6.3 HL Triangle Congruence
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1 Name lass ate 6.3 HL Triangle ongruence Essential Question: What does the HL Triangle ongruence Theorem tell you about two triangles? Explore Is There a Side-Side-ngle ongruence Theorem? Resource Locker You have already seen several theorems for proving that triangles are congruent. In this Explore, you will investigate whether there is a SS Triangle ongruence Theorem. Follow these steps to draw such that m = 30, = 6 cm, and = 4 cm. The goal is to determine whether two side lengths and the measure of a non-included angle (SS) determine a unique triangle. Use a protractor to draw a large 30 angle on a separate sheet of paper. Label it. Use a ruler to locate point on one ray of so that = 6 cm. 30 Now draw so _ that = 4 cm. To do this, open a compass to a distance of 4 cm. Place the point of the compass on point and draw an arc. Plot point where the arc intersects the side of. raw _ to complete. 6 cm 30 What do you notice? Is it possible to draw only one with the given side length? Explain. 6 cm 4 cm Houghton ifflin Harcourt Publishing ompany Reflect 1. o you think that SS is sufficient to prove congruence? Why or why not?. iscussion Your friend said that there is a special case where SS can be used to prove congruence. Namely, when the non-included angle was a right angle. Is your friend right? Explain. 30 odule 6 95 Lesson 3
2 Explain 1 ustifying the Hypotenuse-Leg ongruence Theorem In a right triangle, the side opposite the right angle is the hypotenuse. The two sides that form the sides of the right angle are the legs. hypotenuse You have learned four ways to prove that triangles are congruent. legs ngle-side-ngle (S) ongruence Theorem Side-Side-Side (SSS) ongruence Theorem Side-ngle-Side (SS) ongruence Theorem ngle-ngle-side (S) ongruence Theorem The Hypotenuse-Leg (HL) Triangle ongruence Theorem is a special case that allows you to show that two right triangles are congruent. Hypotenuse-Leg (HL) Triangle ongruence Theorem If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of another right triangle, then the triangles are congruent. Example 1 Prove the HL Triangle ongruence Theorem. E Given: and EF are right triangles; and F are right angles. _ _ E and EF a b c d F e f Prove: EF y the Pythagorean Theorem, a + b = c and + = f. It is given that _ _ E, so = E and c = ƒ. Therefore, c = f and a + b = +. It is given that _ _ EF, so = EF and a = d. Substituting a for d in the above equation, a + b = +. Subtracting a from each side shows that b =, and taking the square root of each side, b =. This shows that _. Therefore, EF by. Your Turn 3. etermine whether there is enough information to prove that triangles VWX and YXW are congruent. Explain. W V Z Y X Houghton ifflin Harcourt Publishing ompany odule 6 96 Lesson 3
3 Explain pplying the HL Triangle ongruence Theorem Example Use the HL ongruence Theorem to prove that the triangles are congruent. Given: P and R are right angles. _ PS _ RQ Prove: PQS RSQ P S Q R 1. P and R are right angles. 1. Given. _ PS _ RQ. Given 3. _ SQ _ SQ 3. Reflexive Property of ongruence 4. PQS RSQ 4. HL Triangle ongruence Theorem Given: and L are right angles. is the midpoint of _ L and _ N. Prove: N L L N 1. and L are right angles. 1.. is the midpoint of _ L and _ N.. 3. _ L and N _ N L 4. Houghton ifflin Harcourt Publishing ompany Reflect 4. Is it possible to write the proof in Part without using the HL Triangle ongruence Theorem? Explain. Your Turn Use the HL ongruence Theorem to prove that the triangles are congruent. 5. Given: and are right angles. _ _ Prove: odule 6 97 Lesson 3
4 Elaborate 6. You draw a right triangle with a hypotenuse that is 5 inches long. friend also draws a right triangle with a hypotenuse that is 5 inches long. an you conclude that the triangles are congruent using the HL ongruence Theorem? If not, what else would you need to know in order to conclude that the triangles are congruent? 7. Essential Question heck-in How is the HL Triangle ongruence Theorem similar to and different from the S, SS, SSS, and S Triangle ongruence Theorems? Evaluate: Homework and Practice 1. Tyrell used geometry software to construct so that m = 0. Then he dragged point so that = 6 cm. He used the software s compass tool to construct a circle centered at point with radius 3 cm. ased on this construction, is there a unique with m = 0, = 6 cm, and = 3 cm? Explain. Online Homework Hints and Help Extra Practice etermine whether enough information is given to prove that the triangles are congruent. Explain your answer.. and 3. PQR and STU R P Q T S U Houghton ifflin Harcourt Publishing ompany odule 6 98 Lesson 3
5 4. G and HG G H 5. EFG and SQR F R S E G Q Write a two-column proof, using the HL ongruence Theorem, to prove that the triangles are congruent. 6. Given: and are right angles. Prove: 7. Given: FGH and H are right angles. H is the midpoint of _ G. _ FH _ Prove: FGH H F G H Houghton ifflin Harcourt Publishing ompany 8. Given: P is perpendicular to QR. N is the midpoint of P. _ QP R Prove: NR PNQ Q N P R odule 6 99 Lesson 3
6 9. Given: and are right angles. Prove: lgebra What value of x will make the given triangles congruent? Explain. 10. L and L 11. and x + x + 8 5x x STV and UVT S 4x + T 13. PQ and PN V 6x - 7 U 7x - 5 Q P N 4x + 5 Houghton ifflin Harcourt Publishing ompany odule Lesson 3
7 lgebra Use the HL Triangle ongruence Theorem to show that EF. (Hint: Use the istance Formula to show that appropriate sides are congruent. Use the slope formula to show that appropriate angles are right angles.) y E x F y 4 E x F -4 Houghton ifflin Harcourt Publishing ompany 16. ommunicate athematical Ideas vertical tower is supported by two guy wires, as shown. The guy wires are both 58 feet long. Is it possible to determine the distance from the bottom of guy wire _ to the bottom of the tower? If so, find the distance. If not, explain why not. 34 ft odule Lesson 3
8 17. carpenter built a truss, as shown, to support the roof of a doghouse. L a. The carpenter knows that. an the carpenter conclude that L L? Why or why not? b. What If? Suppose the carpenter also knows that L is a right angle. an the carpenter now conclude that L L? Explain. 18. ounterexamples enise said that if two right triangles share a common hypotenuse, then the triangles must be congruent. Sketch a figure that serves as a counterexample to show that enise s statement is not true. 19. ulti-step The front of a tent is covered by a triangular flap of material. The figure represents the front of the tent, with PS QR and PQ PR. onah needs to determine the perimeter of PQR so that he can replace the zipper on the tent. Find the perimeter. Explain your steps. Q 5 ft P S 4 ft R Houghton ifflin Harcourt Publishing ompany Image redits: (t) acek Tarczyski/Panther edia/age fotostock odule 6 30 Lesson 3
9 0. student is asked to write a two-column proof for the following. Given: and are right angles. _ _ Prove: _ _ ssuming the student writes the proof correctly, which of the following will appear as a statement or reason in the proof? Select all that apply.. S Triangle ongruence Theorem. Reflexive Property of ongruence. _ _ E. PT. F. HL Triangle ongruence Theorem H.O.T. Focus on Higher Order Thinking 1. nalyze Relationships Is it possible for a right triangle with a leg that is 10 inches long and a hypotenuse that is 6 inches long to be congruent to a right triangle with a leg that is 4 inches long and a hypotenuse that is 6 inches long? Explain. Houghton ifflin Harcourt Publishing ompany. ommunicate athematical Ideas In the figure, _ _ L, _ _ L, and and L are right angles. escribe how you could use three different congruence theorems to prove that L. L odule Lesson 3
10 3. ustify Reasoning o you think there is an LL Triangle ongruence Theorem? That is, if the legs of one right triangle are congruent to the legs of another right triangle, are the triangles necessarily congruent? If so, write a proof of the theorem. If not, provide a counterexample. Lesson Performance Task The figure shows kite. a. What would you need to know about the relationship between _ and _ in order to prove that E E and E E by the HL Triangle ongruence Theorem? b. an you prove that and are congruent using the HL Triangle ongruence Theorem? Explain why or why not. E c. How can you prove that the two triangles named in Part b are in fact congruent, even without the additional piece of information? Houghton ifflin Harcourt Publishing ompany odule Lesson 3
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