5.3 Proving Triangles are

Transcription

1 roving riangles are ongruent: and Goal how triangles are congruent using and. ey Words vertical angles p. 75 alternate interior angles p. 121 Geo-ctivity 1 raw a segment 3 inches long. abel the endpoints and. reating ongruent riangles 2 raw an angle measuring 45 at point raw an angle measuring 30 at point. 4 abel the point where the angle rays intersect as point ompare your triangle to the triangles drawn by your classmates. re the triangles congruent? he Geo-ctivity above suggests the following postulate. Visualize It! In this triangle, &* is the included side between a and a. included side O 14 ngle-ide-ngle ongruence ostulate () Words If two angles and the included side of one triangle are congruent to two angles and the included side of a second triangle, then the two triangles are congruent. ymbols If ngle a ca, and ide &* c &*, and ngle a ca, then c. 250 hapter 5 ongruent riangles

2 X 1 etermine When o se ongruence ased on the diagram, can you use the ongruence ostulate to show that the triangles are congruent? xplain your reasoning. a. b. tudent elp Y I You can use the ongruence ostulate because &* and &* are included between the congruent angles. a. You are given that a ca, a ca, and &*c &*. You can use the ongruence ostulate to show that c. X Y b. You are given that a cay and a cax. You know that &* c Y &*, but these sides are not included between the congruent angles, so you cannot use the ongruence ostulate. O 5.1 ngle-ngle-ide ongruence heorem () Words If two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of a second triangle, then the two triangles are congruent. ymbols If ngle a ca, and ngle a ca, and ide &* c &*, then c. X 2 etermine What Information is issing What additional congruence is needed to show that c by the ongruence heorem? You are given &* c &**. ecause a and a are vertical angles, a ca. he angles that make &* and &** the non-included sides are a and a, so you need to know that a ca. 5.3 roving riangles are ongruent: and 251

3 X 3 ecide Whether riangles are ongruent oes the diagram give enough information to show that the triangles are congruent? If so, state the postulate or theorem you would use. a. b. c. G W X tudent elp Y I he sides of Q look parallel but this information is not marked, so you cannot conclude that any angles are congruent. a. &* c &* Given a ca Given ag cag Vertical ngles heorem se the ongruence heorem to conclude that G cg. b. ased on the diagram, you know only that &** c Q &** and &* c &*. You cannot conclude that the triangles are congruent. c. aw caxw lternate Interior ngles heorem W &** c W &** eflexive rop. of ongruence aw caxw lternate Interior ngles heorem se the ongruence ostulate to conclude that W cxw. X 4 rove riangles are ongruent step in the at s radle string game creates the triangles shown. rove that c. ecause a and a are also alternate interior angles, you can show that c by the ongruence heorem. Visualize It! Given rove tatements 1. &* c &* 2. &* &* &* c &*, &* &* c easons 1. Given 2. Given 3. a ca 3. lternate Interior ngles heorem 4. a ca 4. Vertical ngles heorem 5. c 5. ongruence ostulate 252 hapter 5 ongruent riangles

4 ecide Whether riangles are ongruent 1. omplete the statement: You can use the ongruence ostulate when the congruent sides are? between the corresponding congruent angles. oes the diagram give enough information to show that the triangles are congruent? If so, state the postulate or theorem you would use G xercises Guided ractice Vocabulary heck ell whether the side is included or not included between the given angles. 1. G&* is? between a and ag. 2. G&* is? between a and ag. 3. &** is? between a and ag. 4. G & * is? between a and ag. G kill heck What congruence do you need to know in order to use the indicated postulate or theorem to conclude that the triangles are congruent? 5. ongruence ostulate 6. ongruence heorem oes the diagram give enough information to show that the triangles are congruent? If so, state the postulate or theorem you would use. 7. and Q 8. and 9. and 5.3 roving riangles are ongruent: and 253

5 ractice and pplications xtra ractice ee p ecognizing Included ides ell whether the side is included or not included between the given angles. 10. &* is? between a and a. 11. &* is? between a and a. 12. &* is? between a and a. 13. &* is? between a and a. hoosing or ell which postulate or theorem you would use to show that the triangles are congruent. xplain your reasoning howing riangles are ongruent oes the diagram give enough information to show that the triangles are congruent? If so, state the postulate or theorem you would use. xplain your reasoning V X 22. W G Y omework elp xample 1: xs xample 2: xs xample 3: xs xample 4: xs. 34, ea lane In the sea plane at the right, a and a are right angles, &* c Q &*, and a ca. What postulate or theorem allows you to conclude that cq? Q 254 hapter 5 ongruent riangles

6 issing Information etermine what congruence is needed in order to use the indicated postulate or theorem to show that the triangles are congruent. 24. ongruence 25. ongruence 26. ongruence ostulate heorem ostulate Origami Origami he triangles below show a step in folding an origami seal. tate the third congruence needed to prove that c using the indicated postulate or theorem. 27. Given a ca, &* c &* se the ongruence heorem. X W Y 28. Given a ca, &* c &* se the ongruence ostulate. OIGI is the apanese art of folding paper into objects. he seal shown in the photograph was folded from a single piece of square paper. pplication inks O.O 29. Given a ca, a ca se the ongruence ostulate. 30. Given a ca, &* c &* se the ongruence ostulate. Visualize It! se the given information to make a sketch of Q and. ark the triangles with the given information. 31. aq ca 32. a ca 33. aq ca a ca a ca a ca Q &* c &* Q&* c &* Q &* c &* 34. ogical easoning ill in the missing statements and reasons. Given rove G &* c G &* &* &* G cg G tatements easons 1. G &* c G &* 1. Given 2. &* &* 2.? 3. a ca 3.? 4.? 4. Vertical ngles heorem 5. G cg 5.? 5.3 roving riangles are ongruent: and 255

7 35. ogical easoning ill in the missing statements and reasons. Given rove &* c &* &* &* &* &* c tatements easons 1.? 1. Given 2. &* &* 2.? 3. &* &* 3.? 4. a and a are right. 4.? 5.? 5. ight angles are congruent. 6. a ca 6.? 7. c 7.? tandardized est ractice ixed eview 36. ultiple hoice Which theorem or postulate cannot be used to show that Q cq? inding a issing ength ind the unknown side length. ound your answer to the nearest tenth. (esson 4.4) c 15 b a 26 Identifying ongruent arts In the diagram, c. omplete the statement. (esson 5.1) 40. a c? 41. &* c? 42. a c? 43. &* c? 44. a c? 45. &* c? lgebra kills raction Operations ivide. Write your answer in simplest form. (kills eview, p. 659) hapter 5 ongruent riangles