The hypotenuse and a leg of one triangle are congruent to the hypotenuse and a leg of the other triangle. THEOREM 5.2. right triangles, and

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1 5.4 ypotenuse-eg ongruence heorem: oal se the ongruence heorem and summarize congruence postulates and theorems. ey Words hypotenuse p. 192 leg of a right triangle p. 192 he triangles that make up the skateboard ramp below are right triangles. he hypotenuse and a leg of one triangle are congruent to the hypotenuse and a leg of the other triangle. hypotenuse leg hypotenuse leg tud OY I emember that the longest side of a right triangle is called the hypotenuse. hypotenuse leg t lp leg O 5.2 ypotenuse-eg ongruence heorem () Words If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of a second right triangle, then the two triangles are congruent. ymbols If and are right triangles, and &* c &*, and &* c &*, then c. X 1 etermine When o se Is it possible to show that c using the ongruence heorem? xplain your reasoning. olution In the diagram, you are given that and are right triangles. y the eflexive roperty, you know &* c &* (hypotenuse) and you are given that &* c &** (leg). You can use the ongruence heorem to show that c. 5.4 ypotenuse-eg ongruence heorem: 257

2 Itudent elp I ZON.O O X ore examples at classzone.com X 2 se the ongruence heorem se the diagram to prove that Q c. olution iven &* Q&* Q&* c &* rove Q c tatements easons 1. &* Q&* 1. iven 2. aq and a are right. 2. lines form right angles. 3. Q and are right 3. efinition of right triangle triangles. 4. Q&* c &* 5. &* c &* 4. iven 5. eflexive rop. of ongruence 6. Q c 6. ongruence heorem Y IN ONN O N O You have studied five ways to prove that c. ide &* c &* ide &* c &* ide &* c &* ide &* c &* ngle a ca ide &* c &* ngle a ca ide &* c &* ngle a ca ngle a ca ngle a ca ide &* c &* and are right triangles. ypotenuse &* c &* eg &* c &* 258 hapter 5 ongruent riangles

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. tudent elp Y I here is no ongruence heorem or ostulate, so you cannot conclude that the triangles in xample 3(b) are congruent. olution a. rom the diagram, you know that a ca, a ca, and &* c & *. You can use the ongruence heorem to show that c. b. rom the diagram, you know that &* c &**, &* c &*, and a ca. ecause the congruent angles are not included between the congruent sides, you cannot show that c. X 4 rove riangles are ongruent se the information in the diagram to prove that cw. W olution tatements easons 1. a ca 2. &* c W &** 1. iven 2. iven 3. W is equilateral. 3. efinition of equilateral triangle 4. a caw 4. quilateral triangles are equiangular. 5. a ca 5. iven 6. a caw 6. ransitive rop. of ongruence 7. cw 7. ongruence ostulate 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 ypotenuse-eg ongruence heorem: 259

4 5.4 xercises uided ractice ocabulary heck ell whether the segment is a leg or the hypotenuse of the right triangle. 1. &* 4. &* 2. &* 3. &* 5. &* 6. &* kill heck etermine whether you are given enough information to show that the triangles are congruent. xplain your answer W N ractice and pplications xtra ractice ee p ongruence heorem etermine whether you can use the ongruence heorem to show that the triangles are congruent. xplain your reasoning omework elp xample 1: xs , 24, xample 2: x. 32 xample 3: xs , xample 4: x. 32 andscaping o support a tree, you attach wires from the trunk of the tree to stakes in the ground as shown below. 13. What information do you need to know in order to use the ongruence heorem to show that c? 14. uppose is the midpoint of &*. Name a theorem or postulate you could use to show that c. xplain your reasoning. 260 hapter 5 ongruent riangles

5 You be the udge ecide whether enough information is given to show that the triangles are congruent. If so, state the theorem or postulate you would use. xplain your reasoning Y X Z N ogical easoning hree students are given the diagram shown at the right and asked which congruence postulate or theorem can be used to show that c. xplain why all three answers are correct. eghan c by the ongruence ostulate. eith c by the ongruence ostulate. c by the ypotenuse-eg ongruence heorem. ngie Itudent elp I ZON.O OWO xtra help with problem solving in xs is at classzone.com isualize It! se the given information to sketch N and. ark the triangles with the given information. 25. an and a are right 26. &** N &&, &* &**, angles. &** c &*, &** c N&* &** c &*, N&* c &* 27. &** N &&, &* &**, &** c N && c &** c &* 28. &* N&*,&* &* N&* c &*,N &** c &** 5.4 ypotenuse-eg ongruence heorem: 261

6 issing Information What congruence is needed to show that the triangles are congruent? sing that congruence, tell which theorem or postulate you would use to show that the triangles are congruent Z W X Y 32. ogical easoning ill in the missing statements and reasons. iven &* c &* is the midpoint of &*. a and a are right angles. rove c tatements easons 1. &* c &* 1.? 2.? 2. iven 3.? 3. efinition of midpoint 4. a and a are right angles. 4.? 5.? are 5. efinition of right triangle right triangles. 6. c 6.? tandardized est ractice 33. ulti-tep roblem he diagram below is a plan showing the light created by two spotlights. oth spotlights are the same distance from the stage. lights stage a. how that c. ell what theorem or postulate you use and explain your reasoning. b. Is there another way to show that c? If so, tell how. xplain your reasoning. c. re all four right triangles in the diagram congruent? xplain your reasoning. 262 hapter 5 ongruent riangles

7 ixed eview arallel ines ind ma1 and ma2. xplain your reasoning. (esson 3.4) howing ongruence ecide whether enough information is given to show that the triangles are congruent. If so, state the theorem or postulate you would use. xplain your reasoning. (essons 5.2, 5.3) lgebra kills valuating xpressions valuate. (kills eview, p. 670) p p p 2 6 p p 5 2 p p 2 N Quiz 2 ell whether the theorem or postulate can be used to show that N cq. (essons 5.3, 5.4) N ell whether enough information is given to show that the triangles are congruent. If so, tell which theorem or postulate you would use. xplain your reasoning. (essons 5.3, 5.4) ypotenuse-eg ongruence heorem: 263

8 echnology ctivity 5.4 Investigating ongruence Question Is there a side-side-angle congruence postulate or theorem? xplore 1 raw a segment and label it &*. raw a point not on &* and label this point. onstruct. 2 raw a circle centered at point that intersects in two points as shown. abel the intersection points and. 3 raw &* and &*. ide the circle. hink bout It 1. ind the measures of the segments and angles listed in the tables. egments &* &* &* &* &*????? ngles a a a a a a tudent elp I Y rawing overlapping triangles separately makes it easier to see the triangles. n example is on p. 232.?????? 2. ist the sides of that are congruent to the sides of. 3. ist the angles of that are congruent to the angles of. 4. o you think that c? xplain your reasoning. 5. ketch and separately on a piece of paper. ark the corresponding congruences on the triangles. 6. xplain why there is no side-side-angle congruence postulate. 264 hapter 5 ongruent riangles