US Geometr enchmark Stud Guide 1 Given: 1 Given: Prove: Statements 1. 1. Given Reasons.. Refleive Prop. of 3. 3. SSS Post Prove: G.O.1 etermine whether the following statements are rue or alse. ) ) ) ) ) the lternate Interior ngle heorem, and. herefore, must correspond to and to. hat happens in both ) and ) so the are true, while ) and ) are false. etermine whether the following statements are rue or alse. ) ) ) ) ) orresponding parts of congruent triangles are congruent. Using either of the true congruenc statements in ) or ), would correspond to. herefore, the two segments would be congruent and ) is true. G.O.1 Page 1 of 17 M@WUS (US) 11/0/14
US Geometr enchmark Stud Guide 3 b which postulate or theorem? 3 Name the postulate or theorem b which the congruence statement is true. a) he S ongruence Postulate. b) c) d) G.O.1 Page of 17 M@WUS (US) 11/0/14
US Geometr enchmark Stud Guide 4 he sides of a pentagon have lengths, 3,, 9, and 10 units. he longest side of a similar pentagon is 0. ind the perimeter of the second pentagon. omparing the lengths of the longest side of each pentagon: the larger pentagon is twice as large as the smaller pentagon. herefore the perimeter of the larger pentagon is twice the perimeter of the smaller pentagon. 4 he sides of a heagon have lengths 5, 6, 10, 1, 15, and 1. he longest side of a similar heagon is 14. ind the perimeter of the second heagon. OR ind the scale factor of the similar polgons smaller pentagon 10 larger pentagon 0 1 Let 1 the length of the smallest side of the larger pentagon. 1 1 4 1 then 3 1 6 Similarl, 3 16 and 4 1. he perimeter of the second pentagon is 4 6 16 1 0 64 + + + + units. OR he ratio of the perimeters of similar polgons is equal to the scale factor: 1 + 3+ + 9 + 10 1 3 1 3 3 3 3 3 64 64 he perimeter of the second pentagon is 64 units. G.O.1 G.O.1 Page 3 of 17 M@WUS (US) 11/0/14
US Geometr enchmark Stud Guide 5 ind. 6 3 ~ b the SS Similarit heorem 6 3 4 3 and vertical angles are congruent 4, 5 N 4 Q 6 L herefore, 3 3 9 9 G.SR.5 ind. 6 omplete the statement: 10 If, then Justif our reasoning. 5 10 10 +? 10 5 10 + 7 7 ecause, (corresponding s are if the lines are parallel) and for the same reason,. the Similarit postulate, : 10 + 10 7 5 G.O.1 6 a) Write the similarit statement for the triangles that are similar. 10 b) How do ou know the triangles are similar? c) Solve for. 6 5 4 Page 4 of 17 M@WUS (US) 11/0/14
US Geometr enchmark Stud Guide 7 Given: ~ ZYX List all the angles that are congruent from the similarit statement and then write the statement of proportionalit for the corresponding sides. 7 Given: QRS ~ GH List all the angles that are congruent from the similarit statement and then write the statement of proportionalit for the corresponding sides. Z Y X ZY YX ZX G.O.1 List all the possible was ou can prove the triangles below are congruent. List all the possible was ou can prove the triangles below are congruent. SSS ongruence Postulate SS ongruence Postulate G.SR.5 Page 5 of 17 M@WUS (US) 11/0/14
US Geometr enchmark Stud Guide 9 L n 9 J 7 K M N etermine whether each of the following statements must be rue based on the given diagram above. ) NKJ LKM ) NKJ ~ MKL ) n 7 ) LM JN ) LM JN etermine whether each of the following statements must be rue based on the given diagram above. ) ~ ) ) ) ) KN KM (ef. of segments) JKN MKL (Vertical s are ) NJK MLK (SS Postulate) ) Keeping the corresponding parts in the correct order we can rewrite this congruenc statement: NKJ MKL his means ) is false. ) It also means ) is true. ongruent figures are also similar with a scale factor of 1. ) rom the congruenc statement J L, so n 7, and ) is true. ) rom the congruenc statement LM JN, so ) is true. ) We cannot determine whether LM JN, so ) is not necessaril true. herefore ), ), and ) must be true. G.SR.5 Page 6 of 17 M@WUS (US) 11/0/14
US Geometr enchmark Stud Guide 10 Given: - 10 Point P is transformed to form the image P. - a) etermine the coordinates of if point was rotated 90 about the origin. **Rotations are counter clockwise (unless stated otherwise)** - P - - - - - (, 5) b) etermine the coordinates of if point was reflected over the line 3.. If P is rotated 10 about the origin then the coordinates of 4,1. P are - ( 5, ) - 3-5 5-3. If P is rotated 90 clockwise about the origin then the 4, 1. coordinates of P are. If P is translated 4 units to the left and 1 unit down then the, 5. coordinates of P are c) etermine the coordinates of if point was reflected over the line.. If P is reflected across the - ais, then the coordinates of P are ( 1, 4 ). - - (, 5) - -. If P is reflected across the line, then the coordinates of P are ( 4, 1). G.O.5 Page 7 of 17 M@WUS (US) 11/0/14
US Geometr enchmark Stud Guide 11 raw the resulting image after each of the following transformations takes place on this preimage: 11 raw the resulting image after each of the following transformations takes place on this preimage: - - a) 90 clockwise rotation about the origin. - G H - - - - P (a,b) 6,6 ( ) 6, 4, 90 clockwise rotation about the origin - P (b,-a) 6,6 ( 6) ( ),, 4 a) 90 rotation about the origin. - G b) ilation of 1 centered at the origin. H - - - b) ilation of centered at the origin. - - P (a,b) 6,6 ( ) 6, 4, ilation of 1 centered at the origin P 1 1 a, b ( ) ( ) 3,3 3,1,1 - H - G G.O.5, G.SR.1 Page of 17 M@WUS (US) 11/0/14
US Geometr enchmark Stud Guide 1 ind the area of the triangle. 9 1 4 15 or all right triangles, a + b c where c is the length of the hpotenuse. 5 4 a + b c 9 + b 15 1+ b 5 b 5 1 b 144 b 1 he area of a triangle 1 bh 1 ( 9 )( 1) ( 9) ( 6) 54 1 9 15 etermine whether each of the following. 3. 6. rea of 1 sq. un.. rea of 16 sq. un. he area of the triangle is 54 square units. G.SR. 13 Solve for the variables. 45 9 his is a 45 45 90 triangle. Using 45-45 -90 theorem: leg leg 9 hpotenuse legi 9i 9 Using 45-45 -90 leg:leg:hp. ratios: leg leg 1 1 9 9 1:1: leg hp. 1 9 9 13 etermine whether each of the following. a b J b L 45. JKL is a right scalene triangle. rea of JKL 5 sq. un.. Perimeter of JKL ( + ) a 10 K 10 10 un. G.SR..1 Page 9 of 17 M@WUS (US) 11/0/14
US Geometr enchmark Stud Guide 14 Solve for the variables. m n 4 30 14 p Y 60 n X 1 Z his is a 30 60 90 triangle. Using 30-60 -90 theorem: hpotenuse i 4 m 1 m ( short leg ) Using 30-60 -90 short leg:long leg:hpotenuse ratios: 1: 3 : short leg : h hp.: long leg etermine whether each of the following. m X 30. n 6. n 6 3 ( long leg ) ( short leg ) i 3 n 1i 3 n 1 3 1 m 4 m 4 4 3 n n 4 3. p 3. rea of XYZ 4 3 sq. un. m 1 n 1 3 We can check our answer using the Pthagorean theorem: a + b c 1 + 1 3 4 144 + 1 3 576 144 + 144 3 576 144 + 43 576 576 576 G.SR..1 Page 10 of 17 M@WUS (US) 11/0/14
US Geometr enchmark Stud Guide 15 Solve for the variables in the diagram. Round answers to the nearest tenth. 9 64 z he sum of the interior angles of a triangle is 10 : m + m + m 10 64 + + 90 10 15 M etermine whether each of the following. 75 L 5 z N + 154 10 6 6 We can use right triangle trigonometr to solve for and z... sin 5 tan 5 opposite sinθ hpotenuse sin 64 9 9sin 64 adjacent cosθ hpotenuse z cos 64 9 9cos 64 z. z sin 5. <.1 3.9 z We can check our answer using the tangent ratio. opposite tanθ adjacent tan 64 z.1 tan 64 3.9.05.0 hese values are etremel close and we should epect some error on the right side of the equation because.1 and 3.9 were alread rounded values. G.SR. Page 11 of 17 M@WUS (US) 11/0/14
US Geometr enchmark Stud Guide 16 Solve for the variables in the diagram. Round answers to the nearest tenth. 16 X 7 w 9 n 11 a Y c m Y Z Using the Pthagorean heorem: 7 + n 11 X n 15 Z + n 49 11 n 7 etermine whether each of the following n 7 n.5 We can use right triangle trigonometr to solve for a b setting up an equation using the sine function:. n + m 90. 1 15 m tan opposite sinθ hpotenuse. n 6 7 sin a 11 1 1 7 sin ( sin a ) sin 11 a 39.5. + 15 c. cos n sin m a 39.5 We can solve for w since the sum of the interior angles of a triangle is 10 : m X + m Y + m Z 10 90 + 39.5 + w 10 w + 19.5 10 w 50.5 w 50.5 G.SR. Page 1 of 17 M@WUS (US) 11/0/14
US Geometr enchmark Stud Guide 17 Solve for the variables in the diagram. hen find the area of the triangle. Round answers to the nearest tenth. c z 17 z 30 6 60 70 his is not a right triangle, so we are going to have to use the law of sines or the law of cosines. Since we don t know an of the side lengths across from a known angle measure, we are going to have to use law of cosines using angle : c a + b ab cos c 6 + 6 cos 60 c 36 + 64 96cos 60 c 100 96cos 60 c 100 96cos 60 c 7. fter finding, we can use the law of sines to find the other variables. sin sin sin a b c etermine whether each of the following. is an acute scalene triangle... sin 30 sin 70 sin 30 sin 70 z 64 z z cos 70 +. rea of 4 sin 70 sin sin b c sin z sin 60 6 7. sin z 0.10 6 sin z 0.70 ( z ) 1 1 sin sin sin 0.70 z 46.1 z 46.1 sin sin a c sin sin 60 7. sin z 0.10 sin z 0.960 ( z ) 1 1 sin sin sin 0.960 z 73.7 z 73.7 rea: 1 rea absin 1 rea ( 6 )( ) sin 60 rea 4sin 60 rea 0. herefore the area of the triangle is 0. square units. G.SR.9, G.SR.10, G.SR.11 nd of Stud Guide Page 13 of 17 M@WUS (US) 11/0/14
US Geometr enchmark Stud Guide 1 Statements 1.. You r Solutions: 3. Reasons 1. Given. Vertical angles are congruent 3. S ongruence Postulate 4 he scale factor of the second heagon to the first is 14. 1 3 perimeter of nd 3 perimeter of 1st 5 + 6 + 10 + 1 + 15 + 1 3 69 69 3 i3i3 3 3 3 46 (Refleive Prop.) and is a right angle so the triangles are congruent b the HL heorem. Since and, then. herefore, ) is true and ), ), and ) are false. Since corresponds to in the congruenc statement, then b P. ) is true as well. 5 he perimeter of the second heagon is 46 units. Q ~ NL b the Similarit 6 10 3 a. HL ongruence heorem b. SSS ongruence Postulate c. SS ongruence Postulate d. S ongruence Postulate Q Q Q N NL 6 10 4 10 4. N L Page 14 of 17 M@WUS (US) 11/0/14
US Geometr enchmark Stud Guide 6 7 a) b) b the SS Similarit heorem 10 c) 15 10 6 6 1 6 3 1 9 QRS ~ Q G R S H 15 1 and GH 1 QR RS S Q G H H G 9 (Refleive prop. of ) (SSS Postulate) Keeping the corresponding parts in the correct order we can rewrite this congruenc statement: his means ) is false since the congruent parts don t correspond. We can also rewrite the congruenc statement this wa: his means ) is false and ) is true. rom an of the congruenc statements above, so ) is true. Note that for angles onl the verte of the angle matters in the congruenc statement. rom an of the congruenc statements above. If two lines are cut b a transversal and alternate interior angles are congruent, then the two lines are parallel so and ) is true. SS ongruence Postulate S ongruence Postulate S ongruence heorem Page 15 of 17 M@WUS (US) 11/0/14
US Geometr enchmark Stud Guide 10. If P is rotated 10 about the origin then 4,1. the coordinates of P are 11 a) 90 rotation about the origin.. If P is rotated 90 clockwise about the origin then the coordinates of P are 4, 1. - G G. If P is translated 4 units to the left and 1 unit down then the coordinates of P are, 5. H - H. If P is reflected across the -ais, then the 1, 4. coordinates of P are b) ilation of centered at the origin.. If P is reflected across the line, then the coordinates of P are ( 4, 1). - G G H H - 1 etermine whether each of the following. 3. 6. rea of 1 sq. un.. rea of 16 sq. un. Page 16 of 17 M@WUS (US) 11/0/14
US Geometr enchmark Stud Guide 13 etermine whether each of the following 16 etermine whether each of the following. a b. n + m 90. JKL is a right scalene triangle. rea of JKL 5 sq. un.. 1 15 m tan. Perimeter of JKL ( + ) 10 10 un.. n 6. + 15 c 14 etermine whether each of the following. cos n sin m. m X 30. n 6 17 etermine whether each of the following. n 6 3. is an acute scalene triangle 15. p 3. rea of XYZ 4 3 sq. un. etermine whether each of the following. 75... sin 30 sin 70 sin 30 sin 70 z 64 z z cos 70 +. rea of 4 sin 70. sin 5. tan 5. z sin 5. < Page 17 of 17 M@WUS (US) 11/0/14