is the perpendicular bisector of BC. E is the midpoint of BC. The point of concurrency of the three perpendicular bisectors is called the circumcenter.