Auxiliary line: A line that is added to a figure to aid in a proof.  
Symbol: --

This line can be called AC.

Interior angle: An angle formed by two sides of a polygon.
Symbol: ∠

This angle can be called ∠A.

Exterior angle: An angle formed by one side of a polygon and the extension of an adjacent side.
Symbol: ∠

This angle can be called ∠ABC or ∠B.

Vertex angle: The angle formed by the legs of an isosceles triangle. 
Symbol: ∠

This angle can be called ∠A or ∠ABC.

Base angle: The two angles that have the base as a side.
Symbol: ∠

This angle can be called ∠B or ∠ABC

Included angle: An angle formed by two adjacent sides of a polygon.
Symbol: ∠

This angle can be called ∠C or ∠BAC.

Included side: The common side of two consecutive angles in a polygon.
Symbol: --

This line can be called line BC.

Median: A segment whose endpoints are a vertex of the triangle and the midpoint of the other side.
Symbol: --

This line can be called line AD.

Altitude: A perpendicular segment from a vertex to the line containing the opposite side.
Symbol: --

This line can be called line AD.

Midsegment: A segment that joins the midpoints of the two sides of the triangle.
Symbol: --

This line can be called line DE.

Angle Bisector: A ray that divides an angle into two congruent angles.
Symbol: --

This line can be called line AD.

Perpendicular Bisector: A line that passes through the midpoint of a segment that is perpendicular to the segment.
Symbol: --

This line can be called line DE.

The reasons I chose these terms were because they were the hardest to understand and the most relevant to math. All these terms involve triangles, which is very important in geometry and in math.