How to Prove Triangle Theorems

Common Core: HSG-CO.C.10

Theorems include: measures of interior angles of a triangle sum to 180° base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.

Proving the Triangle Sum Theorem
The sum of the interior angles of a triangle is 180 degrees. Proof: The Isosceles Triangle Theorem
If the two sides of a triangle are congruent then the angles opposite the sides are congruent. The Triangle Midsegment Theorem
The midsegment joining the midpoints of two sides of a triangle is parallel to and half as long as the third side. The Triangle Proportionality Theorem
If a line parallel to one side of a triangle intersects the other two sides, then it divides those sides proportionally. The segment joining midpoints of two sides of a triangle is parallel to the third side and half the length.

The Triangle Angle Bisector Theorem
If a ray bisects the angle of a triangle, it divides the opposite side into segments proportional to the lengths of the other two sides i.e. the base angles of isosceles triangles are congruent. The Medians of a Triangle Are Concurrent: A Visual Proof
A median is the line from the midpoint of a side of a triangle to the opposite vertex. A formal proof is given for the concurrence of the medians in a triangle in a point. That point is called the centroid or barycenter; it is the center of mass of the triangle. The centroid is two thirds the distance from each vertex to the midpoint of the opposite side.


Proof: The Exterior Angles Theorem
The sum of the remote interior angles is equal to the non-adjacent exterior angle
This video provides a two column proof of the exterior angles theorem. Proof: The Sum of the Exterior Angles of a Triangle is 360 Degrees. Proof: The Angle of a Triangle Opposite The Longest Side is the Largest Angle
This video proves if the one side of a triangle is longer than another, then the angle opposite the longer side is greater than the angle opposite the shorter side.

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