Appetizer Draw label and cut out a large

Appetizer Draw, label, and cut out a large triangle; it does not matter what type of triangle. Label (on the inside), the vertices A, B, and C. • Fold A onto C to find the midpoint of AC. Do the same for BC. Label the midpoints L and N, respectively. Draw LN. • Fold each triangle on LN. • Fold A to C; fold B to C. Discussion 1. How does LN compare to AB? Explain. 2. Make a conjecture about how the segment joining the midpoints of two sides of a triangle is related to the third side of the triangle.

Section 5 -1: Midsegments of Triangles Triangle Midsegment Theorem: If a segment joins the midpoints of two sides of a triangle, then the segment is parallel to the third side, and is half its length.

Section 5 -2: Bisectors in Triangles Key Terms The distance between two objects is the length of the shortest path between them. Postulate: A line segment is the shortest path between two points. If two points are the same distance from a third point, then that third point is equidistant from the first two points. B Say what? A C

Key Terms (cont. ) The perpendicular bisector of a segment is the line that bisects and is perpendicular to the segment. C Say what? A D B

Perpendicular Bisector Theorem (PBT) Theorem: If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of that segment. Say what? X A B Y any point (along the perpendicular bisector) endpoints

Converse of the Perpendicular Bisector Theorem (CPBT) Theorem: If two points are each equidistant from the endpoints of a segment, then the two points determine the perpendicular bisector of that segment. Say what? X A B Y two points endpoints

More Key Terms! The distance from a point to a line is the length of the perpendicular segment from the point to the line. X A D Y

Angle Bisector Theorem (ABT) Theorem: If a point is on the bisector of an angle, then it is equidistant from the sides of that angle. B A X Say what? D C Y any point (along the angle bisector) sides

Converse of the Angle Bisector Theorem (CABT) Theorem: If a point in the interior of an angle is equidistant from the sides of the angle, then it is on the angle bisector. B A X Say what? D C Y any point sides angle bisector