Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Euler angles wikipedia , lookup
Reuleaux triangle wikipedia , lookup
Noether's theorem wikipedia , lookup
Riemann–Roch theorem wikipedia , lookup
Four color theorem wikipedia , lookup
Rational trigonometry wikipedia , lookup
Brouwer fixed-point theorem wikipedia , lookup
Trigonometric functions wikipedia , lookup
History of trigonometry wikipedia , lookup
Euclidean geometry wikipedia , lookup
LESSON 6.3 Name HL Triangle Congruence 6.3 G.6.B Explore Prove two triangles are congruent by applying the . . . Hypotenuse-Leg congruence conditions. Also G.2.B, G.5.C Is There a Side-Side-Angle Congruence Theorem? Follow these steps to draw △ABC such that m∠A = 30°, AB = 6 cm, and BC = 4 cm. The goal is to determine whether two side lengths and the measure of a non-included angle (SSA) determine a unique triangle. G.1.F Analyze mathematical relationships to connect and communicate mathematical ideas. Use a protractor to draw a large 30° angle on a separate sheet of paper. Label it ∠A. Use a ruler to locate point B on one ray of ∠A so that AB = 6 cm. Language Objective 1.A, 2.C.3, 2.C.4, 2.D.1 Explain the HL Congruence Theorem in your own words. A _ Now draw BC so that BC = 4 cm. To do this, open a compass to a distance of 4 cm. Place the point of the compass on point B and draw _ an arc. Plot point C where the arc intersects the side of ∠A. Draw BC ENGAGE View the Engage section online. Discuss the photo, asking students to describe the shape of the kite in terms of angles, triangles, and any other geometrical terms that seem relevant. Then preview the Lesson Performance Task. What do you notice? Is it possible to draw only one △ABC with the given side length? Explain. → ‾ in two places. So, it is If extended, the arc would intersect AC © Houghton Mifflin Harcourt Publishing Company possible to draw two different triangles with side length BC. 6 cm A 30° B 6 cm A 30° 4 cm C Reflect 1. Do you think that SSA is sufficient to prove congruence? Why or why not? No. SSA is not sufficient to determine congruence because a given set of values does not necessarily describe a unique triangle. It is possible to draw two different triangles that have two congruent sides and a congruent non-included angle. 2. Discussion Your friend said that there is a special case where SSA can be used to prove congruence—namely, if the non-included angle is a right angle. Is your friend right? Explain. Yes; if the congruent non-included angle were a right angle, then SSA would work. Given a right angle, one set of congruent sides would be legs and the other set the hypotenuses. Given a leg and the hypotenuse of a right triangle, the Pythagorean theorem guarantees a unique triangle. Module 6 ges EDIT--Chan DO NOT Key=TX-B Correction must be Lesson 3 331 gh "File info" made throu Date Class Name HL Triangle e Congruenc about two em tell you uence Theor le Congr HL Triang congruence does the nuse-Leg ion: What g the ... Hypote triangles? ent by applyin es are congru two triangl G.6.B Prove G.5.C Also G.2.B, conditions. -Side-Angle HARDCOVER PAGES 275282 Resource Locker Quest Essential a Side ? Is There e Theorem ent. In this Congruenc les are congru em. Theor g that triang Explore GE_MTXESE353886_U2M06L3 331 ms for provin Triangle Congruence several theore is a SSA BC = 4 cm. already seen gate whether there = 6 cm, and You have cluded will investi = 30°, AB re of a non-in that m∠A Explore, you the measu △ABC such lengths and steps to draw er two side Follow these determine wheth triangle. is to a unique The goal sheet of paper. determine (SSA) a separate angle angle on ctor Use a protra Label it ∠A. to draw a large 30° A Turn to these pages to find this lesson in the hardcover student edition. 30° AB = 6 cm. ∠A so that one ray of point B on to locate 6 cm to a a compass do this, open B and draw _ 30° _ that BC = 4 cm. To ss on point BC A BC so of the compa side of ∠A. Draw Now draw the point cts the 4 cm. Place the arc interse distance of point C where an arc. Plot △ABC. the to complete 6 cm △ABC with only one draw to le 30° it is A ? Is it possib you notice → in two places. So, n. What do ‾ AC Explai ? ect length d inters h BC. given side side lengt the arc woul les with If extended, ent triang two differ to draw possible B Use a ruler B 4 cm C y g Compan why not? s does not ? Why or set of value congruence se a given les that ent to prove ruence becau ent triang SSA is suffici mine cong two differ think that ient to deter ble to draw 1. Do you It is possi is not suffic . No. SSA triangle. ed angle a unique non-includ describe to prove necessarily a congruent can be used sides and Explain. right where SSA congruent friend right? . Given a special case your a have two Is work is d angle. that there SSA woul is a right friend said angle, then Given a leg cluded angle ssion Your tenuses. if the non-in were a right 2. Discu —namely, ed angle set the hypo e triangle. congruence the other non-includ legs and ntees a uniqu congruent would be em guara Yes; if the ruent sides gorean theor set of cong Lesson 3 le, the Pytha angle, one a right triang of e hypotenus and the © Houghto n Mifflin Harcour t Publishin Reflect 331 Module 6 6L3 331 86_U2M0 ESE3538 GE_MTX Lesson 6.3 30° B to complete △ABC. 6.3 331 Resource Locker You have already seen several theorems for proving that triangles are congruent. In this Explore, you will investigate whether there is a SSA Triangle Congruence Theorem. Mathematical Processes PREVIEW: LESSON PERFORMANCE TASK HL Triangle Congruence G.6.B Prove two triangles are congruent by applying the ... Hypotenuse-Leg congruence conditions. Also G.2.B, G.5.C The student is expected to: If a leg and the hypotenuse of one right triangle are congruent to the corresponding leg and hypotenuse of another right triangle, the triangles are congruent. Date Essential Question: What does the HL Triangle Congruence Theorem tell you about two triangles? Texas Math Standards Essential Question: What does the HL Triangle Congruence Theorem tell you about two triangles? Class 04/02/15 11:54 AM 04/02/15 11:54 AM Explain 1 Justifying the Hypotenuse-Leg Congruence 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. EXPLORE hypotenuse You have learned four ways to prove that triangles are congruent. Is there a Side-Side-Angle Congruence Theorem? legs • Angle-Side-Angle (ASA) Congruence Theorem • Side-Angle-Side (SAS) Congruence Theorem • Side-Side-Side (SSS) Congruence Theorem • Angle-Angle-Side (AAS) Congruence Theorem INTEGRATE TECHNOLOGY The Hypotenuse-Leg (HL) Triangle Congruence Theorem is a special case that allows you to show that two right triangles are congruent. Students can use geometry software to explore what you can do with two sides and a non-included angle. Hypotenuse-Leg (HL) Triangle Congruence 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 QUESTIONING STRATEGIES B Prove the HL Triangle Congruence Theorem. E c a Given: △ABC and △DEF are right triangles; ∠C and ∠F are right angles. _ _ _ _ AB ≅ DE and BC ≅ EF C b d A Could you use SAS on the triangles we are discussing here? Explain. No; SAS requires the angle to be included between the two pairs of congruent sides. f F D e Prove: △ABC ≅ △DEF 2 2 By the Pythagorean Theorem, a 2 + b 2 = c 2 and d + e = f 2. It is given that 2 2 _ _ AB ≅ DE, so AB = DE and c = ƒ. Therefore, c 2 = f 2 and a 2 + b 2 = d + e . It is given that _ _ BC ≅ EF, so BC = EF and a = d. Substituting a for d in the above equation, a 2 + b 2 = a 2 2 + e EXPLAIN 1 2 2 the SSS Triangle Congruence Theorem . Your Turn 3. V Determine whether there is enough information to prove that triangles △VWX and △YXW are congruent. Explain. Yes. △VWX and △YXW are _right _ triangles _ that share hypotenuse WX . WX ≅ WX by the Reflexive of Congruence. It is given _ Property _ that WV ≅ XY , therefore △VWX ≅ △YXW by the HL Triangle Congruence Theorem. Module 6 332 W Z Y X © Houghton Mifflin Harcourt Publishing Company Therefore, △ABC ≅ △DEF by Justifying the Hypotenuse-Leg Congruence Theorem . Subtracting a from each side shows that b = e , and taking the square root of each side, b = e . _ _ This shows that AC ≅ DF . 2 AVOID COMMON ERRORS Students may use hypotenuse to describe the longest side of any triangle. Remind them that the term is used only with right triangles. QUESTIONING STRATEGIES With what kind of triangles can you use the Pythagorean Theorem? right triangles only Lesson 3 PROFESSIONAL DEVELOPMENT GE_MTXESE353886_U2M06L3.indd 332 Integrate Mathematical Processes 1/22/15 4:07 AM This lesson provides an opportunity to address Mathematical Process TEKS G.1.F, which calls for students to “analyze mathematical relationships to connect and communicate mathematical ideas.” Students look at pairs of triangles that have two congruent sides and congruent non-included angles. They analyze these relationships to determine that this information is sufficient only to prove right triangles congruent. HL Triangle Congruence 332 Explain 2 EXPLAIN 2 Example 2 Applying the Hypotenuse-Leg Congruence Theorem Use the HL Congruence Theorem to prove that the triangles are congruent. _ _ Given: ∠P and ∠R are right angles. PS ≅ RQ P Q S R Prove: △PQS ≅ △RSQ QUESTIONING STRATEGIES Statements Are all right angles congruent? Explain. Yes; a right angle is an angle that measures 90°, so all right angles have the same measure. This means that all right angles are congruent. AVOID COMMON ERRORS Students may assume that the hypotenuses of two given right triangles are congruent. Remind them not to assume anything is true, and to use only the information that is given. Reasons 1. ∠P and ∠R are right angles. _ _ 2. PS ≅ RQ _ _ 3. SQ ≅ SQ 1. Given 4. △PQS ≅ △RSQ 4. HL Triangle Congruence Theorem 2. Given 3. Reflexive Property of Congruence Given: ∠J and ∠L are right angles. K is the midpoint _ _ of JL and MN. Prove: △JKN ≅ △LKM M J K L N Statements © Houghton Mifflin Harcourt Publishing Company CONNECT VOCABULARY Have students label the parts of a right triangle, identifying the hypotenuse, the legs, and the right angle, and then measure and write the measures of the other two angles. Applying the HL Triangle Congruence Theorem Reasons 1. ∠J and ∠L are right angles. _ _ 2. K is the midpoint of JL and MN . _ _ _ _ 3. JK ≅ LK and MK ≅ NK 1. Given 4. △JKN ≅ △LKM 4. HL Triangle Congruence Theorem 2. Given 3. Definition of midpoint Reflect 4. Is it possible to write the proof in Part B without using the HL Triangle Congruence Theorem? Explain. _ _ Yes, you can use the SAS Triangle Congruence Theorem (∠J ≅ ∠L, JK ≅ LK, and vertical angles JKN and LKM are congruent) or the SAS Triangle Congruence _ _ _ _ Theorem (JK ≅ LK and MK ≅ NK, and vertical angles JKN and LKM are congruent). Your Turn Use the HL Congruence Theorem to prove that the triangles are congruent. _ _ 5. Given: ∠CAB and ∠DBA are right angles. AD ≅ BC A Prove: △ABC ≅ △BAD _ _ It is given that and ∠CAB and ∠DBA are right angles and AD ≅ BC. _ _ AB ≅ AB by the Reflexive Property of Congruence. Then C D Module 6 Lesson 3 B △ABC ≅ △BAD by the HL Triangle Congruence Theorem. 333 COLLABORATIVE LEARNING GE_MTXESE353886_U2M06L3.indd 333 Small Group Activity Have students work in small groups to illustrate the differences between the HL and SAS Triangle Congruence Theorems. They may convey the information in any way, including making a poster, writing an essay, or creating a model. Have each group present their project to the class. 333 Lesson 6.3 1/22/15 4:07 AM