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TRIANGLE CONGRUENCE SSS (Side-Side-Side) Postulate 1.) Using straws and string, construct a triangle that has sides of 2 inches, 2 inches, and 3 inches. Tie the string tightly so that there is no slack. 2.) Remember, you only know the measures of the three sides of the triangle. You do not have any information about the measures of the angles. 3.) Compare your triangle with your partner’s triangle. Place the corresponding sides on top of one another. Are the triangles congruent? 4.) Based on your findings… THE SSS Postulate for Triangle Congruence SAYS THAT IF YOU KNOW THAT THE _____________ OF ONE TRIANGLE ARE CONGRUENT TO THAT OF ANOTHER TRIANGLE, THEN THE TRIANGLES ARE CONGRUENT. SAS (Side-Angle-Side) Postulate 1.) Draw a segment that is 2 inches long on a sheet of notebook paper. Label it AB. At Point A, draw a second segment that is also 2 inches long. such that a 45˚ angle is formed and label it AC. (See the diagram below.) 2in. 45˚ A 2in. B 2.) Extend the side with the 45˚ angle. Then, draw in the third side. Remember, you only know the measure of two sides and their included angle (the angle in between those sides). You do not know any other information. 3.) Compare your triangle with your partner’s triangle. Place the corresponding points on top of one another. Hold the triangles up to the light. Are the triangles congruent? 4.) Based on your findings… THE SAS Postulate for Triangle Congruence SAYS THAT IF YOU KNOW THAT TWO ______________ OF A TRIANGLE AND ITS _____________ ____________ ARE CONGRUENT TO THAT OF ANOTHER TRIANGLE, THEN THE TRIANGLES ARE CONGRUENT. ASA (Angle-Side-Angle) Postulate 1.) Draw a segment that is 2 inches long on a sheet of notebook paper. Label it AB. At Point A, construct a 30˚ angle. At Point B, construct a 50˚ angle. (See the diagram below.) 30˚ 50˚ 2in. A B 2.) Extend the sides making the 30˚ and 50˚ angles to make a triangle. Remember, you know the measures of two angles and their included side (the side between the two angles). You do not know any other information. 3.) Compare your triangle with your partner’s triangle. Place the corresponding points on top of one another. Hold the triangles up to the light. Are the triangles congruent? 4.) Based on your findings… THE ASA Postulate for Triangle Congruence SAYS THAT IF YOU KNOW THAT TWO ______________ OF A TRIANGLE AND ITS _____________ ____________ ARE CONGRUENT TO THAT OF ANOTHER TRIANGLE, THEN THE TRIANGLES ARE CONGRUENT. AAS (Angle-Angle-Side) Postulate 1.) The AAS Postulate is much easier to demonstrate using Geometer’s Sketchpad than using hands-on manipulatives. 2.) Construct a segment on Geometer’s Sketchpad that is 2 inches long. Then, construct a second adjacent segment off of one end of the original segment that is any length, and create a 30˚ angle where the two segments meet. (See the diagram below.) Off of the end of the second unmeasured segment, create a 100˚ angle. Finally, connect the two segments by drawing in a third segment to form a triangle. You do not know any other information. 100˚ 30˚ 2in. 3.) Construct a second triangle that includes the same three pieces that must be measured as the first triangle (one segment and two angles following that segment); however, do not measure the other two sides or the last angle. Try dragging the triangles on top of one another. Are the triangles congruent? 4.) Based on your findings... THE AAS Postulate for Triangle Congruence SAYS THAT IF YOU KNOW THAT TWO ______________ OF A TRIANGLE AND AN ADJACENT _______________ ARE CONGRUENT TO THAT OF ANOTHER TRIANGLE, THEN THE TRIANGLES ARE CONGRUENT. AAA (Angle-Angle-Angle) 1.) Using Geometer’s Sketchpad, construct a triangle making sides of any measures. Measure the three angles in the triangle using the Geometer’s Sketchpad measuring tools. 2.) Try constructing a second triangle that is either larger or smaller than the first triangle but that has the same three angle measures. Is it possible to make a triangle that has the same angle measures but is a different size? 3.) Based on your findings... IS IT POSSIBLE TO PROVE TRIANGLES CONGRUENT IF THE THREE ANGLES IN ONE TRIANGLE HAVE THE SAME MEASURES AS THE THREE ANGLES IN A SECOND TRIANGLE? SSA (Side-Side-Angle) 1.) Using Geometer’s Sketchpad, construct a segment that is 2 inches long, a second adjacent segment that is 1.5 inches long, and an adjacent angle off of the end of the second segment that is an x˚ angle. (See Diagram #1 below. The angle will have be whatever degree measure is needed to form a triangle based on the way the sides were laid out.) Measure the x˚ angle. DIAGRAM #1 DIAGRAM #2 2in. 2in. 1.5 in. 1.5 in. x˚ x˚ 2.) Try constructing a second triangle that also has two adjacent sides measuring 2 inches long and 1.5 inches long and that has the same non-included angle measuring x˚. However, place the sides in a different set up than the last triangle (See Diagram #2 above.) Is it possible to make two triangles that have the given measurements in the requested locations but that are not congruent? 3.) Based on your findings... IS IT POSSIBLE TO PROVE TRIANGLES CONGRUENT IF TWO CONSECUTIVE SIDES AND A NON-INCLUDED ANGLE OF ONE TRIANGLE HAVE THE SAME MEASURES AS TWO CONSECUTIVE SIDES AND A NON-INCLUDED ANGLE OF A SECOND TRIANGLE?