MMS9_Prep book_Unit7.4.qxp 7/7/09 10:56 AM Page 278 7.4 Skill Builder Sum of the Angles in a Triangle In any triangle, the sum of the angle measures is 180°. So, to find an unknown angle measure: • start with 180° • subtract the known measures An isosceles triangle has 2 equal sides and To find the measure of the third angle, subtract the measure of the equal angles twice. 2 equal angles. To find the measure of each equal angle, subtract the known angle from 180°, then divide by 2. G A 40° C 50° B !A ! 180° " 50° " 50° ! 80° J H Sum of equal angles is: 180° " 40° ! 140° Measure of each equal angle: 140° # 2 ! 70° Check 1. Find the measure of the third angle. a) D b) P 60° Q E 65° 50° R F 50 " _____ 60 !E ! 180° " _____ 70 ! _____ 180-65-65 !Q ! ___________________ 50 ! _____ 2. Find the measure of each equal angle. S Sum of equal angles is: 110 70 ! _____ 180° " _____ 70° Measure of each equal angle: T 110 # 2 ! 55 _____ _____ U 278 MMS9_Prep book_Unit7.4.qxp 7/7/09 10:56 AM Page 279 7.4 Similar Triangles FOCUS Use the properties of similar triangles to solve problems. A triangle is a special polygon. When two triangles are similar: • matching angles are equal OR • matching sides are proportional The order in which similar triangles are named gives a lot of information. Suppose !ABC ~ !DEF. The symbol ~ means “is similar to.” Then, "A ! "D, "B ! "E, and "C ! "F Similarly, AB matches DE, BC matches EF, and AC matches DF. Example 1 Identifying Similar Triangles Name the similar triangles. 4.8 cm D 3.0 cm E 3.6 cm F Solution Y 2.0 cm X 2.4 cm 3.2 cm Z Angle measures are not given. So, find out if matching sides are proportional. In !DEF, order the sides from shortest to longest: FD , EF , DE In !XYZ, order the sides from shortest to longest: XY , YZ , ZX Find the scale factors of matching sides. length of FD length of XY ! 3.0 cm 2.0 cm ! 1.5 length of EF length of YZ ! 3.6 cm 2.4 cm ! 1.5 length of DE length of ZX ! 4.8 cm 3.2 cm ! 1.5 Since all scale factors are the same, the triangles are similar. The longest and shortest sides meet at vertices: D and X The two longer sides meet at vertices: E and Z The two shorter sides meet at vertices: F and Y So, !DEF ~ !XZY Read the letters down the columns. 279 MMS9_Prep book_Unit7.4.qxp 7/7/09 10:56 AM Page 280 Check 1. In each diagram, name two similar triangles. a) Two angles in each triangle are given. The measure of the third angle in each triangle is: 36 - 68 = 76 180° ! _________________ A R B 68° 36° 36° 68° C Q P List matching angles: !A " _____ <P " _____ 76 <Q " _____ !B " _____ 68 !C " _____ 36 <R " _____ ARE equal. Matching angles _______ ARE similar. So, the triangles _______ To name the triangles, order the letters so matching angles correspond. PQR "ABC ~ "_______ b) Find out if matching sides are proportional. In "DEF, order the sides from shortest to longest: D J 5.4 cm EF, DE, FD _______________ In "JKL, order the sides from shortest to longest: JK, KL, LJ _______________ Find the scale factors of matching sides. length of EF length of JK CM " 2.8 1.4 CM length of DE length of KL " length of length of FD " LJ 3.2 cm 2.7 cm K 1.6 cm L E 2 " ____ 3.2 CM 2 " ____ 1.6 CM 5.4 CM 2 2.7 CM " ____ THE SAME. So, the triangles _______________ ARE SIMILAR . All scale factors are ___________ D and ____ L The two longer sides meet at vertices: ____ K The two shorter sides meet at vertices: E and ____ ____ K F and ____ The longest and shortest sides meet at vertices: ____ So, "DEF ~ "______ LKJ 280 1.4 cm 2.8 cm F MMS9_Prep book_Unit7.4.qxp 7/7/09 Example 2 10:56 AM Page 281 Using Similar Triangles to Determine a Length These two triangles are similar. Find the length of TU. S ° 6 cm Q 3 cm ° T U R 2 cm P Solution List matching angles: !S ! !P !T ! !Q So, "STU ~ "PQR !U ! !R "STU is an enlargement of "PQR. Choose a pair of matching sides whose lengths are both known: SU ! 6 cm and PR ! 2 cm Scale factor ! length on enlargement length on original Consider the triangle with the unknown length as a reduction or enlargement of the other triangle. ! 6 cm 2 cm !3 The scale factor is 3. Use the scale factor to find the length of TU. TU and QR are matching sides. Length of QR: 3 cm Scale factor: 3 Length of TU: 3 " 3 cm ! 9 cm So, TU has length 9 cm. 281 MMS9_Prep book_Unit7.4.qxp 7/7/09 10:56 AM Page 282 Check 1. These two triangles are similar. Find the length of XV. V F 20 cm X ° W 2 cm 10 cm H G ° List matching angles: <X <V !F ! _____ !G ! _____ XVW So, "FGH ~ "_______ <W !H ! _____ XVW is a reduction of _________ FGH . _________ Choose a pair of matching sides whose lengths are both known: FH = 10 CM AND XW = 2 CM __________________________________ Scale factor ! length on reduction length on original ! 2 CM 10 CM ! _____ 0.2 0.2 . The scale factor is _____ Use the scale factor to find the length of XV. XV and FG are matching sides. 20 CM Length of FG: ________ 0.2 Scale factor: ______ 0.2 X 20 CM = 4 CM Length of XV: _________________________ 4 CM . So, XV has length ______ 282 MMS9_Prep book_Unit7.4.qxp 7/7/09 10:56 AM Page 283 Practice T 1. In each diagram, name two similar triangles. 40° F a) Two angles in each triangle are given. The measure of the third angle 110 - 40 = 30 in each triangle is: 180° ! ___________________ 40° G U 110° 110° H S List matching angles: 110 40 <S " _____ <T " _____ <U !F " _____ " _____ !G " _____ !H " _____ 30 ARE equal, so, the triangles _______ ARE similar. Matching angles _______ UTS To name the triangles, order the letters so matching angles correspond. "FGH ~ "______ b) Find out if matching sides are proportional. JK, KL, LJ In "JKL, order the sides from shortest to longest: ______________ QR, SQ, RS In "QRS, order the sides from shortest to longest: ______________ Find the scale factors of matching sides. length of JK length of QR 3.3 CM " 2.2 CM length of length of KL 4.8 CM SQ " 3.2 CM length of length of LJ " RS K 1.5 " ____ Q 4.8 cm 3.2 cm L 3.3 cm S 1.5 " ____ 5.7 cm 2.2 cm 3.8 cm R J 5.7 CM 1.5 3.8 CM " ____ ARE SIMILAR EQUAL All scale factors are ____________ . So, the triangles ____________ . R J and ____ The longest and shortest sides meet at vertices: ____ Q K and ____ The two shorter sides meet at vertices: ____ The two longer sides meet at vertices: S L and ____ ____ RQS So, "JKL ~ "______ 2. Are these two triangles similar? P In "PQR, order the sides from shortest to longest: QR, RP, PQ ________________ In "BCD, order the sides from shortest to longest: CD, DB, BC ________________ Find the scale factors of matching sides. length of QR length of CD B 12 cm 8 cm Q 6 cm R 4 cm 5 cm D 2 cm C 6 CM 3 " ____ 2 CM 8 CM length of RP " " ____ 2 length of DN 4 CM 12 CM length of PQ 2.4 " " ____ length of BC 5 CM " ARE NOT SIMILAR. DIFFERENT . So, the triangles __________________ All scale factors are ______________ 283 MMS9_Prep book_Unit7.4.qxp 7/7/09 10:56 AM Page 284 3. These two triangles are similar. Find the length of EC. G C List matching angles: D <H <G !C ! ______ !D ! ______ GHF So, "CDE ~ "_______ 2 cm <F !E ! ______ ° 4 cm E F ° H 5 cm GHF . CDE is a reduction of ________ ________ Choose a pair of matching sides whose lengths are both known: DE = 2M AND HF = 5 CM __________________________________ Scale factor ! length on reduction length on original ! 2 CM 5 CM ! _____ 0.4 0.4 . The scale factor is _____ Use the scale factor to find the length of EC. FG are matching sides. EC and _____ 4 CM FG : _______ Length of _____ 0.4 Scale factor: _____ 0.4 X 4 CM = 1.6 CM Length of EC: __________________________ 1.6 CM . So, EC has length ________ 4. At a certain time of day, two trees cast shadows. Find the height of the taller tree. Y B EQUAL . Matching angles are _________ XYZ So, "ABC ~ "_______ ENLARGEMENT "XYZ is an ________________ of "ABC. ZX = 3.6 M AND CA = 2M Use sides _____________________________ to find the scale factor. length on enlargement length on original ! 5.4 m 3m 55° A 2m C 3.6 M 2M 1.8 ! _____ The scale factor is 1.8. Use the scale factor to find the height of the taller tree, YZ. BC and YZ are matching sides. 1.8 3M Length of BC: ______ Scale factor: _____ 1.8 X 3 M = 5.4 M Length of YZ: _____________________ 5.4M So, the height of the taller tree is _____________. 284 X 55° 3.6 m Z MMS9_Prep book_Unit7.4.qxp 7/7/09 10:56 AM Page 285 5. The two triangles in this diagram are similar. Find the length of DE. A 1.8 cm E D ° 1.2 cm To better see the individual triangles, we draw the triangles separately. B ° 4.0 cm A C A 1.8 cm + 1.2 cm = —— cm D ° ——— E cm B ° —— cm C <D <E <A !A ! ______ !B ! ______ !C ! ______ ADE So, "ABC ~ "______ ABC ADE . _________ is a reduction of _________ Choose a pair of matching sides whose lengths are both known: AD = 1.8 CM AND AB AB = 3.0 CM __________________________________ Scale factor ! length on reduction length on original 1.8 CM ! 3.0 CM 0.6 ! _____ 0.6 . The scale factor is _____ Use the scale factor to find the length of DE. BC are matching sides. DE and _____ _____ 4.0 CM BC : __________ Length of _____ 0.6 Scale factor: _____ 0.6 X 4.0 CM = 2.4 CM Length of DE: __________________________ So, DE has length _________ 2.4 CM . 285

© Copyright 2018