1. 5.1 Midsegments of Triangles
-Midsegment: connects the midpoint of two sides
-Theorem 5.1: If a segments joins the midpoints of 2 sides of a triangle, then the segment is parallel to the 3rd side, and is half its length

2. 5.2 Perpendicular and angle bisectors
-Theorem 5.4: If a point is on the bisector of an angle, then the point is equidistant from the sides of the angle.
-Theorem 5.5: If a point is equidistant from
the sides of the angle, then
the point is on the
angle bisector

3. 5.3 bisectors in triangles
-Theorem 5.6: The perpendicular bisectors of the sides of a triangle are concurrent at a point equidistant from the vertices.

4. 5.3 bisectors in triangles
-Circumcenter: Point of concurrency of perpendicular bisectors of a triangle (is center of circle circumscribed about triangle)

5. 5.3 bisectors in triangles
-Theorem 5.7: The bisectors of the angles of a
triangle are concurrent at a
point equidistant from the
Sides of the triangle.
-Incenter: Point of concurrency of the
angle bisectors of a triangle (is center of
circle inscribed in triangle)

6. 5.4 Medians and altitudes
-Theorem 5.8: The medians of a triangle are concurrent at a point 2/3 distant from each vertex to the midpoint of the opposite side.

7. 5.4 Medians and altitudes
-Altitude: perpendicular segment from a vertex of a triangle to the line containing opposite side.
-Theorem 5.9: The lines that contain the altitudes of a triangle are concurrent.

8. 5.5 Indirect Reasoning -Indirect Reasoning: All
possibilities are considered and
then all but one are proved false
-Indirect proof: state an assumption as the opposite of what you’re trying to prove. Show the assumption leads to a contradiction, therefore proving your assumption false and what you wanted true

9. 5.6 Inequalities in triangles
-Comparison Property of Inequality:
If a = b + c and c > 0, then a > b
-Corollary to Triangle Exterior Angle Theorem: The exterior angle of a triangle is greater than either of its remote interior angles
-Theorem 5.10: If 2 sides of a triangle are not congruent, then the larger angle lies opposite the longer side
-Theorem 5.11: If 2 angles of a triangle are not congruent, then the longer side lies opposite the
larger angle
-Theorem 5.12: The Triangle Inequality Theorem
says the sum of 2 legs of any 2 sides of a
triangle is greater than the length of the 3rd side