Theorems and Postulates for Congruent Triangles

The goal of this dynamic worksheet is to explore the theorems for congruent triangles.

  1. First you need to learn how this applet works. Click on some angles and sides of the given triangle ABC. When an element of the triangle is selected it will change its color and a free congruent element will appear to the right of triangle ABC.
  2. The free elements to the right have the functionalities listed below. Try  each of them.

    • The segments can be moved to a different location on the screen by dragging the solid point.
    • The segments can be rotated by dragging the empty point.
    • You can construct a circle with center at the solid point and a radius equal to the length of the segment by clicking the empty point.

    • The angles can be moved to a different location by dragging the thicker side.
    • The angles can be rotated by dragging the empty point or the vertex.
    • The sides of the angles can be extended by dragging the endpoints.

  3. Constructions:
    • Select three elements of the given triangle ABC (any combination of sides and angles) by clicking on them.
    • Using these elements try to construct a triangle congruent to the given triangle ABC.
    • If your construction has sides parallel to the corresponding sides of the given triangle, you can check your construction by dragging triangle ABC over the constructed triangle.
  4. Will all possible combinations of three elements work?
  5. There are detailed instructions and videos below the applet.

Irina Boyadzhiev, Created with GeoGebra

Consider the following cases:
  1. SSS - click on all sides of triangle ABC. Use the three free sides to the right of the triangle to construct a congruent triangle to the given one. You can see a short video of the construction here

  2. ASA - click on any two angles and the included side. Use the three elements to the right of triangle ABC to construct a triangle, congruent to the given triangle ABC.

  3. SAS - click on any two sides and the included angle. Use the three elements to the right of triangle ABC to construct a triangle, congruent to the given triangle ABC. Notice that with the given elements you will be able to find just the endpoints of the third side. If you want to connect them, you can click on the corresponding side of the triangle ABC and use the free segment to connect the points.

  4. SSA - click on any two sides and one non-included angle. Use the three elements to the right of triangle ABC and try to construct a triangle, congruent to the given triangle ABC. Will this always work? You can see a short video here.

  5. A Special case 1 of SSA - HL.  Click on the button under the triangle to change angle β to a right angle. Try to construct a congruent triangle to the given one by using a leg, the hypotenuse and the right angle. You can see a video of the construction here.

  6. A Special case 2 of SSA - click on two sides and the angle across the longer of the selected sides. Use these three elements to construct a triangle congruent to the given triangle ABC. You can see a video of the construction here.

  7. AAA- Is there exactly one triangle with these angles?  Is this a theorem for congruence of triangles?
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