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Triangle Definitions Help (page 3)

By — McGraw-Hill Professional
Updated on Oct 3, 2011

Inverse Congruence Facts

Here are two “mirror images” of the facts just stated. They concern triangles that are inversely congruent. The wording is almost (but not quite) the same!

Inverse Congruence Fact 1: Corresponding Sides

  • If two triangles are inversely congruent, then their corresponding sides have equal lengths as you proceed around the triangles in opposite directions. The converse of this is also true. If two triangles have corresponding sides with equal lengths as you proceed around them in opposite directions, then the two triangles are inversely congruent.

Inverse Congruence Fact 2: Corresponding Interior Angles

  • If two triangles are inversely congruent, then their corresponding interior angles have equal measures as you proceed around the triangles in opposite directions. The converse of this is not necessarily true. It is possible for two triangles to have corresponding interior angles with equal measures as you proceed around them in opposite directions, and yet the two triangles are not inversely congruent.

Point-Point-Point Principle

Let P, Q , and R be three distinct points that do not all lie on the same straight line. Then the following statements are true (Fig. 2-4):

  • P, Q , and R lie at the vertices of some triangle T
  • T is the only triangle having vertices P, Q , and R

 

Triangles Triangle Definitions Point-point-point Principle

Fig. 2-4 . The three-point principle; side-side-side triangles.

Triangle Definition Practice Problems

PROBLEM 1

Suppose you have a perfectly rectangular field surrounded by four straight lengths of fence. You build a straight fence diagonally across this field, so the diagonal fence divides the field into two triangles. Are these triangles directly congruent? If they are not congruent, are they directly similar?

SOLUTION 1

It helps to draw a diagram of this situation. If you do this, you can see that the two triangles are directly congruent. Consider the theoretical images of the triangles (which, unlike the fences, you can move around in your imagination). You can rotate one of these theoretical triangles exactly 180° (π rad), either clockwise or counterclockwise, and move it a short distance upward and to the side, and it will fit exactly over the other one.

PROBLEM 2

Suppose you have a telescope equipped with a camera. You focus on a distant, triangular sign and take a photograph of it. Then you double the magnification of the telescope and, making sure the whole sign fits into the field of view of the camera, you take another photograph. When you get the photos developed, you see triangles in each photograph. Are these triangles directly congruent? If not, are they directly similar?

SOLUTION 2

In the photos, one triangle looks larger than the other. But unless there is something wrong with the telescope, or you use a star diagonal when taking one photograph and not when taking the other (a star diagonal renders an image laterally inverted), the two triangle images have the same shape in the same rotational sense. They are not directly congruent, but they are directly similar.

Practice problems for these concepts can be found at: Triangle Practice Test.

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