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4.2 1 PLAN PACING Basic: 2 days Average: 2 days Advanced: 2 days Block Schedule: 0.5 block with 4.1 0.5 block with 4.3 What you should learn GOAL 1 Identify congruent figures and corresponding parts. GOAL 1 IDENTIFYING CONGRUENT FIGURES Two geometric figures are congruent if they have exactly the same size and shape. Each of the red figures is congruent to the other red figures. None of the blue figures is congruent to another blue figure. GOAL 2 Prove that two triangles are congruent. Congruent Not congruent Why you should learn it FE 䉲 To identify and describe congruent figures in real-life objects, such as the sculpture described L AL I in Example 1. RE LESSON OPENER VISUAL APPROACH An alternative way to approach Lesson 4.2 is to use the Visual Approach Lesson Opener: •Blackline Master (Chapter 4 Resource Book, p. 26) • Transparency (p. 22) Congruence and Triangles MEETING INDIVIDUAL NEEDS • Chapter 4 Resource Book Prerequisite Skills Review (p. 5) Practice Level A (p. 30) Practice Level B (p. 31) Practice Level C (p. 32) Reteaching with Practice (p. 33) Absent Student Catch-Up (p. 35) Challenge (p. 38) • Resources in Spanish • Personal Student Tutor When two figures are congruent, there is a correspondence between their angles and sides such that corresponding angles are congruent and corresponding sides are congruent. For the triangles below, you can write ¤ABC £ ¤PQR, which is read “triangle ABC is congruent to triangle PQR.” The notation shows the congruence and the correspondence. NEW-TEACHER SUPPORT See the Tips for New Teachers on pp. 1–2 of the Chapter 4 Resource Book for additional notes about Lesson 4.2. There is more than one way to write a congruence statement, but it is important to list the corresponding angles in the same order. For example, you can also write ¤BCA £ ¤QRP. Corresponding angles 5. right scalene triangle yes Two Open Triangles Up Gyratory II by George Rickey Æ Æ Æ Æ B ™B £ ™Q BC £ QR ™C £ ™R CA £ RP A C q SOLUTION STUDENT HELP Study Tip Notice that single, double, and triple arcs are used to show congruent angles. R Naming Congruent Parts The congruent triangles represent the triangles in the photo above. Write a congruence statement. Identify all pairs of congruent corresponding parts. 202 202 Æ P WARM-UP EXERCISES no Æ AB £ PQ ™A £ ™P EXAMPLE 1 Transparency Available Tell whether it is possible to draw each triangle. 1. acute scalene triangle yes 2. obtuse equilateral triangle no 3. right isosceles triangle yes 4. scalene equiangular triangle Corresponding sides F R S E The diagram indicates that ¤DEF £ ¤RST. The congruent angles and sides are as follows. Angles: Sides: ™D £ ™R, ™E £ ™S, ™F £ ™T Æ Æ Æ Æ Æ Æ DE £ RS, EF £ ST, FD £ TR Chapter 4 Congruent Triangles D T xy Using Algebra EXAMPLE 2 Using Properties of Congruent Figures 2 TEACH In the diagram, NPLM £ EFGH. F M 8m a. Find the value of x. L b. Find the value of y. 110ⴗ (2x ⴚ 3) m 87ⴗ Æ (7y ⴙ 9)ⴗ 72ⴗ 10 m P SOLUTION G N E H EXTRA EXAMPLE 1 Write a congruence statement for the triangles below. Identify all pairs of congruent corresponding parts. See below. H Æ a. You know that LM £ GH. So, LM = GH. 8 = 2x º 3 b. You know that ™N £ ™E. Y So, m™N = m™E. 72° = (7y + 9)° 11 = 2x K Z X 63 = 7y 5.5 = x ......... 9=y J EXTRA EXAMPLE 2 In the diagram, ABCD ⬵ KJHL. a. Find the value of x. 3 b. Find the value of y. 25 The Third Angles Theorem below follows from the Triangle Sum Theorem. You are asked to prove the Third Angles Theorem in Exercise 35. A 9 cm (4x ⫺ 3) cm K B L 91° THEOREM Third Angles Theorem THEOREM 4.3 B If two angles of one triangle are congruent to two angles of another triangle, then the third angles are also congruent. D E If ™A £ ™D and ™B £ ™E, then ™C £ ™F. H (5y ⫺ 12)° J EXTRA EXAMPLE 3 Find the value of x. 14 C A 86° 113° 6 cm C B F D (4x ⫹ 15)° 87° 22° A F C THEOREM D EXAMPLE 3 Using the Third Angles Theorem Find the value of x. CHECKPOINT EXERCISES M R For use after Example 1: 1. Identify all pairs of congruent corresponding parts. T (2x ⴙ 30)ⴗ SOLUTION INT STUDENT HELP NE ER T HOMEWORK HELP Visit our Web site www.mcdougallittell.com for extra examples. In the diagram, ™N £ ™R and ™L £ ™S. From the Third Angles Theorem, you know that ™M £ ™T. So, m™M = m™T. From the Triangle Sum Theorem, m™M = 180° º 55° º 65° = 60°. m™M = m™T 60° = (2x + 30)° åA • åR; åB • åQ; åC • åP; A 苶B 苶 • R苶Q 苶; B 苶C苶 • Q 苶P苶 ; A 苶C苶 • R苶P苶 55ⴗ N 65ⴗ L R A S B Q Third Angles Theorem P C Substitute. 30 = 2x Subtract 30 from each side. 15 = x Divide each side by 2. 4.2 Congruence and Triangles For use after Examples 2 and 3: 2. Find the value of x. 11 G K 203 J F Extra Example 1 Sample answer: †XYZ • †HKJ; åX • åH; åY • åK; åZ • åJ; X苶Y苶 • H 苶K苶; Y苶Z苶 • K苶J苶; X苶Z苶 • H 苶J苶 E 100° 45° H I (4x ⫺ 9)° 203 GOAL 2 EXTRA EXAMPLE 4 Decide whether the triangles are congruent. Justify your reasoning. Determining Whether Triangles are Congruent EXAMPLE 4 H E PROVING TRIANGLES ARE CONGRUENT Decide whether the triangles are congruent. Justify your reasoning. 58° R 92ⴗ G 58° J F Proof From the diagram, you are given that all three pairs of corresponding sides are congruent: E苶F苶 • J苶H 苶, E苶G 苶 • J苶G 苶, F苶G 苶• H 苶G 苶. Because åF and åH have the same measure, åF • åH. By the Vertical Angles Theorem, you know that åEGF • åJGH. By the Third Angles Theorem, åE • åJ. So all three sides and all three angles are congruent. By the definition of congruent triangles, †EFG • †JHG. q M P SOLUTION Paragraph Proof From the diagram, you are given that all three pairs of corresponding sides are congruent. Æ Æ Æ Æ Æ Æ RP £ MN, PQ £ NQ, and QR £ QM Because ™P and ™N have the same measure, ™P £ ™N. By the Vertical Angles Theorem, you know that ™PQR £ ™NQM. By the Third Angles Theorem, ™R £ ™M. 䉴 So, all three pairs of corresponding sides and all three pairs of corresponding angles are congruent. By the definition of congruent triangles, ¤PQR £ ¤NQM. Proving Two Triangles are Congruent EXAMPLE 5 EXTRA EXAMPLE 5 Given: M 苶N 苶⬵Q 苶P苶, M 苶N 苶 储 P苶Q 苶, O is the midpoint of 苶M 苶Q 苶 and 苶P苶N 苶. Prove: 䉭MNO ⬵ 䉭QPO M N 92ⴗ FOCUS ON APPLICATIONS The diagram represents the triangular stamps shown in the photo. Prove that ¤AEB £ ¤DEC. Æ Æ Æ A Æ GIVEN 䉴 AB ∞ DC , AB £ DC , Æ B E Æ E is the midpoint of BC and AD. N PROVE 䉴 ¤AEB £ ¤DEC O C D Plan for Proof Use the fact that ™AEB and ™DEC are vertical angles to show Æ that those angles are congruent. Use the fact that BC intersects parallel segments Æ Æ AB and DC to identify other pairs of angles that are congruent. Q L AL I FE Statements (Reasons) 1. M 苶N 苶• Q 苶P苶, M 苶N 苶 ‚ P苶Q 苶, (Given) 2. åOMN • åOQP, åMNO • åQPO (Alternate Interior Angles Theorem) 3. åMON • åQOP (Vertical Angles Theorem) 4. O is the midpoint of M 苶Q 苶 and P苶N 苶. (Given) 5. M 苶O 苶• Q 苶O 苶, P苶O 苶• N 苶O 苶 (Def. of midpoint) 6. †MNO • †QPO (Def. of • †s ) RE P TRIANGULAR STAMP When these stamps were issued in 1997, Postmaster General Marvin Runyon said, “Since 1847, when the first U.S. postage stamps were issued, stamps have been rectangular in shape. We want the American public to know stamps aren’t ‘square.’” SOLUTION Statements Æ 1. Given 2. 2. Alternate Interior Angles Theorem Æ 3. 4. 6. 204 204 Reasons 1. AB ∞ DC, 5. Checkpoint Exercises for Examples 4 and 5 on next page. Æ Æ AB £ DC ™EAB £ ™EDC, ™ABE £ ™DCE ™AEB £ ™DEC Æ E is the midpoint of AD, Æ E is the midpoint of BC. Æ Æ Æ Æ AE £ DE, BE £ CE ¤AEB £ ¤DEC Chapter 4 Congruent Triangles 3. Vertical Angles Theorem 4. Given 5. Definition of midpoint 6. Definition of congruent triangles In this lesson, you have learned to prove that two triangles are congruent by the definition of congruence—that is, by showing that all pairs of corresponding angles and corresponding sides are congruent. In upcoming lessons, you will learn more efficient ways of proving that triangles are congruent. The properties below will be useful in such proofs. CHECKPOINT EXERCISES For use after Examples 4 and 5: 1. Decide whether the triangles are congruent. Justify your reasoning. C THEOREM THEOREM 4.4 Properties of Congruent Triangles 20° Q 20° B REFLEXIVE PROPERTY OF CONGRUENT TRIANGLES 120° A Every triangle is congruent to itself. C E SYMMETRIC PROPERTY OF CONGRUENT TRIANGLES If ¤ABC £ ¤DEF, then ¤DEF £ ¤ABC. D F K TRANSITIVE PROPERTY OF CONGRUENT TRIANGLES If ¤ABC £ ¤DEF and ¤DEF £ ¤JKL, then ¤ABC £ ¤JKL. J L GUIDED PRACTICE Vocabulary Check Concept Check ✓ ✓ 2. Third Angles Theorem 3. No; corresponding sides are not congruent. Skill Check ✓ 1. Copy the congruent triangles shown at the right. Then label the vertices of your triangles so that ¤JKL £ ¤RST. Identify all pairs of congruent corresponding angles and corresponding sides. ™J and ™R, ™K and Æ Æ Æ Æ Æ Æ D B 2. How does the student know that the corresponding angles are congruent? 3. Is ¤ABC £ ¤ADE? Explain your answer. A C E q N P 6. What is the measure of ™R? 30° 7. What is the measure of ™N? 30° Æ Æ Æ Æ 9. Which side is congruent to LN? PR CLOSURE QUESTION If 䉭GHJ ⬵ 䉭MNP, what corresponding angles and sides are congruent? 3 square units 105ⴗ L From the diagram, you are given that all three pairs of corresponding sides are congruent: C苶E苶 • Q 苶P苶, E苶D 苶 • P苶D 苶, C苶D 苶• Q 苶D 苶. Because åC and åQ have the same measure, and åE and åP have the same measure, åC • åQ and åE • åP. By the Third Angles Theorem, åCDE • åQDP. So all three sides and all three angles are congruent. By the definition of congruent triangles, †CED • †QPD. DAILY PUZZLER The design shown is created entirely from congruent triangles. If the area of the design is 96 square units, what is the area of one triangle? 45ⴗ 5. What is the measure of ™M? 45° 8. Which side is congruent to QR? MN P åG • åM, åH • åN, åJ • åP, G 苶H 苶• M 苶N 苶, H 苶J苶 • N 苶P苶, G 苶J苶 • M 苶P苶 Use the diagram at the right, where ¤LMN £ ¤PQR. 4. What is the measure of ™P? 105° 120° D FOCUS ON VOCABULARY Draw two figures that are similar but not congruent. Point out that these figures have corresponding sides. Make sure students understand that the term corresponding sides does not imply that the sides are necessarily congruent. ™S, ™L and ™T, JK and RS , KL and ST , JL and RT . ERROR ANALYSIS Use the information and the diagram below. On an exam, a student says that ¤ABC £ ¤ADE because the corresponding angles of the triangles are congruent. E M 4.2 Congruence and Triangles R 205 205