Triangle Congruence and Similarity Help (page 2)
Introduction to Triangle Congruence and Similarity
There are four criteria that can be used to define sets of triangles that are directly congruent. These are called the side–side–side (SSS), side–angle–side (SAS), angle–side–angle (ASA), and angle–angle–side (AAS) principles. The last of these can also be called side–angle–angle (SAA). A fifth principle, called angle–angle–angle (AAA), can be used to define sets of triangles that are directly similar.
Let S, T , and U be defined, specific line segments. Let s, t , and u be the lengths of those three line segments, respectively. Suppose that S, T , and U are joined at their end points P, Q , and R (Fig. 2-4). Then the following statements hold true:
- Line segments S, T , and U determine a triangle
- This is the only triangle that has sides S, T , and U in this order, as you proceed around the triangle in the same rotational sense
- All triangles having sides of lengths s, t , and u in this order, as you proceed around the triangles in the same rotational sense, are directly congruent
Let S and T be two distinct line segments. Let P be a point that lies at the ends of both of these line segments. Denote the lengths of S and T by their lowercase counterparts s and t , respectively. Suppose S and T subtend an angle x , expressed in the counterclockwise sense, at point P (Fig. 2-5). Then the following statements are all true:
- S, T , and x determine a triangle
- This is the only triangle having sides S and T that subtend an angle x , measured counterclockwise from S to T , at point P
- All triangles containing two sides of lengths s and t that subtend an angle x , measured counterclockwise from the side of length s to the side of length t , are directly congruent
Let S be a line segment having length s , and whose end points are P and Q . Let x and y be the angles subtended relative to S by two lines L and M that run through P and Q , respectively (Fig. 2-6), such that both angles are expressed in the counterclockwise sense. Then the following statements are all true:
- x , S , and y determine a triangle
- This is the only triangle determined by x , S , and y , proceeding from left to right
- All triangles containing one side of length s , and whose other two sides subtend angles of x and y relative to the side whose length is s , with x on the left and y on the right and both angles expressed in the counterclockwise sense, are directly congruent
Angle–Angle–Side (aas) Or Side–Angle–Angle (saa)
Let S be a line segment having length s , and whose end points are P and Q . Let x and y be angles, one adjacent to S and one opposite, and both expressed in the counterclockwise sense (Fig. 2-7). Then the following statements are all true:
- S, x , and y determine a triangle
- This is the only triangle determined by S, x and y in the counterclockwise sense
- All triangles containing one side of length s , and two angles x and y expressed and proceeding in the counterclockwise sense, are directly congruent
Let L, M , and N be lines that lie in a common plane and intersect in three points as illustrated in Fig. 2-8. Let the angles at these points, all expressed in the counterclockwise sense, be x , y , and z . Then the following statements are all true:
- There are infinitely many triangles with interior angles x , y , and z , in this order and proceeding in the counterclockwise sense
- All triangles with interior angles x , y , and z , in this order, expressed and proceeding in the counterclockwise sense, are directly similar
Let It Be So!
Are you wondering why the word “let” is used so often? For example, “Let P, Q , and R be three distinct points.” This sort of language is customary. You’ll find it all the time in mathematical literature. When you are admonished to “let” things be a certain way, you are in effect being asked to imagine, or suppose, that things are such, to set the scene in your mind for statements or problems that follow.
Triangle Congruence and Similarity Practice Problems
Refer to Fig. 2-6. Suppose x and y both measure 60°. If the resulting triangle is reversed from left to right—that is, flipped over around a vertical axis—will the resulting triangle be directly similar to the original? Will it be directly congruent to the original?
This is a special case in which a triangle can be flipped over and the result is not only inversely congruent, but also directly congruent, to the original. This is the case because the triangle is symmetrical with respect to a straight-line axis. To clarify this, draw a triangle after the pattern in Fig. 2-6, but using a protractor to generate 60° angles for both x and y. (As it is drawn in this book, the figure is not symmetrical and the angles are not both 60°.) Then look at the image you have drawn, both directly and while standing in front of a mirror. The two mirror-image triangles are, in this particular case, identical.
Suppose, in the situation of Problem 1 you split the triangle, whose angles x and y both measure 60°, right down the middle. You do this by dropping a vertical line from the top vertex so it intersects line segment PQ at its midpoint. Are the resulting two triangles, each comprising half of the original, directly similar? Are they directly congruent? Are they inversely similar? Are they inversely congruent?
These triangles are mirror images of each other, but you cannot magnify, reduce, and/or rotate one of these triangles to make it fit exactly over the other. The triangles are not directly similar, nor are they directly congruent, even though, in a sense, they are the same size and shape.
Remember that for two triangles to be directly similar, the lengths of their sides must be in the same proportion, in order, as you proceed in the same rotational sense (counterclockwise or clockwise) around them both. In order to be directly congruent, their sides must have identical lengths, in order, as you proceed in the same rotational sense, around both.
These two triangles are inversely similar and inversely congruent, because they are mirror images of each other and are the same size.
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