Isosceles and Equilateral Triangles

Transcription

1 Locker LESSON 7.2 Isosceles and Equilateral Triangles Texas Math Standards The student is expected to: G.6.D Verify theorems about the relationships in triangles, including... base angles of isosceles triangles... and apply these relationships to solve problems. Mathematical Processes G.1.G Display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication. Language Objective 2..4, 2.D.1, 2.D.2, 3..3, 3.H.3 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. 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? G.6.D Verify theorems about the relationships in triangles, including base angles of isosceles triangles and apply these relationships to solve problems. 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 Using a compass, place the point on the vertex and draw an arc that intersects the sides of the angle. Label the points and. ase ase angles heck students construtions. Legs Resource Locker 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. Module 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.6.D Verify theorems about the relationships in triangles, including base angles of isosceles triangles and apply these relationships to solve problems. 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 Turn to these pages to find this lesson in the hardcover student edition. Module Lesson Lesson 7.2

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 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 Lesson 2 PROFESSIONL DEVELOPMENT 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. 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. 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. Isosceles and Equilateral Triangles 368

3 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 Given Reflexive Property of ongruence Symmetric Property of Equality SS Triangle ongruence Theorem PT Step 2 omplete the statement of the onverse of the Isosceles Triangle Theorem. 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 Given Reflexive Property of ongruence Symmetric Property of Equality S Triangle ongruence Theorem 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 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. 369 Lesson 7.2

4 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 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 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? a triangle is equilateral, then it is also equiangular. 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 Lesson 2 DIFFERENTITE INSTRUTION Visual ues Visually represent the Equilateral Triangle Theorem and its converse: Equilateral Triangle Theorem onverse then then Isosceles and Equilateral Triangles 370

5 EXPLIN 3 Using Properties of Isosceles and Equilateral Triangles INTEGRTE MTHEMTIL PROESSES Focus on Math onnections 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. QUESTIONING STRTEGIES 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: = 60. Houghton Mifflin Harcourt Publishing ompany Image redits: Nelvin. epeda/zum Press/orbis 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? R 6x - 5 4x + 7 To find the length of each side of the triangle, first find the value of x. onverse of the Equilateral Triangle Theorem m T = 6x 5 = 4x + 7 x = 6 Solve for x. x Definition of congruence Substitute 6 for x into either 6x 5 or 4x + 7. Substitution Property of Equality 6 (6) 5 = 36 5 = 31 or 4 (6) + 7 = = 31 So, the length of each side of the triangle is 31 cm. T 3x To find the measure of the vertex angle of the triangle, first find the value of x. m R = m S = x m R + m S + m T = 180 Triangle Sum Theorem x + x + 3x = 180 S Isosceles Triangle Substitution Property of Equality Theorem 5x = 180 ddition Property of Equality x = 36 Division Property of Equality 36 So, m T = 3x = 3( ) = 108. Module 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. 371 Lesson 7.2

6 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) = 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 = y Therefore, 3 10 y + 9 = 3 (20) + 9 = = The length of each side is 15 cm. 3 y y 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 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 Then 2 sides 2 angles 2 angles 2 sides Equilateral Triangle Then Theorem 3 sides 3 angles onverse 3 angles 3 sides Isosceles and Equilateral Triangles 372