# Lesson 4.1 Triangle Sum Conjecture

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1 Lesson 4.1 ringle um onjecture Nme eriod te n ercises 1 9, determine the ngle mesures. 1. p, q 2., y 3., b p 98 q y b 4. r, s, 5., y 6. y t t s r y y s 8. m 9. m s m c c b 10. Find the mesure of. 11. Find the sum of the mesures of the mrked ngles. 12. se the digrm to eplin why 13. se the digrm to eplin why nd re complementry. m m m m. 24 H 4 iscovering Geometry rctice our kills 2008 Kendll Hunt ublishing

2 Lesson 4.2 roperties of sosceles ringles Nme eriod te n ercises 1 3, find the ngle mesures. 1. m 2. m G N G n ercises 4 6, find the mesures. 4. m, perimeter 5. he perimeter of LO 6. he perimeter of is of is 536 m. L, 344 cm. m, m 7 cm cm m L m O y 31 cm68 y 7.. Nme the ngle(s) congruent to. b. Nme the ngle(s) congruent to. c. Wht cn you conclude bout nd? Why? 8., y 9. nd. 10. se the digrm to eplin f m 120, wht is why is isosceles. m? 4y y 79 iscovering Geometry rctice our kills H Kendll Hunt ublishing

3 Lesson 4.3 ringle nequlities Nme eriod te n ercises 1 nd 2, determine whether it is possible to drw tringle with sides of the given mesures. f it is possible, write yes. f it is not possible, write no nd mke sketch demonstrting why it is not possible cm, 30 cm, 45 cm 2. 9 km, 17 km, 28 km 3. f 17 nd 36 re the lengths of two sides of tringle, wht is the rnge of possible vlues for the length of the third side? n ercises 4 6, rrnge the unknown mesures in order from gretest to lest c b b c b c 40 d Wht s wrong with this picture? plin why is isosceles. n ercises 11 nd 12, use compss nd strightedge to construct tringle with the given sides. f it is not possible, eplin why not H 4 iscovering Geometry rctice our kills 2008 Kendll Hunt ublishing

4 Lesson 4.4 re here ongruence hortcuts? Nme eriod te n ercises 1 3, nme the conjecture tht leds to ech congruence JN 3. bisects,, nd n ercises 4 9, nme tringle congruent to the given tringle nd stte the congruence conjecture. f you cnnot show ny tringles to be congruent from the informtion given, write cnnot be determined nd redrw the tringles so tht they re clerly not congruent. 4. is the midpoint of 5. K is kite with K. 6. nd. K J N 6 K 7. ON O y N G 10 8 O O n ercises 10 12, use compss nd strightedge or ptty pper nd strightedge to construct tringle with the given prts. hen, if possible, construct different (noncongruent) tringle with the sme prts. f it is not possible, eplin why not iscovering Geometry rctice our kills H Kendll Hunt ublishing

5 Lesson 4.5 re here Other ongruence hortcuts? Nme eriod te n ercises 1 6, nme tringle congruent to the given tringle nd stte the congruence conjecture. f you cnnot show ny tringles to be congruent from the informtion given, write cnnot be determined nd eplin why VW 3. O V W 4. is the ngle bisector 5. N 6. FGH is prllelogrm. of. G. L K G F N L H 7. he perimeter of is 350 cm. 8. he perimeter of V is 95 cm. s OL? plin. s V WV? plin. L V O W n ercises 9 nd 10, construct tringle with the given prts. hen, if possible, construct different (noncongruent) tringle with the sme prts. f it is not possible, eplin why not H 4 iscovering Geometry rctice our kills 2008 Kendll Hunt ublishing

6 Lesson 4.6 orresponding rts of ongruent ringles Nme eriod te 1. Give the shorthnd nme for ech of the four tringle congruence conjectures. n ercises 2 5, use the figure t right to eplin why ech congruence is true. W is prllelogrm. 2. W 3. W W 4. W 5. W For ercises 6 nd 7, mrk the figures with the given informtion. o demonstrte whether the segments or the ngles indicted re congruent, determine tht two tringles re congruent. hen stte which conjecture proves them congruent. 6. is the midpoint of W nd 7. is isosceles nd is the bisector. s W? Why? of the verte ngle. s? Why? W n ercises 8 nd 9, use the figure t right to write prgrph proof for ech sttement. 8. F 9. F F 10. is n isosceles trpezoid with nd. Write prgrph proof eplining why. iscovering Geometry rctice our kills H Kendll Hunt ublishing

7 Lesson 4.7 Flowchrt hinking Nme eriod te omplete the flowchrt for ech proof. 1. : nd how: Flowchrt roof 2. : Kite K with K K how: K bisects K nd K Flowchrt roof K K K K K is kite K efinition of bisect 3. : is prllelogrm how: Flowchrt roof is prllelogrm efinition of me segment 30 H 4 iscovering Geometry rctice our kills 2008 Kendll Hunt ublishing

8 Lesson 4.8 roving pecil ringle onjectures Nme eriod te n ercises 1 3, use the figure t right. 1. is medin, perimeter 60, nd is n ngle bisector, nd m 54. m 3. is n ltitude, perimeter 42, m 38, nd 8. m, 4. is equilterl. 5. NG is equingulr m nd perimeter NG 51. N 6. is equilterl, is isosceles with bse, perimeter 66, nd perimeter 82. erimeter 7. omplete flowchrt proof for this conjecture: n n isosceles tringle, the ltitude from the verte ngle is the medin to the bse. : sosceles with nd ltitude how: is medin Flowchrt roof is n ltitude nd re right ngles efinition of ltitude 8. Write flowchrt proof for this conjecture: n n isosceles tringle, the medin to the bse is lso the ngle bisector of the verte ngle. : sosceles with nd medin how: bisects iscovering Geometry rctice our kills H Kendll Hunt ublishing

9 LON 3.8 he entroid cm, 5.7 cm, 4.8 cm G LON 4.2 roperties of sosceles ringles 1. m m G m 39, perimeter of 46 cm 5. L 163 m, m m 44, b. c. by the onverse of the onjecture , y m m 55 by V, which mkes m 55 by the ringle um onjecture. o, is isosceles by the onverse of the sosceles ringle onjecture. LON 4.3 ringle nequlities 6 cm 10 cm 1. es 2. No 4. (3, 4) 5. 16, L 8, 15, ncenter b. entroid c. ircumcenter d. ircumcenter e. Orthocenter f. ncenter g. entroid LON 4.1 ringle um onjecture 1. p 67, q , y , b r 40, s 40, t , y y s m m 10. m cm 12. he sum of the mesures of nd is 90 becuse m is 90 nd ll three ngles must be 180. o, nd re complementry. 13. m m becuse they re verticl ngles. ecuse the mesures of ll three ngles in ech tringle dd to 180, if equl mesures re subtrcted from ech, wht remins will be equl b c 5. b c 6. c d b he interior ngle t is 60. he interior ngle t is 20. ut now the sum of the mesures of the tringle is not y the terior ngles onjecture, 2 m. o,m. o, by the onverse of the sosceles ringle onjecture, is isosceles. 11. Not possible km 9 km 28 km LON 4.4 re here ongruence hortcuts? 1. or () 5. () iscovering Geometry rctice our kills NW Kendll Hunt ublishing

10 6. nnot be determined, s shown by the figure. 9. ll tringles will be congruent by. ossible tringle: 7. NO () 8. nnot be determined, s shown by the figure. 9. OG () 10. Only one tringle becuse of. 11. wo possible tringles. 12. Only one tringle becuse of. LON 4.5 re here Other ongruence hortcuts? 1. nnot be determined 10. ll tringles will be congruent by. ossible procedure: se nd to construct nd then copy nd t the ends of. LON 4.6 orresponding rts of ongruent ringles 1.,,, 2. W, onjecture 3. W, onjecture W by. W by. 7. by. by. 8. ossible nswer: nd F re both the distnce between nd. ecuse the lines re prllel, the distnces re equl. o, F. 9. ossible nswer: F (see ercise 8). F F becuse both re right ngles, F F becuse they re the sme segment. o, F F by. F by. 10. ossible nswer: t is given tht nd, nd becuse they re the sme segment. o by nd by. LON 4.7 Flowchrt hinking 2. () 3. ( or ) 4. () 5. N () 6. GK ( or ) 7. es, OL by. 8. No, corresponding sides V nd WV re not congruent. 1. (ee flowchrt proof t bottom of pge 101.) 2. (ee flowchrt proof t bottom of pge 101.) 3. (ee flowchrt proof t bottom of pge 101.) LON 4.8 roving pecil ringle onjectures m m 52, m N erimeter NW iscovering Geometry rctice our kills 2008 Kendll Hunt ublishing

11 7. (ee flowchrt proof t bottom of pge 102.) 8. Flowchrt roof is medin efinition of medin onjecture bisects efinition of bisect me segment ; 36 sides sides m 150 LON 5.2 terior ngles of olygon sides sides 3. 4 sides 4. 6 sides 5. 64, b , b , b 132, c , b 40, c 105, d LON 5.1 olygon um onjecture , b 103, c 97, d 83, e , b 44, c 51, d 85, e 44, f 136 Lesson 4.7, ercises 1, 2, 3 1. onjecture onjecture me segment 2. K K K K K is kite efinition of kite K K onjecture K K K bisects K nd efinition of bisect K K me segment 3. onjecture is prllelogrm efinition of prllelogrm me segment onjecture efinition of prllelogrm onjecture iscovering Geometry rctice our kills NW Kendll Hunt ublishing

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