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.

Check out our most popular games!

We welcome your feedback, comments and questions about this site or page. Please submit your feedback or enquiries via our Feedback page.