Download TRIANGLE CONGRUENCE

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?