Essential Question How can you use congruent triangles to make an indirect measurement?

Transcription

1 5.7 Using ongruent riangles ssential uestion How can you use congruent triangles to make an indirect measurement? easuring the Width of a iver IIUI H OI O OH o be proficient in math, you need to listen to or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments. Work with a partner. he figure shows how a surveyor can measure the width of a river by making measurements on only one side of the river. a. tudy the figure. hen explain how the surveyor can find the width of the river. b. Write a proof to verify that the method you described in part (a) is valid. iven is a right angle, is a right angle, c. xchange proofs with your partner and discuss the reasoning used. easuring the Width of a iver Work with a partner. It was reported that one of apoleon s officers estimated the width of a river as follows. he officer stood on the bank of the river and lowered the visor on his cap until the farthest thing visible was the edge of the bank on the other side. He then turned and noted the point on his side that was in line with the tip of his visor and his eye. he officer then paced the distance to this point and concluded that distance was the width of the river. a. tudy the figure. hen explain how the officer concluded that the width of the river is. b. Write a proof to verify that the conclusion the officer made is correct. iven is a right angle, is a right angle, c. xchange proofs with your partner and discuss the reasoning used. ommunicate Your nswer 3. How can you use congruent triangles to make an indirect measurement? 4. Why do you think the types of measurements described in xplorations and are called indirect measurements? ection 5.7 Using ongruent riangles 77

2 5.7 esson ore Vocabulary revious congruent figures corresponding parts construction What You Will earn Use congruent triangles. rove constructions. Using ongruent riangles ongruent triangles have congruent corresponding parts. o, if you can prove that two triangles are congruent, then you know that their corresponding parts must be congruent as well. Using ongruent riangles xplain how you can use the given information to prove that the hang glider parts are congruent. iven, rove OUIO If you can show that, then you will know that. irst, copy the diagram and mark the given information. hen mark the information that you can deduce. In this case, and are supplementary to congruent angles, so. lso, by the eflexive roperty of ongruence (heorem.). ark given information. ark deduced information. wo angle pairs and a non-included side are congruent, so by the ongruence heorem (heorem 5.),. ecause corresponding parts of congruent triangles are congruent,. onitoring rogress. xplain how you can prove that. Help in nglish and panish at igideasath.com 78 hapter 5 ongruent riangles

3 Using ongruent riangles for easurement I O O When you cannot easily measure a length directly, you can make conclusions about the length indirectly, usually by calculations based on known lengths. Use the following method to find the distance across a river, from point to point. lace a stake at on the near side so that. ind, the midpoint of. ocate the point so that and,, and are collinear. xplain how this plan allows you to find the distance. OUIO ecause and, and are congruent right angles. ecause is the midpoint of,. he vertical angles and are congruent. o, by the ongruence heorem (heorem 5.0). hen because corresponding parts of congruent triangles are congruent,. o, you can find the distance across the river by measuring. lanning a roof Involving airs of riangles Use the given information to write a plan for proof. iven, 3 4 rove 4 3 OUIO In and, you know that and. If you can show that, then you can use the ongruence heorem (heorem 5.5). o prove that, you can first prove that. You are given and 3 4. by the eflexive roperty of ongruence (heorem.). You can use the ongruence heorem (heorem 5.0) to prove that. lan for roof Use the ongruence heorem (heorem 5.0) to prove that. hen state that. Use the ongruence heorem (heorem 5.5) to prove that. onitoring rogress Help in nglish and panish at igideasath.com. In xample, does it matter how far from point you place a stake at point? xplain. 3. Write a plan to prove that U U. U ection 5.7 Using ongruent riangles 79

4 roving onstructions ecall that you can use a compass and a straightedge to copy an angle. he construction is shown below. You can use congruent triangles to prove that this construction is valid. tep tep tep 3 raw a segment and arcs o copy, draw a segment with initial point. raw an arc with center. Using the same radius, draw an arc with center. abel points,, and. raw an arc raw an arc with radius and center. abel the intersection. raw a ray raw. In xample 4, you will prove that. roving a onstruction Write a proof to verify that the construction for copying an angle is valid. OUIO dd and to the diagram. In the construction, one compass setting determines,,, and, and another compass setting determines and. o, you can assume the following as given statements. iven,, rove lan for roof lan in ction how that, so you can conclude that the corresponding parts and are congruent..,, O. iven.. ongruence heorem (heorem 5.8) orresponding parts of congruent triangles are congruent. onitoring rogress Help in nglish and panish at igideasath.com 4. Use the construction of an angle bisector on page 4. What segments can you assume are congruent? 80 hapter 5 ongruent riangles

5 5.7 xercises ynamic olutions available at igideasath.com Vocabulary and ore oncept heck. O H parts of congruent triangles are congruent.. WII escribe a situation in which you might choose to use indirect measurement with congruent triangles to find a measure rather than measuring directly. onitoring rogress and odeling with athematics In xercises 3 8, explain how to prove that the statement is true. (ee xample.) In xercises 3 and 4, write a proof to verify that the construction is valid. (ee xample 4.) H 8. W V H V U W 3. ine perpendicular to a line through a point not on the line lan for roof how that by the ongruence heorem (heorem 5.8). hen show that using the ongruence heorem (heorem 5.5). Use corresponding parts of congruent triangles to show that and are right angles. 4. ine perpendicular to a line through a point on the line In xercises 9, write a plan to prove that. (ee xample 3.) H lan for roof how that by the ongruence heorem (heorem 5.8). Use corresponding parts of congruent triangles to show that and are right angles. In xercises 5 and 6, use the information given in the diagram to write a proof. 5. rove H H ection 5.7 Using ongruent riangles 8

6 6. rove UX Y W X U Y V 0. HOUH OVOI he ermuda riangle is a region in the tlantic Ocean in which many ships and planes have mysteriously disappeared. he vertices are iami, an uan, and ermuda. Use the Internet or some other resource to find the side lengths, the perimeter, and the area of this triangle (in miles). hen create a congruent triangle on land using cities as vertices. ermuda 7. OI WIH HI xplain how to find the distance across the canyon. (ee xample.) iami, an uan, uerto ico 8. HOW O YOU I? Use the tangram puzzle. a. Which triangle(s) have an area that is twice the area of the purple triangle? b. How many times greater is the area of the orange triangle than the area of the purple triangle? 9. OO rove that the green triangles in the amaican flag are congruent if and is the midpoint of.. I U Your friend claims that WZY can be proven congruent to YXW using the H ongruence W heorem (hm. 5.9). Is your friend correct? xplain your reasoning. Z. II HII etermine whether each conditional statement is true or false. If the statement is false, rewrite it as a true statement using the converse, inverse, or contrapositive. a. If two triangles have the same perimeter, then they are congruent. b. If two triangles are congruent, then they have the same area. 3. I O IIO Which triangles are congruent to? elect all that apply. X Y aintaining athematical roficiency ind the perimeter of the polygon with the given vertices. (ection.4) 4. (, ), (4, ), (4, ), (, ) 5. ( 5, 3), (, ), (3, 4) eviewing what you learned in previous grades and lessons H 8 hapter 5 ongruent riangles