How to Prove Triangle Theorems
Videos, solutions, and lessons to help High School students learn how to prove
theorems about triangles.
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.
Common Core: HSG-CO.C.10
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
This video provides a two column proof of the exterior angles
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
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.