20.1 Exploring What Makes Triangles Congruent

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1 ame lass ate 20.1 xploring What akes Triangles ongruent ssential Question: How can you show that two triangles are congruent? esource ocker xplore Transforming Triangles with ongruent orresponding arts You can apply what you ve learned about corresponding parts of congruent figures to write the following true statement about triangles. If two triangles are congruent, then the corresponding parts of the triangles are congruent. The statement is sometimes referred to as T. The converse of T can be stated as follows. If all corresponding parts of two triangles are congruent, then the triangles are congruent. Use a straightedge and tracing paper to explore this converse statement. Trace the angles and segments shown to draw. epeat the process to draw on a separate piece of tracing paper. abel the triangles. and and and and Houghton ifflin Harcourt ublishing ompany and What must you do to show that the triangles are congruent? and lip the piece of tracing paper with and arrange the two triangles on a desk as shown in the figure. Then move the tracing paper with so that point maps to point. ame the rigid motion that you used. odule esson 1

2 ame a rigid motion you can use to map point to point. How can you be sure the image of is? ame a rigid motion you can use to map point to point. To show that the image of point is point, complete the following. is reflected across, so the measure of the angle is preserved. Since, you can conclude that the image of lies on. It is given that _, so the image of point must be. What sequence of rigid motions maps onto? eflect 1. iscussion Is there another sequence of rigid motions that maps onto? xplain. 2. iscussion Is the converse of T always true when you apply it to triangles? xplain why or why not based on the results of the xplore. Houghton ifflin Harcourt ublishing ompany odule esson 1

3 xplain 1 eciding If Triangles are ongruent by omparing orresponding arts biconditional is a statement that can be written in the form p if and only if q. You can combine what you learned in the xplore with the fact that corresponding parts of congruent triangles are congruent to write the following true biconditional. Two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. To decide whether two triangles are congruent, you can compare the corresponding parts. If they are congruent, the triangles are congruent. If any of the corresponding parts are not congruent, then the triangles are not congruent. xample 1 etermine whether the given triangles are congruent. xplain. 2 m m 1 m 60 J 60 1 m H 1.7 m 30 2 m ompare corresponding sides to decide if they are congruent. H = = 1.7 m, HJ = = 1 m, and J = = 2 m. So, H, HJ, and J. ompare corresponding angles to decide if they are congruent. m = m = 30, m H = m = 90, and m J = m = 60. So,, H, and J. Houghton ifflin Harcourt ublishing ompany HJ because all pairs of corresponding parts are congruent cm 7.9 cm 8.5 cm 8.4 cm 7.9 cm 12.1 cm ompare corresponding sides to decide if they are congruent. = = cm, so _. = = cm, so _. However,, so _ is not congruent to. The triangles are not congruent because odule esson 1

4 eflect 3. ritique easoning The contrapositive of a conditional statement if p, then q is the statement If not q, then not p. The contrapositive of a true statement is always true. Janelle says that you can justify art using the contrapositive of T. Is this accurate? xplain your reasoning. Your Turn etermine whether the given triangles are congruent. xplain your reasoning S U T V W xplain pplying roperties of ongruent Triangles Triangles are part of many interesting designs. You can ensure that triangles are congruent by making corresponding sides congruent and corresponding angles congruent. To do this, you may have to use the Triangle Sum Theorem, which states that the sum of the measures of the angles of a triangle is 180. You will explore this theorem in more detail later in this course. Houghton ifflin Harcourt ublishing ompany Image redits: ichael Zegers/Westend61/orbis odule esson 1

5 xample 2 ind the value of the variable that results in congruent triangles J 55 (5x + 30) Step 1 Identify corresponding angles. corresponds to J, because they have the same measure and they are formed by congruent corresponding sides. Similarly, corresponds to. So, corresponds to. Step 2 ind m. Triangle Sum Theorem m J + m + m = 180 Substitute m = 180 Simplify m = 180 Subtract 100 from each side. m = 80 Step 3 Write an equation to find the value of x. Set corresponding measures equal. m = m Substitute. 5x + 30 = 80 Subtract 30 from each side. 5x = 50 ivide each side by 5. x = 10 Houghton ifflin Harcourt ublishing ompany 50 mm 36 mm 30 mm (2y + 20) mm 30 mm 50 mm Step 1 Identify corresponding sides, beginning with side _.,, and,, so _ corresponds to. Step 2 Write an equation to find the value of y. Set corresponding measures equal. = mm Substitute. 2y + 20 = Subtract 20 from each side. 2y = ivide each side by 2. y = odule esson 1

6 eflect 6. The measures of two angles of QS are 18 and 84. The measures of two angles of TUV are 18 and 76. Is it possible for the triangles to be congruent? xplain. Your Turn ind the value of the variable that results in congruent triangles. 7. (x + 17) 8. J (4y + 12) mm 22 mm 15 mm 22 mm 15 mm 32 mm laborate 9. ll three angles of measure 60 and all three sides are 4 inches long. ll three angles of Q measure 60 and all three sides are 4 inches long. an you conclude that the triangles are congruent? Why or why not? 10. Use the concept of rigid motion to explain why two triangles cannot be congruent if any pair of corresponding parts is not congruent. 11. ssential Question heck-in Q and STU have six pairs of congruent corresponding parts and Q can be mapped onto STU by a translation followed by a rotation. How are the triangles related? xplain your reasoning. Houghton ifflin Harcourt ublishing ompany odule esson 1

7 valuate: Homework and ractice 1. escribe a sequence of rigid motions that maps onto Q to show that Q. Online Homework Hints and Help xtra ractice Q etermine whether the given triangles are congruent. xplain your reasoning. 2. Q S ft 70º 7.2 ft 20º 6.8 ft 6.8 ft 20º 7.2 ft 70º 2.5 ft in. T S V 7 in. 82 U ft J 32 ft 32 ft 32 ft 17 ft 32 ft Houghton ifflin Harcourt ublishing ompany W ind the value of the variable that results in congruent triangles cm 2.2 cm 50 (2x + 5) H 3.1 cm 3.1 cm T (3z - 3) cm S odule esson 1

8 8. 9. (7y - 10) mm 31 mm 31 mm 24 mm H X 24 mm 25 mm J 22 Y 36 Z (4w + 14) Q etermine whether the given triangles are congruent. xplain y x y Q x 4 Houghton ifflin Harcourt ublishing ompany odule esson 1

9 12. S y T -2 X U x W 18.4 V y 2 4 x Houghton ifflin Harcourt ublishing ompany Image redits: eonard de Selva/orbis 14. H represents an artist s initial work on a design for a new postage stamp. What must be the values of x, y, and z in order for the artist s stamp to be congruent to? (2x + 5) mm (3z + 14) mm (5y) mm H 35 mm 25 mm 25 mm odule esson 1

10 15. ulti-step ind the values of the variables that result in congruent triangles. (3x + 24)º (7x - 4)º (2x + y)º Q etermine whether each statement is always, sometimes, or never true. xplain your reasoning. 16. If has angles that measure 10 and 40, and has angles that measure 40 and 120, then. 17. Two triangles with different perimeters are congruent. 18. If J, then m = m. 19. Two triangles that each contain a right angle are congruent. 20. Tenaya designed the earrings shown. She wants to be sure they are congruent. She knows that the three pairs of corresponding angles are congruent. What additional measurements should she make? xplain. 37 mm 35 mm Houghton ifflin Harcourt ublishing ompany 12 mm odule esson 1

11 21. etermine whether J and Q are congruent or not congruent based on the given information. Select the correct answer for each lettered part. a. m J = m = m = 60, m = m Q = m = 60, J = = J = 1.2 cm, Q = Q = = 1.5 cm ongruent ot congruent b. m J _ = _ 48, _ m _ = _ 93, _ m = 48, m = 39, J Q, Q, J ongruent ot congruent c. J, Q,, J = Q = 22 in., = Q = 34 in., J = = 28 in. ongruent ot congruent d. m J = 51, m = 77, m = 51, m = 53 ongruent ot congruent e. m J = 45, m = 80, m Q = 80, m = 55, J = Q = 1.5 mm, = Q = 1.3 mm, J = = 1.8 mm ongruent ot congruent H.O.T. ocus on Higher Order Thinking 22. ounterexamples Isaiah says it is not necessary to check all six pairs of congruent corresponding parts to decide whether two triangles are congruent. He says that it is enough to check that the corresponding angles are congruent. Sketch a counterexample. xplain your counterexample. Houghton ifflin Harcourt ublishing ompany 23. ritique easoning elly was asked to determine whether is congruent to. She noted that,, and that the three pairs of corresponding angles are congruent. She said that this is only five pairs of congruent corresponding parts, so it is not possible to conclude that is congruent to. o you agree? xplain. 44 in. 72 in. 72 in. 44 in. 24. nalyze elationships avid uses software to draw two triangles. He finds that he can use a rotation and a reflection to map one triangle onto the other, and he finds that the image of vertex is vertex, the image of vertex V is vertex, and the image of vertex W is vertex Y. In how many different ways can avid write a congruence statement for the triangles? xplain. odule esson 1

12 esson erformance Task or enny s science project, he is studying whether honeybees favor one color of eight-petal flowers over other colors. or his display, he is making eight-petal flowers from paper in various colors. or each flower, he ll cut out eight triangles like the one in the figure. x y y a. ind x, the measure in degrees of the top angle of each triangle. xplain how you found x. b. ind y, the measure in degrees of the two base angles of each triangle. xplain how you found y. c. xplain how enny could confirm that one of his triangles is congruent to the other seven. Houghton ifflin Harcourt ublishing ompany odule esson 1