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Congruent Triangles
Introduction: We examine two triangles which are congruent because all corresponding angles and sides have the same measures. We discuss circumstances which guarantee that two triangles are congruent.

Definition: Two triangles are said to be congruent if the corresponding angles and sides have the same measurements. This means that there are six corresponding parts with the same measurements. Under certain circumstances, it is only necessary to know that three corresponding parts have the same measures in order to guarantee that two triangles are congruent. We summarize these situations below without proof.
  1. SSS – Two triangles are congruent if all three sides have equal measures.
  2. SAS – Two triangles are congruent if two sides and the included angle have the same measures.
  3. ASA – Two triangles are congruent if two angles and the side between them have the same measures.
  4. AAS – Two triangles are congruent if two angles and a third side have the same measure.
  5. HL – Two right triangles are congruent if their hypotenuses and one leg have the same measure.
  6. HA – Two triangles are congruent if their hypotenuses and one of the acute angles have the same measure.
As soon as the above facts are established, we know that the two triangles are congruent and also that all six of their corresponding angles and sides have the same measures. When two triangles such as ABC and XYZ are congruent and angles correspond as follows: A→X, B→Y, and C→Z we write this congruence as . We illustrate some of these situations below.
Let's Practice:
  1. For situation #1 SSS, suppose that in the diagram below we have sides QR = AB, RP = BC, and sides PQ = CA.
These triangles must be congruent, and therefore the corresponding angles A→Q, B→R, and C→P must also be congruent or have the same measure. We have
  1. In the diagram below, there are two overlapping triangles AQP and BPR. If we know that , are these triangles congruent?
Since these triangles share a common side PQ, in this situation, we have AAS as in #4. This is because two corresponding angles are equal and a third side PQ, shared by both triangles, must be the same in both triangles. Therefore the other three corresponding parts must also be equal. For example, sides AQ and BP have the same measure. We have .
  1. In the diagrams below, assume that each hypotenuse of these right triangles, XY and PQ, has a measure of 4 feet and that one of the acute angles in each has a measure of 43º. What else do we know about these triangles?
Since the hypotenuses and one acute angle of each triangle are equal, we have HA as in #6 above. Therefore, these triangles are congruent and all six corresponding parts are equal in measure. For example, both angles X and P must have measure 47º in order for the acute angles to be complementary and for the three angles of these triangles to have a sum of 180º. We have .

Examples
Example Consider two triangles ABC and XYZ. Suppose that angles A and X both have measure 32º and angles B and Y both have measures 79º. If we also know that side AB = XY, what else do we know about these two triangles?
What is your answer?
 
Example In right triangles PQR and ABC, suppose that the hypotenuse AB has the same measure as the hypotenuse PQ. Further, suppose that leg QR = BC. What else do we know about these triangles?
What is your answer?
 



M Ransom

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