Lesson 15 Trigonometric Laws Unit 2 Review Unit 2 Performance Task

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2 Contents Unit 1 Congruence, Proof, and Constructions Lesson 1 Transformations and Congruence Lesson Translations Lesson Reflections Lesson Rotations Lesson 5 Smmetr and Sequences of Transformations Lesson Congruent Triangles Lesson 7 Using Congruence to Prove Theorems Lesson 8 Constructions of Lines and Angles Lesson 9 Constructions of Polgons Unit 1 Review Unit 1 Performance Task Unit Similarit, Proof, and Trigonometr Lesson 10 Dilations and Similarit Lesson 11 Similar Triangles Lesson 1 Trigonometric Ratios Lesson 1 Relationships between Trigonometric Functions Lesson 1 Solving Problems with Right Triangles Lesson 15 Trigonometric Laws Unit Review Unit Performance Task Common New Arizona's York Core College State Standards Common Core Career Learning Read Standards for Mathematics G.CO.1, G.CO., G.CO. HS.G.CO.1, G.CO., G.CO., G.CO., G.CO., G.CO.5 G.CO.5 HS.G.CO.1, G.CO., G.CO., G.CO., G.CO., G.CO.5 G.CO.5 HS.G.CO.1, G.CO., G.CO., G.CO., G.CO., G.CO.5 G.CO.5 HS.G.CO., G.CO.5 G.CO.5 HS.G.CO.7, G.CO.8 G.CO.8 HS.G.CO.9, G.CO.10, G.CO.11 G.CO.11 HS.G.CO.1 HS.G.CO.1 HS.G.SRT.1a, G.SRT.1b, G.SRT. G.SRT. HS.G.SRT., G.SRT., G.SRT., G.SRT.5 HS.G.SRT. HS.G.SRT.7 HS.G.SRT.8, G.SRT.9, G.SRT.9, G.MG.1, G.MG.1, G.MG., G.MG. G.MG., G.MG. HS.G.SRT.10, G.SRT.11 Unit Etending to Three Dimensions Lesson 1 Circumference and Area of Circles Lesson 17 Three-Dimensional Figures Lesson 18 Volume Formulas Unit Review Unit Performance Task HS.G.GMD.1, G.MG.1 G.MG.1 HS.G.GMD., G.MG.1 G.MG.1 HS.G.GMD.1, G.GMD., G.MG.1 G.MG.1 Duplicating this page is prohibited b law. 015 Triumph Learning, LLC Problem Solving Performance Task

3 Unit Connecting Algebra and Geometr through Coordinates Lesson 19 Parallel and Perpendicular Lines Lesson 0 Distance in the Plane Lesson 1 Dividing Line Segments Lesson Equations of Parabolas Lesson Using Coordinate Geometr to Prove Theorems Unit Review Unit Performance Task Unit 5 Circles With and Without Coordinates Lesson Circles and Line Segments Lesson 5 Circles, Angles, and Arcs Lesson Arc Lengths and Areas of Sectors Lesson 7 Constructions with Circles Lesson 8 Equations of Circles Lesson 9 Using Coordinates to Prove Theorems about Circles Unit 5 Review Unit 5 Performance Task Common New Arizona's York Core College State Standards Common Core Career Learning Read Standards for Mathematics HS.G.GPE., G.GPE.5 G.GPE.5 HS.G.GPE., G.GPE.7 G.GPE.7 HS.G.GPE., G.GPE. G.GPE. HS.G.GPE. HS.G.GPE. HS.G.C.1, G.C., G.C., G.MG.1 G.MG.1 HS.G.C., G.C. G.C. HS.G.C.5, G.MG.1 G.MG.1 HS.G.C., G.C., G.C., G.MG.1 G.MG.1 HS.G.GPE.1 HS.G.GPE. Duplicating this page is prohibited b law. 015 Triumph Learning, LLC Unit Applications of Probabilit Lesson 0 Set Theor Lesson 1 Modeling Probabilit Lesson Permutations and Combinations Lesson Calculating Probabilit Lesson Conditional Probabilit Lesson 5 Using Probabilit to Make Decisions Unit Review Unit Performance Task Glossar Formula Sheet Math Tools S.CP.1 HS.S.CP.1 S.CP. HS.S.CP. S.CP.9 HS.S.CP.9 S.CP., HS.S.CP., S.CP.7, S.CP.7, S.CP.8 S.CP.8 S.CP., HS.S.CP., S.CP.5, S.CP.5, S.CP., S.CP., S.CP.8 S.CP.8 S.MD., HS.S.MD., S.MD.7 S.MD.7

4 LESSON Congruent Triangles Congruence with Multiple Sides UNDERSTAND Two triangles are congruent if all of their corresponding angles are congruent and all of their corresponding sides are congruent. However, ou do not need to know the measures of ever side and angle to show that two triangles are congruent. nabc is formed from three line segments: AB, BC, and AC. Suppose those segments were pulled apart and used to build another triangle, such as na9b9c9 shown. This new triangle could be formed b reflecting nabc over a vertical line. A reflection is a rigid motion, so na9b9c9 must be congruent to nabc. In fact, an triangle built with these segments could be produced b A A performing rigid motions on nabc, so all such triangles would be congruent to nabc. B C C B So, knowing all of the side lengths of two triangles is enough information to determine if the are congruent. Side-Side-Side (SSS) Postulate: If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent. UNDERSTAND You can also use the Side-Angle-Side (SAS) Postulate to prove that two triangles are congruent. Side-Angle-Side (SAS) Postulate: If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent. Look at ndef on the coordinate plane. Appling rigid motions to two of its sides produced new line segments. Sides DE and EF were translated 7 units to the right to form GH and HI. DE and EF were rotated 180 about the origin to form JK and KL. DE and EF were reflected over the -ais to form MN and NO. Each rigid motion preserved the lengths of the segments as well as the angle between them. In all three images, onl one segment can be drawn to complete each triangle. Those line segments, shown as dotted lines, are congruent to DF. So, ndef nghi njkl nmno. The smbol means "is congruent to." D E 0 M N F O G L K H I J Duplicating this page is prohibited b law. 015 Triumph Learning, LLC 8 Unit 1: Congruence, Proof, and Constructions

5 Connect The coordinate plane on the right shows nabc and ndef. Use the SAS Postulate to show that the triangles are congruent. Then, identif rigid motions that could transform nabc into ndef. D B A 0 F C E 1 Make a plan. Duplicating this page is prohibited b law. 015 Triumph Learning, LLC Each triangle has one horizontal side and one vertical side that intersect, so both are right triangles. Since all right angles are congruent, the triangles have at least one pair of corresponding congruent angles. To prove the triangles congruent b the SAS Postulate, find the lengths of the adjacent sides (legs) that form the right angles. Identif rigid motions that can transform nabc to ndef. Stud the shapes of the triangles. AC corresponds to DF and is parallel to it. Verte B is above AC on the left, while verte E lies below DF and is also on the left. n ABC could be reflected over the -ais. After such a reflection, n A9B9C9 would have vertices A9(, 1), B9(, ), and C9(, 1). To transform this image to ndef, translate it units to the left. DISCUSS Find and compare the lengths of the corresponding legs. Count the units to find the length of each leg. AB 5 DE 5 AC 5 5 DF 5 5 AB DE and AC DF The triangles have two pairs of corresponding congruent sides, and their included angles are congruent right angles. So, according to the SAS Postulate, nabc ndef. If nabc had instead been rotated 90 and then translated down 1 unit, would the resulting image be congruent to nabc? How could ou prove our answer? Lesson : Congruent Triangles 9

6 Congruence with Multiple Angles UNDERSTAND A third method for proving that triangles are congruent is the Angle-Side-Angle Theorem. Angle-Side-Angle (ASA) Theorem: If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent. Look at the coordinate planes below. Triangle GHJ is shown on the left. On the right, sides HJ and GJ of the triangle have been replaced b ras. G G J H 0 H 0 The coordinate plane on the right of this paragraph shows three transformations of GH and the ras etending from points G and H. KL is a translation of GH units to the right. ON is a 180 rotation of GH. RQ is a reflection of GH over the -ais. Each of those rigid motions has carried the angle-side-angle combination to a new location. The segments KL, ON, and RQ are congruent to GH. Rigid motions also preserved the angles formed b the segment and each ra. G H 0 R Q K L N O The coordinate plane on the lower right shows the triangles formed b etending the ras until the intersect. In each case, the ras can onl intersect at one point and thus can form onl one triangle. Each of these triangles is congruent to nghj. G J H 0 Q S R M N K L P O Duplicating this page is prohibited b law. 015 Triumph Learning, LLC Duplicating this page is prohibited b law. 015 Triumph Learning, LLC 0 Unit 1: Congruence, : Proof, Proof, and and Constructions

7 Connect Triangles PQR and STU are shown on the coordinate plane on the right. Given that /P /S and /R /U, prove that the triangles are congruent. Then identif rigid motions that can transform npqr into nstu. U T 0 S Q P R 1 Make a plan. You alread know that two pairs of corresponding angles are congruent. If ou can show that the included sides are the same length, then the triangles are congruent b the ASA Theorem. Find the lengths of the included sides. Duplicating this page is prohibited b law. 015 Triumph Learning, LLC Duplicating this page is prohibited b law. 015 Triumph Learning, LLC Identif rigid motions that could transform npqr into nstu. Stud the shapes of the triangles. Side PR is horizontal, and corresponding side SU is vertical. It appears that npqr was rotated 90 counterclockwise. If that rotation were around the origin, np9q9r9 would have vertices at P9(, 1), Q9(, ), and R9(, ). To transform this image to nstu, translate it units down. Triangle STU can be produced b rotating npqr 90 counterclockwise and then translating the image down units. DISCUSS In npqr, the included side of /P and /R is PR. Find the length of PR b counting units. PR 5 5 In nstu, the included side for /S and /U is SU. Find the length of SU b counting units. SU 5 5 Since two pairs of corresponding angles and the included sides are congruent, npqr nstu b the ASA Theorem. In triangles ABC and DEF, /A 5 /D 5 0, /B 5 /E 5 0, and /C 5 /F Are triangles ABC and DEF congruent? Lesson : Congruent Triangles 1

8 EXAMPLE A Show that ntuv is congruent to nxyz. Y X T U Z V 0 1 Make a plan. To use the SSS Postulate, ou need to show that each side on nxyz has a corresponding congruent side on ntuv. To do so, show that each side of nxyz is a rigid-motion transformation of a side of ntuv. Compare the endpoints of XY and TU. DISCUSS Follow the same process for the other two pairs of corresponding sides. The endpoints of YZ and UV have the same -coordinates but opposite -coordinates. The endpoints of XZ and TV also have the same -coordinates but opposite -coordinates. So, YZ is a reflection of UV over the -ais, and XZ is a reflection of TV over the -ais. How does /Z compare to /V? Could one angle have a greater measure than the other? Points T and X have the same -coordinates but opposite -coordinates. The same is true for points U and Y. This indicates that TU can be reflected over the -ais to form XY. As a result, ou know that XY TU. Draw a conclusion. Since each side of nxyz is a reflection of the corresponding side of ntuv, the corresponding sides of the triangles are congruent. According to the SSS Postulate, ntuv nxyz. Duplicating this page is prohibited b law. 015 Triumph Learning, LLC Unit 1: Congruence, : Proof, Proof, and and Constructions

9 EXAMPLE B Ian is studing wing designs for airplanes. He compares two wings whose triangular cross sections both contain one 0 angle, an adjacent side that measures feet, and a non-adjacent side that measures feet. Determine if the triangular cross sections are identical Make a plan. Identical triangles are congruent, so rigid motions should carr one of the figures onto the other. Transform one of the figures so that known congruent parts line up, and compare the other parts. Attempt to align the angle and adjacent side. The angles are alread aligned, but the -foot sides are not. Rotate the second triangle 0 counterclockwise so that its -foot side is also horizontal. Re-align the angles. The angles are no longer aligned, so reflect the image verticall to bring them back into alignment. 0 Duplicating this page is prohibited b law. 015 Triumph Learning, LLC DISCUSS 0 If two triangles have two pairs of congruent sides and one pair of congruent angles, can ou prove that the triangles are congruent? Translate the image onto the other triangle. 0 The angle and adjacent side are aligned, but the remaining sides and angles do not match. The cross sections are not congruent. Lesson : Congruent Triangles

10 Practice Use the coordinate plane below for questions 1. L O M 0 N 1. /LON and /MON both measure degrees.. LO 5 OM 5 units. nlon and nmon share side.. Triangles LON and MON are congruent b the Postulate. HINT If two triangles share a side, that line segment is the same length in both triangles. Choose the best answer. 5. Which pair of rigid motions shows that nabc and na9b9c9 are congruent? C B A A 0 B C A. reflection across the -ais followed b a translation of units up B. reflection across the -ais followed b a translation of units down C. rotation of 180 about the origin followed b a reflection over the -ais D. rotation of 90 counterclockwise about the origin followed b a translation of units down Duplicating this page is prohibited b law. 015 Triumph Learning, LLC Unit 1: Congruence, Proof, and Constructions

11 . The coordinate plane on the right shows isosceles right triangles HIJ and KLM. Use the ASA Postulate to prove that nhij and nklm are congruent. Identif rigid motions that could transform nhij into nklm. I H 0 K J M L 7. Triangle ABC was reflected horizontall, reflected verticall, and then translated to form triangle A9B9C9. Identif the lengths and angle measures below. A9B9 5 AC 5 m/b9 5 m/a 5 A 1 B C C 5 5 B 15 A Use the following information for questions 8 and 9. Right triangle NOP has vertices N(1, 1), O(1, 5), and P(, 1). Right triangle QRS has vertices Q(, 1), R(, 5), and S(1, 1). 8. SKETCH Sketch both triangles on the coordinate plane. Duplicating this page is prohibited b law. 015 Triumph Learning, LLC 9. PROVE Prove that nnop and nqrs are congruent. 0 Lesson : Congruent Triangles 5

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