This post was contributed by a community member. The views expressed here are the author's own.

Health & Fitness

Proving 2 triangles are congruent (Geometry)

Ways to prove that 2 triangles are congruent to each other.

Mathnasium - Math Learning Center's profile picture
Mathnasium - Math Learning Center, Neighbor

In one of the chapters in the geometry curriculum, students will be asked to prove that 2 triangles are congruent.  By congruent, it means that the triangles are identical.  The lengths of all three sides of the triangle A are the same as the lengths of the three sides of triangle B.  It also means that the measure of all three angles inside triangle A are identical to the three angles inside triangle B.

There are 5 ways that the students have to remember that triangles can be proven to be congruent.

  1. Side-Side-Side (SSS)
  2. Side-Angle-Side (SAS)
  3. Angle-Side-Angle (ASA)
  4. Angle-Angle-Side (AAS)
  5. Hypotenuse-Leg (HL) only applies to right triangles
In all the explanations below, we will utilize the examples of triangle 1 (with coordinates of ABC) and triangle 2 (with coordinates of XYZ).  If triangle 1 and triangle 2 are said to be congruent, it means that the following are all true:
  • angle A and angle X are congruent AND
  • angle B and angle Y are congruent AND
  • angle C and angle Z are congruent AND
  • segment AB and segment XY are congruent AND
  • segment BiC and segment YZ are congruent AND
  • segment AC and segment XZ are congruent
Side-Side-Side (SSS)

If the length of 3 sides of a triangle is congruent to the length of 3 sides of another triangle, then the triangles are said to be congruent.  In our example, if triangle

ABC's AB is congruent to XYZ's XY AND
ABC's BC is congruent to XYZ's YZ AND
ABC's AC is congruent to ZYZ's XZ THEN

Triangle ABC is congruent to Triangle XYZ

Side-Angle-Side (SAS)

If the length of 2 sides of a triangle is congruent to the length of 2 sides of another triangle AND the angles between the 2 sides of the triangle are congruent, then the triangles are said to be congruent.  NOTE:  the congruent angle has to be "sandwiched" between the 2 sides of the triangle.  In our example, if triangle

ABC's AB is congruent to XYZ's XY AND
ABC's BC is congruent to XYZ's YZ AND
ABC's angle B is congruent to XYZ's angle Y THEN

Triangle ABC is congruent to Triangle XYZ

Angle-Side-Angle (ASA)

If one side of a triangle is congruent to another side of a triangle, and the base angles of the side of the triangles are congruent, then the 2 triangles are said to be congruent.

In our example, if triangle

ABC's AB is congruent to XYZ's XY AND
ABC's angle A is congruent to XYZ's angle X AND
ABC's angle B is congruent to XYZ's angle Y THEN

Triangle ABC is congruent to Triangle XYZ

Angle-Angle-Side (AAS)

If the 2 angles of a triangle is congruent to 2 angles of another triangle, and 1 side of the triangle is the base of one of the 2 angles, and this side is congruent to the side of another triangle, then the triangles are said to be congruent.

In our example, if triangle

ABC's angle A is congruent to XYZ's angle X AND
ABC's angle B is congruent to XYZ's angle Y AND
ABC's BC is congruent to XYZ's YZ THEN

Triangle ABC is congruent to Triangle XYZ

Hypotenuse-Leg

This test of triangle congruency only applies to comparing 2 right triangle.  If the length of the hypotenuse of a right triangle is congruent to the length of the hypotenuse of a second right triangle, and if one of the legs of the right triangle is congruent to another leg of the second right triangle, then the triangles are said to be congruent. 

If you have any question regarding this type of problems, please feel free to reach out to me or any of the instructors in my center.

Michael Huang
Center Director
Mathnasium of Glen Rock/Ridgewood
T:     201-444-8020   
E:  glenrock@mathnasium.com
www.mathnasium.com/glenrock






The views expressed in this post are the author's own. Want to post on Patch?

More from Ridgewood-Glen Rock