Isosceles and Equilateral Triangles

OMMON ORE Locker LESSON ommon ore Math Standards The student is expected to: OMMON ORE G-O..10 Prove theorems about triangles. Mathematical Practices OMMON ORE 7.2 Isosceles and Equilateral Triangles MP.3 Logic Language Objective Explain to a partner what you can deduce about a triangle if it has two sides with the same length. ENGGE Essential Question: What are the special relationships among angles and sides in isosceles and equilateral triangles? In an isosceles triangle, the angles opposite the sides are. In an equilateral triangle, all the sides and angles are, and the measure of each angle is 60. PREVIEW: LESSON PERFORMNE TSK View the Engage section online. Discuss the photo, explaining that the instrument is a sextant and that long ago it was used to measure the elevation of the sun and stars, allowing one s position on Earth s surface to be calculated. Then preview the Lesson Performance Task. Houghton Mifflin Harcourt Publishing ompany Name lass Date 7.2 Isosceles and Equilateral Triangles Essential Question: What are the special relationships among angles and sides in isosceles and equilateral triangles? Explore Investigating Isosceles Triangles n isosceles triangle is a triangle with at least two sides. The sides are called the legs of the triangle. The angle formed by the legs is the vertex angle. The side opposite the vertex angle is the base. The angles that have the base as a side are the base angles. In this activity, you will construct isosceles triangles and investigate other potential characteristics/properties of these special triangles. Do your work in the space provided. Use a straightedge to draw an angle. Label your angle, as shown in the figure. Using a compass, place the point on the vertex and draw an arc that intersects the sides of the angle. Label the points and. Vertex angle ase ase angles heck students construtions. Legs Resource Locker Module 7 327 Lesson 2 Name lass Date 7.2 Isosceles and Equilateral Triangles Essential Question: What are the special relationships among angles and sides in isosceles and equilateral triangles? Houghton Mifflin Harcourt Publishing ompany G-O..10 Prove theorems about triangles. Explore Investigating Isosceles Triangles n isosceles triangle is a triangle with at least two sides. The sides are called the legs of the triangle. The angle formed by the legs is the vertex angle. The side opposite the vertex angle is the base. The angles that have the base as a side are the base angles. In this activity, you will construct isosceles triangles and investigate other potential characteristics/properties of these special triangles. Do your work in the space provided. Use a straightedge to draw an angle. Label your angle, as shown in the figure. Vertex angle Legs ase ase angles heck students construtions. Using a compass, place the point on the vertex and draw an arc that intersects the sides of the angle. Label the points and. Resource HRDOVER PGES 283 292 Turn to these pages to find this lesson in the hardcover student edition. Module 7 327 Lesson 2 327 Lesson 7.2

Use the straightedge to draw line segment. EXPLORE Investigating Isosceles Triangles D Use a protractor to measure each angle. Record the measures in the table under the column for Triangle 1. INTEGRTE TEHNOLOGY Students have the option of completing the isosceles triangle activity either in the book or online. m m m Triangle 1 Triangle 2 Triangle 3 Triangle 4 QUESTIONING STRTEGIES What must be true about the triangles you construct in order for them to be isosceles triangles? They must have two sides. E Possible answer for Triangle 1: m = 70 ; m = 55 ; m = 55. Repeat steps D at least two more times and record the results in the table. Make sure is a different size each time. Reflect 1. How do you know the triangles you constructed are isosceles triangles? The compass marks equal lengths on both sides of ; therefore,. 2. Make a onjecture Looking at your results, what conjecture can be made about the base angles, and? The base angles are. Explain 1 Proving the Isosceles Triangle Theorem and Its onverse In the Explore, you made a conjecture that the base angles of an isosceles triangle are. This conjecture can be proven so it can be stated as a theorem. Isosceles Triangle Theorem If two sides of a triangle are, then the two angles opposite the sides are. This theorem is sometimes called the ase ngles Theorem and can also be stated as ase angles of an isosceles triangle are. Module 7 328 Lesson 2 PROFESSIONL DEVELOPMENT Houghton Mifflin Harcourt Publishing ompany How could you draw isosceles triangles without using a compass? Possible answer: Draw and plot point on one side of. Then use a ruler to measure and plot point on the other side of so that =. EXPLIN 1 Proving the Isosceles Triangle Theorem and Its onverse ONNET VOULRY sk a volunteer to define isosceles triangle and have students give real-world examples of them. If possible, show the class a baseball pennant or other flag in the shape of an isosceles triangle. Tell students they will be proving theorems about isosceles triangles and investigating their properties in this lesson. Learning Progressions In this lesson, students add to their prior knowledge of isosceles and equilateral triangles by investigating the Isosceles Triangle Theorem from both an inductive and deductive perspective. The opening activity leads students to make a conjecture about the measures of the base angles of an isosceles triangle. Students prove their conjecture and its converse later in the lesson. They also prove the Equilateral Triangle Theorem and its converse, and use the properties of both types of triangles to find the unknown measure of angles and sides in a triangle. ll students should develop fluency with various types of triangles as they continue their study of geometry. Isosceles and Equilateral Triangles 328

QUESTIONING STRTEGIES What can you say about an isosceles triangle,, with base angles and, if you know that m = 100? Explain. y the Isosceles Triangle Theorem,, and m + m = 80 by the Triangle Sum Theorem, so m = m = 40. What can you say about the angles of an isosceles right triangle? The angles of the triangle measure 90, 45, and 45. Example 1 Prove the Isosceles Triangle Theorem and its converse. Step 1 omplete the proof of the Isosceles Triangle Theorem. Given: Prove: Statements Reasons 1. 1. Given 2. 2. Reflexive Property of ongruence 3. 3. Symmetric Property of Equality 4. 4. SS Triangle ongruence Theorem 5. 5. PT Step 2 omplete the statement of the onverse of the Isosceles Triangle Theorem. If two angles of a triangle are, then the two sides opposite those angles are. Step 3 omplete the proof of the onverse of the Isosceles Triangle Theorem. Given: Prove: Houghton Mifflin Harcourt Publishing ompany Statements Reasons 1. 1. Given 2. 2. Reflexive Property of ongruence 3. 3. Symmetric Property of Equality 4. 4. S Triangle ongruence Theorem 5. 5. PT Reflect 3. Discussion In the proofs of the Isosceles Triangle Theorem and its converse, how might it help to sketch a reflection of the given triangle next to the original triangle, so that vertex is on the right? Possible answer: Sketching a copy of the triangle makes it easier to see the two pairs of corresponding sides and the two pairs of corresponding angles. Module 7 329 Lesson 2 OLLORTIVE LERNING Small Group ctivity Geometry software allows students to explore the theorems in this lesson. For the Isosceles Triangle Theorem (or the Equilateral Triangle Theorem), students should construct an isosceles (or equilateral) triangle and measure the angles. s students drag the vertices of the triangle to change its size or shape, the individual base angle measures will change (for isosceles only), but the relationship between the lengths of the sides and the measures of the angles will remain the same. 329 Lesson 7.2

Explain 2 Proving the Equilateral Triangle Theorem and Its onverse n equilateral triangle is a triangle with three sides. n equiangular triangle is a triangle with three angles. Equilateral Triangle Theorem EXPLIN 2 Proving the Equilateral Triangle Theorem and Its onverse If a triangle is equilateral, then it is equiangular. Example 2 Prove the Equilateral Triangle Theorem and its converse. Step 1 omplete the proof of the Equilateral Triangle Theorem. Given: Prove: Given that we know that by the Isosceles Triangle Theorem. It is also known that by the Isosceles Triangle Theorem, since. Therefore, by substitution. Finally, by the Transitive Property of ongruence. OLLORTIVE LERNING The converse of this theorem is proved interactively using a paragraph proof. Have small groups of students discuss the proof and highlight the important statements (steps) and reasons for the statements. sk them how they would present the same proof using the two-column method. QUESTIONING STRTEGIES The converse of the Equilateral Triangle Theorem is also true. onverse of the Equilateral Triangle Theorem If a triangle is equiangular, then it is equilateral. Step 2 omplete the proof of the onverse of the Equilateral Triangle Theorem. Reflect Given: Prove: ecause, by the onverse of the Isosceles Triangle Theorem. by the onverse of the Isosceles Triangle Theorem because. Thus, by the Transitive Property of ongruence,, and therefore,. Houghton Mifflin Harcourt Publishing ompany What is the connection between equilateral triangles and equiangular triangles? If a triangle is equilateral, then it is also equiangular. If a triangle is equiangular, then it is also equilateral. VOID OMMON ERRORS Some students may confuse the theorems in this lesson because they are so similar. Have students draw and label diagrams to illustrate the theorems and then add visual cues, if needed, to help them remember how the theorems are applied. 4. To prove the Equilateral Triangle Theorem, you applied the theorems of isosceles triangles. What can be concluded about the relationship between equilateral triangles and isosceles triangles? Possible answer: Equilateral/equiangular triangles are a special type of isosceles triangles. Module 7 330 Lesson 2 DIFFERENTITE INSTRUTION Visual ues Visually represent the Equilateral Triangle Theorem and its converse: Equilateral Triangle Theorem onverse If then If then Isosceles and Equilateral Triangles 330

EXPLIN 3 Using Properties of Isosceles and Equilateral Triangles INTEGRTE MTHEMTIL PRTIES Focus on Math onnections MP.1 Encourage students to discuss how the Triangle Sum Theorem and the theorems in this lesson help them solve for the unknown angles and sides of an isosceles or equilateral triangle. Have them share their ideas about the best method to use to solve for the unknown quantities in each problem. Explain 3 Using Properties of Isosceles and Equilateral Triangles You can use the properties of isosceles and equilateral triangles to solve problems involving these theorems. Example 3 Find the indicated measure. Katie is stitching the center inlay onto a banner that she created to represent her new tutorial service. It is an equilateral triangle with the following dimensions in centimeters. What is the length of each side of the triangle? 4x + 7 To find the length of each side of the triangle, first find the value of x. onverse of the Equilateral Triangle Theorem = 6x 5 = 4x + 7 6x - 5 Definition of congruence Substitution Property of Equality QUESTIONING STRTEGIES If the triangle is equiangular, how do you find the measure of one of its angles? Divide the sum of the interior angles by the number of interior angles: 180 3 = 60. Houghton Mifflin Harcourt Publishing ompany Image redits: Nelvin. epeda/zum Press/orbis x = 6 Solve for x. Substitute 6 for x into either 6x 5 or 4x + 7. 6 (6) 5 = 36 5 = 31 or 4 (6) + 7 = 24 + 7 = 31 So, the length of each side of the triangle is 31 cm. m T T 3x x R S To find the measure of the vertex angle of the triangle, first find the value of x. m R = m S = x Isosceles Triangle Theorem m R + m S + m T = 180 Triangle Sum Theorem x + x + 3x = 180 Substitution Property of Equality 5x = 180 ddition Property of Equality x = 36 Division Property of Equality 36 So, m T = 3x = 3( ) = 108. Module 7 331 Lesson 2 LNGUGE SUPPORT onnect Vocabulary Help students understand the meanings of isosceles, equilateral, and equiangular by having them make a poster showing each type of triangle along with its definition. n isosceles triangle has two sides, an equilateral triangle has three sides, and an equiangular triangle has three angles. Relate the prefix equi- to equal to help students make connections between the terms. 331 Lesson 7.2

Your Turn 5. Find m P. ELORTE P (3x + 3) R Q (5x - 2) m P = m Q = (3x + 3) 2 (3x + 3) + (5x - 2) = 180 x = 16 m P = (3x + 3) = (3 (16) + 3) = 51 6. Katie s tutorial service is going so well that she is having shirts made with the equilateral triangle emblem. She has given the t-shirt company these dimensions. What is the length of each side of the triangle in centimeters? = 3 10 y + 9 = 4 5 y - 1 20 = y Therefore, 3 10 y + 9 = 3 (20) + 9 = 6 + 9 = 15 10 The length of each side is 15 cm. 3 y + 9 10 4 y - 1 5 QUESTIONING STRTEGIES How do you use the Isosceles Triangle Theorem to find the measures of the base angles of an isosceles triangle, given a known value for the measure of the vertex angle? Subtract the measure of the vertex angle from 180, and then divide the answer by 2 to find the measure of each base angle. How do you use the Equilateral Triangle Theorem to find the measures of the angles of an equilateral triangle? The theorem says that the triangle is equiangular, so each angle must measure 60. Elaborate 7. Discussion onsider the vertex and base angles of an isosceles triangle. an they be right angles? an they be obtuse? Explain. The vertex angle of an isosceles triangle can be acute, right, or obtuse as long as its measure is less than 180. The base angles of an isosceles triangle can only be acute, meaning they have a measurement less than 90. because otherwise they would cause the sum of the base angles to be 180 before adding in the third angle, which contradicts the Triangle Sum Theorem. 8. Essential Question heck-in Discuss how the sides of an isosceles triangle relate to its angles. The legs of an isosceles triangle are opposite from the base angles and because the base angles are, the legs are also because of the onverse of the Isosceles Triangle Theorem. Module 7 332 Lesson 2 Houghton Mifflin Harcourt Publishing ompany SUMMRIZE THE LESSON Have students fill out charts for the two theorems and their converses. Sample: Theorem onverse Isosceles Triangle If Then 2 sides 2 angles 2 angles 2 sides Equilateral Triangle If Then Theorem 3 sides 3 angles onverse 3 angles 3 sides Isosceles and Equilateral Triangles 332

EVLUTE SSIGNMENT GUIDE oncepts and Skills Explore Investigating Isosceles Triangles Example 1 Proving the Isosceles Triangle Theorem and Its onverse Example 2 Proving the Equilateral Triangle Theorem and Its onverse Example 3 Using Properties of Isosceles and Equilateral Triangles Practice Exercise 1 Exercise 2 Exercise 3 Exercises 4 13 Evaluate: Homework and Practice 1. Use a straightedge. Draw a line. Draw an acute angle with vertex along the line. Then use a compass to copy the angle. Place the compass point at another point along the line and draw the copied angle so that the angle faces the original angle. Label the intersection of the angle sides as point. Look at the triangle you have formed. What is true about the two base angles of? What do you know about and? What kind of triangle did you form? Explain your reasoning., so opposite sides and are. Therefore, it is an isosceles triangle. 2. Prove the Isosceles Triangle Theorem as a paragraph proof. Given: Prove: Proof: It is given that. y the Reflexive Property of ongruence,. Given that, then by the SS Triangle ongruence Theorem. Therefore, because corresponding parts of triangles are. Online Homework Hints and Help Extra Practice INTEGRTE MTHEMTIL PRTIES Focus on Reasoning MP.2 Have students analyze the following statement: If a triangle is equilateral, then the triangle is isosceles. Is the statement true? (yes) Is the converse of the statement true? (no) Have them use the properties of isosceles and equilateral triangles to justify their answers. Houghton Mifflin Harcourt Publishing ompany 3. omplete the flow proof of the Equilateral Triangle Theorem. Given: Prove: Given Given Isosceles Triangle Theorem Isosceles Triangle Theorem Substitution Transitive Property of ongruence Module 7 333 Lesson 2 Exercise Depth of Knowledge (D.O.K.) OMMON ORE Mathematical Practices 1 2 Skills/oncepts MP.6 Precision 2 3 2 Skills/oncepts MP.4 Modeling 4 11 2 Skills/oncepts MP.2 Reasoning 12 20 2 Skills/oncepts MP.4 Modeling 21 3 Strategic Thinking MP.3 Logic 22 3 Strategic Thinking MP.5 Using Tools 23 3 Strategic Thinking MP.5 Using Tools 333 Lesson 7.2

Find the measure of the indicated angle. 4. m 5. m R S 46 (2x + 2) T m = m = 46 m S = m T = (2x + 2) m = 180 - m - m m S + m T + m R = 180 m = (180-46 - 46) 2m T + m R = 180 m = 88 2 (2x + 2) + (3x + 1) = 180 4x + 4 + 3x + 1 = 180 7x + 5 = 180 7x = 175 x = 25 (3x + 1) R INTEGRTE MTHEMTIL PRTIES Focus on ommunication MP.3 Discuss each of the following questions about triangles as a class. Have students explain how the theorems in this lesson justify their responses. 1. Each acute angle of an obtuse isosceles triangle must be less than 45. 2. Each angle of an equilateral triangle must be 60. 3. Each acute angle of an isosceles right triangle must be 45. m R = (3x + 1) = (3(25) + 1) = (75 + 1) = 76 6. m O 7. m E D F N (4x + 1) (5x - 4) 7y M O (4y - 15) m O = m M = (4y - 15) m O + m M + m N = 180 2m O + m N = 180 2 (4y - 15) + 7y = 180 8y - 30 + 7y = 180 15y - 30 = 180 y = 14 m O = (4y - 15) = (4(14) - 15) = (56-15) = 41 E m D = m E = (4x + 1) m D + m E + m F = 180 2m E + m F = 180 2 (4x + 1) + (5x - 4) = 180 8x + 2 + 5x - 4 = 180 13x - 2 = 180 13x = 182 x = 14 m E = (4x + 1) = (4 (14) + 1) = (56 + 1) = 57 Houghton Mifflin Harcourt Publishing ompany Module 7 334 Lesson 2 Isosceles and Equilateral Triangles 334

VOID OMMON ERRORS In this lesson, students prove the Isosceles Triangle Theorem by using the SS ongruence riterion. Having seen this congruence criterion, students may assume another congruence criterion can be used to prove the theorem. Encourage students to try other methods of proof, but be sure students understand that some conditions may not apply to isosceles triangles. Find the length of the indicated side. 8. DE D 9. E 3x - 4 DF EF, so DF = EF. 3x - 4 = 5x - 12-4 = 2x - 12 8 = 2x 4 = x DE = DF = EF 5x - 12 DE = 3x - 4 = 3 (4) - 4 = 12-4 = 8 DE = 8 F KL 10x + 3 K J 3x + 24 JK KL, so JK = KL 10x + 3 = 3x + 24 7x + 3 = 24 7x = 21 x = 3 KL = 3x + 24 = 3 (3) + 24 = 33 KL = 33 L 10. 11. 3 x + 4 2 1 x + 9 5 5 y - 1 4 7 y - 2 3 Houghton Mifflin Harcourt Publishing ompany, so =. 3 1 2 x + 4 = 5 x + 9 13 10 x + 4 = 9 13 10 x = 5 50 x = 13 3 = 2 ( 50 13 ) + 4 75 52 = 13 + 13 = 127 13 = 127 13, so = =. 5 7 4 y - 1 = 3 y - 2 15 12 y - 1 = 28 12 y - 2-13 12 y = -1 12 y = 13 5 = 4 ( 12 13 ) - 1 60 52 = 52-52 = 2 13 Module 7 335 Lesson 2 335 Lesson 7.2

12. Given JKL with m J = 63 and m L = 54, is the triangle an acute, isosceles, obtuse, or right triangle? y the Triangle Sum Theorem, m K = 63, so the triangle is an acute isosceles triangle because all angle measures are less than 90. 13. Find x. Explain your reasoning. The horizontal lines are parallel. x 107 y the def. supp., the base angles of the top triangle have a measure of 73. Therefore, the measure of the vertex angle is 34 by the Triangle Sum Theorem. The base angles of the bottom isosceles triangle will also measure 34 by the Vertical ngles Theorem. Thus, x will equal 112 by the Triangle Sum Theorem. PEER-TO-PEER DISUSSION Have students work in pairs to create a poster that shows examples of how to find the unknown sides or angles of an isosceles and an equilateral triangle, when given the measure of one angle or an algebraic expression. 14. Summarize omplete the diagram to show the cause and effect of the theorems covered in the lesson. Explain why the arrows show the direction going both ways. t least two sides Isosceles ase angles are. Triangles are. ll, or three sides are. Equilateral Triangles ll angles are. Possible explanation: The arrows go both ways because each theorem and its converse are both true. 15. plane is flying parallel to the ground along. When the plane is at, an air-traffic controller in tower T measures the angle to the plane as 40. fter the plane has traveled 2.4 miles to, the angle to the plane is 80. How can you find T? y the ngle ddition Postulate, m T = 80-40 = 40. m T = 40 by lt. Int. Thm. T T by the definition of congruence and T by the onverse of the Isosceles Triangle Theorem. Then = T = 2.4 mi. 2.4 mi 80º 40º T Houghton Mifflin Harcourt Publishing ompany Module 7 336 Lesson 2 Isosceles and Equilateral Triangles 336

16. John is building a doghouse. He decides to use the roof truss design shown. If m DF = 35, what is the measure of the vertex angle of the isosceles triangle? D E F (2x + 15) G y the Isosceles Triangle Theorem,, so m = (2x + 15). m DF + m D + m = 180 35 + 90 + (2x + 15) = 180 2x + 140 = 180 2x = 40 x = 20 m = (2 (20) + 15) = 55 m = 180 - m - m = 180-2 (55 ) = 180-110 = 70 The measure of the vertex angle is 70. 17. The measure of the vertex angle of an isosceles triangle is 12 more than 5 times the measure of a base angle. Determine the sum of the measures of the base angles. 2 (measure of base angle) + (measure of vertex angle) = 180 2 (x) + (5x + 12) = 180 x = 24 The sum of the measures of the base angles is 24 + 24 = 48. Houghton Mifflin Harcourt Publishing ompany Image redits: De Klerk/ lamy 18. Justify Reasoning Determine whether each of the following statements is true or false. Select the correct answer for each lettered part. Explain your reasoning. a. ll isosceles triangles have at least True False two acute angles. b. If the perimeter of an equilateral True False triangle is P, then the length of each of its sides is P 3. c. ll isosceles triangles are True False equilateral triangles. d. If you know the length of one of True False the legs of an isosceles triangle, you can determine its perimeter. e. The exterior angle of an equilateral True False triangle is obtuse. a. t least the base angles have to be acute. b. ecause all three sides are equal, dividing P by 3 gives the length of each side. c. n isosceles triangle requires only a minimum of two sides being, not all three. d. You need to know the length of one leg and the base to determine perimeter. e. The exterior angle of a triangle is equal to the sum of its remote interior angles. The exterior angle measures 60 + 60 = 120, which is obtuse. Module 7 337 Lesson 2 337 Lesson 7.2

19. ritical Thinking Prove, given point M is the midpoint of. M Statements Reasons 1. M is the midpoint of. 1. Given 2. M M 2. Definition of midpoint 3. 3. Given 4. M M 4. Reflexive Property of ongruence 5. M M 5. SSS Triangle ongruence Theorem 6. 6. PT INTEGRTE MTHEMTIL PRTIES Focus on Math onnections MP.1 Remind students that an isosceles triangle has at least two sides, and its properties can be used to prove that it also has at least two angles. 20. Given that is an isosceles triangle and D and D are angle bisectors, what is m D? m = m = 70, so m D = m D = 35. Then, m D = 180 (35 + 35 ) = 110. 40 D H.O.T. Focus on Higher Order Thinking 21. nalyze Relationships Isosceles right triangle has a right angle at and. D bisects angle, and point D is on. If D, describe triangles D and D. Explain. HINT: Draw a diagram. D The triangles are isosceles triangles; the bisected right angle results in two 45 angles, and the perpendicular segments result in two right angles, so angles and must also measure 45. Since D D by the Reflexive Property, triangles D and D are by S. Houghton Mifflin Harcourt Publishing ompany Module 7 338 Lesson 2 Isosceles and Equilateral Triangles 338

JOURNL ompare the hypothesis and the conclusion of the Isosceles Triangle Theorem with its converse. Support your comparison with a sketch. ommunicate Mathematical Ideas Follow the method to construct a triangle. Then use what you know about the radius of a circle to explain the congruence of the sides. 22. onstruct an isosceles triangle. Explain how you know that two sides are. Use a compass to draw a circle. Mark two different points on the circle. Use a straightedge to draw a line segment from the center of the circle to each of the two points on the circle (radii). Draw a line segment (chord) between the two points on the circle. I know two sides are because the two line segments I drew from the center each represent the radius of the circle and so are the equal-length sides of an isosceles triangle. Houghton Mifflin Harcourt Publishing ompany 23. onstruct an equilateral triangle. Explain how you know the three sides are. Use a compass to draw a circle. Draw another circle of the same size that goes through the center of the first circle. (oth should have the same radius length.) Mark one point where the circles intersect. Use a straightedge to draw line segments connecting both centers to each other and to the intersection point. I know the three sides are because the three line segments drawn are radii, which have the same length in both circles, since the circles are the same size. Therefore, all of the line segments are and form the three sides of an equilateral triangle. Module 7 339 Lesson 2 339 Lesson 7.2

Lesson Performance Task The control tower at airport is in contact with an airplane flying at point P, when it is 5 miles from the airport, and 30 seconds later when it is at point Q, 4 miles from the airport. The diagram shows the angles the plane makes with the ground at both times. If the plane flies parallel to the ground from P to Q at constant speed, how fast is it traveling? VOID OMMON ERRORS When students have determined that the plane has traveled 4 miles in 30 seconds, they may convert units incorrectly to find the plane s speed in miles per hour. The correct conversion is: 4 mi Q 5 mi P 4 mi 30 sec 60 sec 1 min 60min = 14,400 mi = 480 mi 1 hr 30 hr 1 hr m QP = 80-40 = 40 80 40 QP and T are parallel, so m QP = m PT = 40 because the angles form alternate interior angles. y the converse of the Isosceles Triangle Theorem, QP is isosceles because m QP = m QP = 40. In isosceles QP, QP = Q = 4 miles. It took the plane 30 seconds to travel 4 miles, so it was traveling 4 2 = 8 miles per minute, or 8 60 = 480 miles per hour. T Houghton Mifflin Harcourt Publishing ompany INTEGRTE MTHEMTIL PRTIES Focus on Reasoning MP.2 Have students refer to the diagram in the Lesson Performance Task, changing the angles measuring 40 and 80 to angles measuring a and (2a). sk them to justify the conclusion that no matter how high the plane is, the distance it will travel from P to Q will equal the distance from Q to. Sample answer: m QP = (2a) - a = a. m QP = m PT = a because the angles are alternate interior angles for parallel lines QP and T. QP is isosceles by the converse of the Isosceles Triangle Theorem. Therefore, PQ = Q. Module 7 340 Lesson 2 EXTENSION TIVITY Have students research a technique used by marine navigators called a doubled angle fix or doubled angle on the bow. The problem in the Lesson Performance Task is based on this method. Students should describe how a marine navigator could use the method to find the distance of an object on shore, and how an isosceles triangle would be used in the calculation. Scoring Rubric 2 points:student correctly solves the problem and explains his/her reasoning. 1 point: Student shows good understanding of the problem but does not fully solve or explain. 0 points: Student does not demonstrate understanding of the problem. Isosceles and Equilateral Triangles 340