ASA Triangle Congruence

Transcription

1 Locker LSSON 5.2 S Triangle ongruence Texas Math Standards The student is expected to: G.6. Prove two triangles are congruent by applying the Side-ngle-Side, ngle-side-ngle, Side-Side-Side, ngle-ngle-side, and Hypotenuse- Leg congruence conditions. lso G.3., G.5., G.5. Mathematical Processes G.1. reate and use representations to organize, record, and communicate mathematical ideas. Language Objective 1., 1., 1., 2..4, 3.H, 4. Have students work in pairs to label and color code congruent angles and a side in pairs of triangles. NGG ssential Question: What does the S Triangle ongruence Theorem tell you about triangles? If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, the triangles are congruent. PRVIW: LSSON PRORMN TSK View the ngage section online. xplain that the flags of many countries incorporate geometric objects such as triangles. Then preview the Lesson Performance Task. Houghton Mifflin Harcourt Publishing ompany Name lass ate 5.2 S Triangle ongruence ssential Question: What does the S Triangle ongruence Theorem tell you about triangles? G.6. Prove two triangles are congruent by applying the... ngle-side-ngle... congruence conditions. lso G.3., G.5., G.5. xplore 1 rawing Triangles Given Two ngles and a Side You have seen that two triangles are congruent if they have six pairs of congruent corresponding parts. However, it is not always possible to check all three pairs of corresponding sides and all three pairs of corresponding angles. ortunately, there are shortcuts for determining whether two triangles are congruent. raw a segment that is 4 inches long. Label the endpoints and. Use a protractor to draw a 30 angle so that one side is _ and its vertex is point. Use a protractor to draw a 40 angle so that one side is _ and its vertex is point. Label the point where the sides of the angles intersect as point. Put your triangle and a classmate s triangle beside each other. Is there a sequence of rigid motions that maps one to the other? What does this tell you about the triangles? Yes; the triangles are congruent. 1. In a polygon, the side that connects two consecutive angles is the included side of those two angles. escribe the triangle you drew using the term included side. e as precise as possible. It is a triangle with a 30 angle, a 40 angle, and an included side that is 4 inches long. 2. iscussion ased on your results, how can you decide whether two triangles are congruent without checking that all six pairs of corresponding sides and corresponding angles are congruent? Possible answer: If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent in. 40 Resource Locker Module Lesson 2 Name lass ate 5.2 S Triangle ongruence ssential Question: What does the S Triangle ongruence Theorem tell you about triangles? G.6. Prove two triangles are congruent by applying the... ngle-side-ngle... congruence conditions. lso G.3., G.5., G.5. xplore 1 rawing Triangles Given Two ngles Houghton Mifflin Harcourt Publishing ompany and a Side You have seen that two triangles are congruent if they have six pairs of congruent corresponding parts. However, it is not always possible to check all three pairs of corresponding sides and all three pairs of corresponding angles. ortunately, there are shortcuts for determining whether two triangles are congruent. Use a protractor to draw a 30 angle so that one side is _ and raw a segment that is 4 inches long. Label the endpoints and. its vertex is point. Use a protractor to draw a 40 angle so that one side is _ and its vertex is point. Label the point where the sides of the angles intersect as point. Put your triangle and a classmate s triangle beside each other. Is there a sequence of rigid motions that maps one to the other? What does this tell you about the triangles? Yes; the triangles are congruent. Resource in. 1. In a polygon, the side that connects two consecutive angles is the included side of those two angles. escribe the triangle you drew using the term included side. e as precise as possible. It is a triangle with a 30 angle, a 40 angle, and an included side that is 4 inches long. 2. iscussion ased on your results, how can you decide whether two triangles are congruent without checking that all six pairs of corresponding sides and corresponding angles are congruent? Possible answer: If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent. Module Lesson 2 HROVR PGS Turn to these pages to find this lesson in the hardcover student edition. 261 Lesson 5.2

2 xplore 2 Justifying S Triangle ongruence xplain the results of xplore 1 using transformations. Use tracing paper to make two copies of the triangle from xplore 1 as shown. Identify the corresponding parts you know to be congruent and mark these congruent parts on the figure. _ What can you do to show that these triangles are congruent? ind a sequence of rigid motions that maps one triangle onto the other triangle. translate so that point maps to point. What translation vector did you use? Use a rotation to map point to point. What is the center of the rotation? What is the angle of the rotation? The center of the rotation is point (or ); the angle of the rotation is m. How do you know the image of point is point? It is given that _ _, so the image of point must be point. What rigid motion do you think will map point to point? reflection across G to show that the image of point is point, notice that is reflected across, so the measure of the angle is preserved. Since you can conclude that the image of _ lies on. In particular, the image of point must lie on. y similar reasoning, the image of _ lies on and the image of point must lie on. the only point that lies on both _ and _ is point. H escribe the sequence of rigid motions used to map to. a translation followed by a rotation followed by a reflection _ the vector with initial point and terminal point ( ) 3. iscussion rturo said the argument in the activity works for any triangles with two pairs of congruent corresponding angles, and it is not necessary for the included sides to be congruent. o you agree? xplain. No; the included sides must be congruent to conclude that the image of point is point after a rotation around point. Houghton Mifflin Harcourt Publishing ompany XPLOR 1 rawing Triangles Given Two ngles and a Side INTGRT THNOLOGY Have students explore the ngle-side-ngle theorem using geometry software. QUSTIONING STRTGIS How can you check whether the triangles you draw are congruent to the triangles your classmates draw? Place one student s page on top of the other student s page and check to see if the triangles can be made to coincide exactly. XPLOR 2 Justifying S Triangle ongruence INTGRT MTHMTIL PROSSS ocus on Reasoning Have students respond to this prompt in their math journals. If I know that two pairs of corresponding angles and the included sides of two triangles are congruent, I know. I know this because. Module Lesson 2 PROSSIONL VLOPMNT Math ackground Students know that when triangles are congruent, all pairs of corresponding sides and corresponding angles are congruent. s an extension of that, students to begin to develop converses of the statement that orresponding Parts of ongruent Triangles re ongruent in which they do not need to know that all six pairs of corresponding parts are congruent in order to prove that triangles are congruent. In this lesson, they explore the ngle-side-ngle (S) Theorem. They find that if they can prove that two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, the triangles are congruent. QUSTIONING STRTGIS What is the benefit of using the ngle-side- ngle Theorem instead of PT? You need to find only three pairs of congruent corresponding parts with S, as opposed to six pairs with PT. S Triangle ongruence 262

3 XPLIN 1 eciding Whether Triangles re ongruent Using S Triangle ongruence xplain 1 eciding Whether Triangles re ongruent Using S Triangle ongruence You can state your findings about triangle congruence as a theorem. This theorem can help you decide whether two triangles are congruent. S Triangle ongruence Theorem If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent. ONNT VOULRY Remind students that the S Triangle ongruence Theorem is a shortened form of its full name, the ngle-side-ngle Triangle ongruence Theorem. When you write S, it is helpful to read it aloud as ngle-side-ngle and have students do the same to reinforce what it means. xample 1 etermine whether the triangles are congruent. xplain your reasoning. Step 1 ind m. m + m + m = 180 m = 180 m = 180 m = 45 Step 2 ompare the angle measures and side lengths cm 2.3 cm m = m = 45, = = 2.3 cm, and m = m = 61 QUSTIONING STRTGIS Why do you need to find the measure of the missing angle to use the S Triangle ongruence Theorem? The two sides that you know are congruent need to be the included sides, so you need to know the measures of the angles at their endpoints. LNGUG SUPPORT Give each pair pictures of congruent and non-congruent triangles, highlighters, protractors, and rulers. Instruct them to prove which pairs are congruent by using ngle-side-ngle to prove it. Have students highlight the angles and the side they used to show congruence, and write notes explaining why the triangles are congruent. Houghton Mifflin Harcourt Publishing ompany So,, _ _, and. and include side _, and and include side _. So, by the S Triangle ongruence Theorem. Step 1 ind m P. m M + m N + m P = m P = m P = 180 m P = 111 Step 2 ompare the angle measures and side lengths. J in. K 110 L P 62 in. N None of the angles in MNP has a measure of 110. Therefore, there is/is not a sequence of rigid motions that maps MNP onto JKL, and MNP is/is not congruent to JKL. 4. In Part, do you need to find m K? Why or why not? No; you only need to know that JKL has an angle ( L) that is not congruent to any angle of MNP. t that point, you can conclude that the triangles are not congruent. M Module Lesson 2 OLLORTIV LRNING Small Group ctivity Have students experiment with congruent triangles and triangles that are not congruent but do have some congruent parts. Instruct them to draw a pair of congruent triangles and a pair of non-congruent triangles that meet the following criteria: at least two pairs of congruent sides at least one pair each of congruent sides and congruent angles all three pairs of congruent angles at least two pairs of congruent sides and one pair of congruent angles 263 Lesson 5.2

4 Your Turn etermine whether the triangles are congruent. xplain your reasoning xplain 2 Proving Triangles re ongruent Using S Triangle ongruence The S Triangle ongruence Theorem may be used as a reason in a proof. xample 2, _ _, and since both are right angles. and include side _ and and include side _. So, by the S Triangle ongruence Theorem. Write each proof. Given: MQP NPQ, MPQ NQP Prove: MQP NPQ Q P R U 1 in. T 1 in m R = 180 m R = 70 None of the angles in PQR has a measure of 67. So, PQR is not congruent to STU. Q M S N P XPLIN 2 Proving Triangles re ongruent Using S Triangle ongruence VOI OMMON RRORS Some students may forget to include in their proofs the information that is given in the diagram. Remind them to start the proof by listing the given information. QUSTIONING STRTGIS How do you know when you have enough information to complete the proof? To complete the proof, you need to show that two angles and the included side of one triangle are congruent to the corresponding angles and side of the other triangle. 1. MQP NPQ 1. Given 2. MPQ NQP 2. Given 3. QP QP 3. Reflexive Property of ongruence 4. MQP NPQ 4. S Triangle ongruence Theorem Houghton Mifflin Harcourt Publishing ompany ONNT VOULRY Have students review the different units used to measure triangles to show congruence. Length is measured in linear units such as cm, mm, or inches, while angles are measured in degrees. Module Lesson 2 IRNTIT INSTRUTION Modeling Instruct students to draw and label three triangles according to the following specifications. There is just enough information to prove that they are congruent using the S Triangle ongruence Theorem. There is enough information to prove that they are not congruent. There is some of the information required to prove that they are congruent, but not enough. S Triangle ongruence 264

5 Given:, is the midpoint of _. Prove: is the midpoint of _. 2. Given Given 3. _ _ efinition of midpoint Vertical angles are congruent. S Triangle ongruence Theorem 7. In Part, suppose the length of _ is 8.2 centimeters. an you determine the length of any other segments in the figure? xplain. Yes; = 8.2 cm because _ _ by PT. Your Turn Write each proof. 8. Given: JLM KML, JML KLM Prove: JML KLM J K L M Houghton Mifflin Harcourt Publishing ompany 1. JLM KML 1. Given 2. JML KLM 2. Given 3. LM _ _ LM 3. Reflexive Property of ongruence 4. JML KLM 4. S Triangle ongruence Theorem Module Lesson 2 LNGUG SUPPORT Vocabulary evelopment Make sure students understand the meaning of included side. Sketch R, MT, and SQU on the board. Use color to highlight _ SQ and define what it means for it to be included between S and Q. sk questions such as What side is included between and R? and etween which two angles is _ MT? ontinue to drill students until they can recognize and name included sides fluently. 265 Lesson 5.2

6 9. Given: S and U are right angles, _ RV bisects _ SU. R Prove: RST VUT S T U LORT VISUL US 1. S and U are right angles. 1. Given 2. S U 2. ll right angles are congruent. V Use colored pencils to label congruent sides using ticks and congruent angles using arcs to help students better visualize the angles and sides that are congruent 3. _ RV bisects _ SU. 3. Given 4. _ ST _ UT 4. efinition of bisector 5. RTS VTU 5. Vertical angles are congruent. 6. RST VUT 6. S Triangle ongruence Theorem laborate 10. iscussion Suppose you and a classmate both draw triangles with a 30 angle, a 70 angle, and a side that is 3 inches long. How will they compare? xplain your reasoning. The triangles will be congruent by the S Triangle ongruence Theorem if the 3 inch side is the included side. Otherwise, the triangles will have the same shape but not necessarily the same size. SUMMRIZ TH LSSON Why would you use the S Triangle ongruence Theorem? What do you need to know to use it? You would use the S Triangle ongruence Theorem to prove that two triangles are congruent by using only three pairs of congruent parts. You need to know that two pairs of corresponding angles are congruent and the included sides between those angles are also congruent. 11. iscussion How can a diagram show you that corresponding parts of two triangles are congruent without providing specific angle measures or side lengths? Possible answer: Vertical angles are congruent. Overlapping sides are congruent. Right angles are congruent. ngles or sides marked with congruence symbols are congruent. 12. ssential Question heck-in What must be true in order for you to use the S Triangle ongruence Theorem to prove that triangles are congruent? Two angles and the included side of one triangle must be congruent to two angles and the included side of another triangle. Houghton Mifflin Harcourt Publishing ompany Module Lesson 2 S Triangle ongruence 266