Free Math Interactive | Congruence Theorem

Congruence TheoremThis high school level math interactive was developed to get students to explore conditions that result in congruent triangles. Based on a previous applet that was available on the NCTM Illuminations website, our team was asked to redesign and make the interaction of this applet more user-friendly and intuitive.


Students start by selecting any three elements in the top right corner of the screen. For example, a student may choose three sides, or one side and two angles, or two sides and one angle.

The elements will appear in the blue area of the screen, and students can then click and drag the elements to the appropriate points to create a triangle. To move an element, just click and drag the center dot. To rotate an element, click and drag an endpoint.

When two elements are aligned, they will “snap” together. When all three elements have been arranged to correctly form a triangle, all elements will snap together and the inside of the triangle will be filled in.

After the first triangle is formed, a new set of elements will appear on the screen. Students can then use the congruent elements to make another triangle. They will be asked to identify whether the second triangle is congruent to the first one and click and drag the triangle to move it over the first triangle. The Flip button can also be used to perform a horizontal reflection.

If the two triangles are congruent, the student will be asked if it’s possible to make a triangle that is not congruent to the original. Encourage students to keep exploring and not jump to conclusions from just one example. Ask students to write down and explain which sets of conditions they think lead to congruency.

Grade Levels

Grades 9-12.


If you like this applet, you can find more ideas on how to use it on the NCTM Illuminations website.

Math Curriculum and Math Games by Smart Tutor

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