Congruent Triangles

This manipulative allows you to construct two triangles from various combinations of sides and angles. You can choose to work with any one of four different cases.

You are given three red segments and three blue segments, with each blue segment matching a red segment. Each segment can be dragged and rotated. Drag the end point of one segment to coincide with the end point of another segment of the same color; they will snap together. Then drag the end of the third segment of the same color onto a free end of the first two. By rotating the outer segments of your three, you should be able to get their free ends together. When they are close enough, the two will snap together and you will get a solid triangular region. Repeat the process with the segments of the other color. Now you have a red triangle and a blue triangle, both with matching side lengths. Drag and rotate the triangles to get matching sides together. It may be necessary to press the FLIP button to flip one triangle over before you can match all three sides. If the triangles are congruent, when you have matched sides the colors will switch, ending in a green triangle.

SAS Case (Two sides and the included angle).

You are given two red segments and a red angle, and three matching blue pieces. Drag and rotate the red angle so its vertex is on top of one end point of the red segment and one side of the angle lies along the segment. Then place one end of the other red segment on the vertex of the red angle and rotate the segment until it lies along the other side of the angle. When the segment coincides with the side of the angle, you get a solid red triangular region with the two given sides and included angle. Repeat the process to get a blue triangle. Drag and rotate the triangles to get matching sides together. It may be necessary to press the FLIP button to flip one triangle over before you can match all three sides. If the triangles are congruent, when you have matched sides the colors will switch, ending in a green triangle.

ASA Case (Two angles and the included side).

You are given two red angles and a red segment, and three matching blue pieces. Drag and rotate a red angle so its vertex is on top of an end point of the red segment and one side of the angle lies along the segment. Then place the second red angle at the other end of the segment and rotate the angle until it lies along the segment, so that the free ends of the angles are on the same side of the red segment. You get a solid red triangular region with the two given angles and included side. Repeat the process to get a blue triangle. Drag and rotate the triangles to get matching sides together. It may be necessary to press the FLIP button to flip one triangle over before you can match all three sides. If the triangles are congruent, when you have matched sides the colors will switch, ending in a green triangle.

SSA Case (Two sides and an angle opposite one of them, often called the Ambiguous Case)

You are given two red segments and a red angle, and three matching blue pieces, but in this case the angle is not included between the given sides. Place the red angle at one end of a red segment with one side of the angle along the segment. Then attach the other red segment to the end of the segment opposite the attached angle. The angle determines the line along which the third side must lie. Rotate the free end of your second segment until the free end is on the line. You will get a solid red triangular region with the given sides and angle. Repeat the process to get a blue triangle, but your goal is to rotate the third blue side so that your blue triangle is NOT CONGRUENT to the red triangle. If your triangles are congruent, you get a message inviting you to Click and start over, to construct two noncongruent triangles, each having the given sides and the given angle opposite one of the sides.