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Congruent Right Triangles In this section, you will extend your knowledge of congruence into the realm of right triangles. You will learn shortcuts for proving congruence in right triangles. Then you will use these congruence theorems to discover other new and useful facts. Objectives Experiment with congruent triangles. Explore the right triangle congruence shortcuts theorems — HL, LL, HA, LA. Discover the relationship between the right triangle congruence theorems and the congruence theorems for nonright triangles. Prove the Perpendicular Bisector Theorem. Prove the Angle Bisector Theorem and its Converse. Congruent Right Triangles To begin, recall what you know about a pair of congruent triangles: they have the same size and same shape, and all of their corresponding side lengths and angle measures are equal. What if the two congruent triangles are right triangles? You can experiment with some here. Notice that congruent right triangles have the same properties that were listed on the previous slide — same size and shape, three pairs of congruent sides, and three pairs of congruent angles. Also, recall the congruence postulates and theorems that “work” for all triangles. Can these be simplified for right triangles? After all, two right triangles automatically have one pair of congruent angles already — the right angles. You will try to build a right triangle below by requesting the lengths of the hypotenuse and one leg. You will be given two sets of these pieces so that you can build two right triangles. Each set of pieces includes a right angle, hypotenuse, and leg. Are they congruent? Is it possible to build two triangles that are not congruent? You may have seen that your triangles were always congruent. If two right triangles have congruent hypotenuses and one pair of congruent legs, the triangles are congruent. This is the HL Congruence Theorem. (“HL” stands for “hypotenuse-leg”.) HL Theorem: If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and corresponding leg of another right triangle, then the triangles are congruent. Given: A and Conclusion: D are right angles. ABC DEF Reason: HL Try the experiment again, this time requesting the lengths of two legs of a right triangle. Again, you will be given two sets of the legs you requested, along with a right angle to go with each set of legs. Build two triangles and see if congruence is guaranteed. You may have noticed that if both pairs of legs of right triangles are congruent, the triangles themselves are guaranteed to be congruent. You have discovered the LL Congruence Theorem. (“LL” stands for “leg-leg”.) LL Theorem If the legs of one right triangle are congruent to the corresponding legs of another right triangle, then the triangles are congruent. Given: I and Conclusion: L are right angles. GHI JKL Reason: LL So why does HL guarantee congruence? Recall that when you know two side lengths in a right triangle, you can calculate the third with the Pythagorean Theorem. So, if two pairs of sides are congruent, the third pair must also be congruent. By the same argument you could conclude that LL also guarantees congruence. Since LL means that two pairs of legs are congruent, you know by the Pythagorean Theorem that the hypotenuses are also congruent. Thus, LL is also a special case of SSS. You can verify this for yourself in the example below. Thus, HL is just a special case of SSS congruence! Are there more shortcut congruence theorems for right triangles? Experiment with “hypotenuse-angle”. Request a hypotenuse length and an acute angle measure. Then use those pieces to build two right triangles. Are they guaranteed to be congruent? Indeed, HA, or “hypotenuse-angle”, works. In other words, if two right triangles have congruent hypotenuses and one pair of congruent angles (in addition to the congruent right angles), the triangles are guaranteed to be congruent. HA Theorem: If the hypotenuse and an acute angle of one right triangle are congruent to the hypotenuse and corresponding acute angle of another right triangle, then the triangles are congruent. Given: M and O Conclusion: P are right angles. R MNO Reason: HA PQR Now experiment with “leg-angle.” Note that there are two possible legs to consider. One leg is included between the right angle and the other given angle, and one leg is not. First, experiment with the leg that is included between the two given angles. Are two triangles built with these pieces always congruent? What about the other “leg-angle” case, in which the leg is not included between the given angles? Does this guarantee congruent triangles? Experiment below. You may have noticed that both cases of “leg-angle” guarantee congruence. This is the fourth and final right triangle congruence theorem, the LA Congruence Theorem. LA Theorem: If one leg and an acute angle of one right triangle are congruent to one leg and an acute angle of another right triangle, then the two right triangles are congruent. How do HA and LA relate to the congruence theorems for nonright triangles? When you consider that the right angles are automatically congruent, you can see that they are simplified versions of either ASA or AAS. You have now discovered four congruence theorems for use with right triangles. They are summarized in the table below. Notice that each is really just a special case of an earlier congruence theorem. In addition, each has only two letters. Can you see why, with right triangles, two is enough? Well done. The hypotenuses are congruent, but that is not enough to ensure congruence. HA or HL would work, but just H is not good enough. True or False? Based on the information marked in the diagram, ABC DEF. True Very nice. You have two pairs of sides and their included angles congruent (SAS). But this is also a right triangle, and the congruent sides are both legs, so you could also use LL. Based only on the information marked in the diagram, which two congruence theorems could be given as reasons why ABC DEF? A. HL B. LLC. HAD. LAE. ASAF. SAS Fine work. If you do not look at these as right triangles, you have the AAS case, because the congruent side is not included between the congruent angles. When you do consider them as right triangles, you have hypotenuse-angle ("HA") congruence. Based only on the information marked in the diagram, which two congruence theorems could be given as reasons why GHI JKL? A. HLB. LL C. HAD. LAE. AASF. ASA Excellent. As "non-right" triangles, this is the ASA case, because the congruent sides are included between the congruent angles. As right triangles, this is the leg-angle ("LA") case. Based only on the information marked in the diagram, which two congruence theorems could be given as reasons why MNO PQR? D. LAE. AASF. ASA A. HLB. LLC. HA It's time to put the right triangle congruence theorems you've learned to work for you. The next several pages present four useful new theorems about points, lines, and angles. A key to each proof is congruent right triangles — even though the theorems themselves are not about triangles at all! Before getting to the proofs, be sure you recall that the distance from a point to a line is always measured perpendicularly. This is illustrated here. The first theorem describes how any point on a perpendicular bisector is related to the segment it bisects. Perpendicular Bisector Theorem If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. Experiment to confirm this for yourself. Once you have seen that it seems to be true, click through the proof. The converse of the Perpendicular Bisector Theorem is also true. Perpendicular Bisector Converse: If a point is equidistant from the endpoints of a segment, then it lies on the perpendicular bisector of that segment. Experiment to convince yourself of this, then click through the sequence of buttons to see a proof. Again, you will see that right triangle congruence is a key to the proof. Just as a segment can be bisected by a line, so can an angle. The Angle Bisector Theorem relates any point on the angle bisector to the sides of the angle. Angle Bisector Theorem If a point is on the bisector of an angle, then it is equidistant from the two sides of the angle. Experiment to confirm this theorem. Then click through the sequence of buttons to see a proof. The converse of the Angle Bisector Theorem is true as well. Angle Bisector Converse: If a point is in the interior of an angle and equidistant from the two sides of the angle, then it lies on the bisector of the angle. Experiment to confirm this for yourself. Then click through the sequence of buttons to see a proof. Notice the use, once again, of a right triangle congruence theorem. Nice job! The distance from a point to a line is always measured by the perpendicular segment. (This is also always the shortest distance from the point to the line.) What is the distance from point C to ? A. B. 10.8 10.5 C. 14.6 D. can't be determined Well done. Point S is equidistant from the sides of So, if m SED = 30 , then m What is m RED. Therefore, S is on the angle bisector. RED = 60 . RED? A. 15 B. 17 D. 34 C. 30 E. 60 F. can't be determined Outstanding! There is no guarantee that because you do not know if bisects True or false? Based only on the given information, it is guaranteed that Given: . . ABC False Excellent. In this example, you know that point C is on the perpendicular bisector of you can conclude that C is equidistant from A and B, or, in other words, . . Therefore, True or false? Based only on the given information, it is guaranteed that Given: . ABC True You have seen a great deal about congruence in right triangles in this section. Four congruence theorems, given again in the table below, were presented and then used in proofs. You are now ready to extend this knowledge into the topic of similar right triangles.