Congruence Theorems

  • Congruence Theorems

    Grade: High School

    Investigate congruence by manipulating the parts (sides and angles) of a triangle. If you can create two different triangles with the same parts, then those parts do not prove congruence. Can you prove all the theorems?



    • Each triangle congruence theorem uses three elements (sides and angles) to prove congruence. Select three triangle elements from the top, left menu to start.
      • Note: The tool does not allow you to select more than three elements. If you select the wrong element, simply un-select it before choosing another element. This creates those elements in the work area.
      • On the top of the toolbar, the three elements are listed in order. For example, if you choose side AB, angle A, and angle B, you will be working on Angle – Side – Angle. If instead you choose side AB, angle A, and angle C, you will be working on Angle – Angle – Side. The two theorems are different, even though both involve two angles and one side.
    • Construct your triangle:
      • Move the elements of the triangle so that points labeled with the same letter touch.
      • Click and drag a dot to move the element to a new location.
      • Click and drag a side's endpoint or angle's arrow to rotate the element. The center of rotation is the side's midpoint or the angle's vertex, respectively.
      • To help place elements, points marked with the same letter snap together. When angles snap, the rays are extended to the edge of the work area.
      • When you create a closed triangle, the points merge and center is filled in.
      • Once a triangle is formed with the original three elements, the triangle moves to the bottom right corner of the work area, and congruent elements appear. Try to form a second triangle.
      • If a second triangle is successfully formed, you will be asked if they are congruent. You can test congruence by manipulating either triangle.
        • Click and drag within the triangle to move it to a new location.
        • Click and drag a vertex to rotate the triangle.
        • Use the Flip button to reflect the triangle horizontally. First click on the triangle you would like to reflect, and then click the Flip button.
        • If the two triangles are not congruent, you have successfully disproved a theorem. If the two triangles are congruent, you will be asked if to make a triangle that is not congruent to the original. You will be given the option to try again.
        • The Reset button clears the work area and creates new sides and angles for the selected elements.
        • The New button clears your selection and work area.


        3502 TrianglesFor every arrangement of three elements, it is possible to test for triangle congruence. However, just like two triangle might be mirror images but still congruent, flipping the order creates identical possible theorems. For example Angle – Angle – Side is the same as Side – Angle – Angle because they are the same elements in reverse order.

        How many possibilities are there? Test each of them and organize your results in the table with three columns:

        • Congruence Theorem
        • Does it prove congruence?
        • Sketch

        As you work, remember to try every possibility. When creating your second triangle, see if an element can connect to adjacent elements in more than one place. Also remember, you may have to turn or flip your triangles to see how they are congruent. For example, all the triangles to the right are congruent.

        Did all the possibilities work? How are you certain of your answers? Try comparing your answers to a friend's. If you disagree on any possible theorem, try re-creating your triangles.

        Objectives and Standards

        NCTM Standards and Expectations
        • Geometry / Measurement
        • Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.
        • High School (9-12)
        • Geometry