Triangle Congruence Theorems
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The experiments in the Measuring Triangles
lesson may have convinced you of these Theorems. Remember, though,
that experiments with specific triangles are not a substitute for
a proof for all triangles.
Some basic facts
Here are some basic facts (axioms and theorems) that will be used in the proofs.
Definitions: A line segment is the set of all points on a line between two
points, called the endpoints.
A ray is the set of all points on a line on one side of a point, called its endpoint.
Two lines in the plane that don't intersect are called parallel.
If you have a copy of Geometer's Sketchpad installed, you can download
the original sketch.
Axiom: Two lines which are not parallel intersect in one point.
Theorem: Two rays or line segments which are not parallel
intersect in one point or not at all.
Theorem: A line and a circle intersect in 0, 1, or 2 points. If 1 point, the
line is called tangent to the circle.
Theorem: Two circles intersect in 0, 1, or 2 points.
The Triangle Inequality and the Side-Side-Side Triangle Congruence Theorem.
Theorem: The Triangle Inequality.
(a)
In a triangle, the sum of the lengths
of any two sides is greater than the length of the third side.
(b) If, for three positive real numbers a, b, and c, the sum of any two is greater than the third, then there is a triangle with sides a, b, and c.
The Side-Side-Side Triangle Congruence Theorem. (SSS for short.)
If two triangles have the same side lengths, then the triangles are congruent.
Another way to think of this theorem is that given three sides,
there is only one triangle with these measurements.
Make the triangle disappear by making one side too long. Try it for each side.
(This illustrates the triangle inequality.)
Change the lengths of the three sides to get different triangles.
(This illustrates the SSS theorem.)
If you have a copy of Geometer's Sketchpad installed, you can download
the original sketch.
The Side-Angle-Side Triangle Congruence Theorem. (SAS for short.)
If two triangles have two sides and the included angles equal, respectively, then the triangles are congruent.
Another way to think of this theorem is that given two sides and an included
angle, there is only one triangle with these measurements.
Change segments AB and AC and angle A to get different triangles.
This illustrates the SAS theorem.
If you have a copy of Geometer's Sketchpad installed, you can download
the original sketch.
The Angle-Side-Angle Triangle Congruence Theorem. (ASA for short.)
If two triangles have two angles and the included sides equal, respectively, then the triangles are congruent.
Another way to think of this theorem is that given two angles and an included
side, there is only one triangle with these measurements.
Change segment BC and angles B and C to get different triangles.
This illustrates the ASA theorem.
If you have a copy of Geometer's Sketchpad installed, you can download
the original sketch.
There is no Angle-Side-Side Theorem.
(You can easily rembember this
using the acronym ASS.)
- There is no triangle with those measurements.
- There is exactly one triangle with those measurements.
- There are two non-congruent triangles with those measurements.
So two triangles with two sides and a non-included angle equal may or may
not be congruent.
Use the buttons to see the two different triangles the measurements determine.
Can you adjust the sides and/or angle to make the two triangles become one?
Can you adjust the sides and/or angle to make the triangles disappear?
If you have a copy of Geometer's Sketchpad installed, you can download
the original sketch.
The Angle-Angle Triangle Similarity Theorem.
If two angles on one triangle are equal, respectively, to two angles on another triangle, then the triangles are similar.
Change the length of segment AB to see that you get a similar triangle.
Change the angles A and B to get a different shape of triangle.
This illustrates the AA theorem.
If you have a copy of Geometer's Sketchpad installed, you can download
the original sketch.
Individual problems
"Ruler" means you can use the markings.
- Use a ruler and compass and the technique in the SSS diagram to either construct a triangle with the given meausrements, or to show none exists.
- 4 inches, 5 inches, and 6 inches
- 2 inches, 3 inches, and 6 inches
- Use a ruler and protractor, and the technique in the SAS diagram to either construct a triangle with the given meausrements.
- Use a ruler and protractor and the technique in the ASA diagram to either construct a triangle with the given meausrements, or to show none exists.
- 100o, 3 inches, 80o
- 30o, 3 inches, 50o
- Use a ruler, compass, and protractor and the technique in the ASS diagram to either construct all possible triangles with the given meausrements, or to show none exists.
Make sure you have the angle and sides in the correct positions.
- 30o, 4 inches, 1 inches
- 30o, 4 inches, 2 inches
- 30o, 4 inches, 3 inches
- 30o, 4 inches, 4 inches
- 30o, 4 inches, 5 inches
- Use a ruler and protractor and the technique in the AA diagram to either construct two triangles with the given meausrements, or to show none exists.
- (Extension:) Find precise conditions on the A, S, and S in the ASS
diagram that will allow you to predict whether given measurements will determine
2, 1, or no triangles. Prove your answer if you can. It will help if you use the
interactive sketch on the Web.
This is a prototype of JavaSketchpad, a World-Wide-Web component of The Geometer's Sketchpad. Copyright ©1990-1997 by Key Curriculum Press, Inc. All rights reserved. Portions of this work are being funded by the National Science Foundation (awards DMI 9561674 & 9623018).
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