Videos and lessons with examples and solutions to help High School students
learn how to prove theorems about triangles.

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.

If the two sides of a triangle are congruent then the angles opposite the sides are congruent.

The midsegment joining the midpoints of two sides of a triangle is parallel to and half as long as the third side.

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.

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.

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.

The Medians of a Triangle Are Concurrent: A
Visual Proof from the Wolfram Demonstrations Project by Tomas
Garza

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 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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