1 Geometry Fall 2016 Name: Period: Unit 2 Congruence. Designed to be used together with Mathematics Vision Project Packet (Year 2) Module 5

Transcription

1 1 Geometry Fall 2016 Name: Period: Unit 2 Congruence Designed to be used together with Mathematics Vision Project Packet (Year 2) Module 5 1

2 2 Geometry Homework Guide Unit 2 Period Name Date Learning Target/Standard Homework Entry Task Entry Task Total: Important Dates: 2

3 3 Finding Congruent Triangles Name 1) All triangles are made up of three sides and have three interior angles. Using a protractor, measure all three angles in each of the triangles below measured to the nearest degree: a. "#$ m " = m # = m $ = A B C b. '() m ' = m ( = m ) = c. *+, m * = m + = m, = P Q R d. Make a conjecture about the measures of the angles found in a triangle. What do the measures of the angles in triangles all seem to have in common (We will prove this conjecture to be true later.) 3

4 4 2) Given that there are si parts to every triangle (three sides and three angles) it turns out that when we compare any two of them to see if they are the same triangle congruent, we only need to compare any combination of three of those parts. List all the ways that we can choose three parts of a triangle to compare: A S S A A S 3) When we choose three parts of a triangle and compare them to their corresponding parts of a different triangle, sometimes we find the two triangles are congruent and sometimes we find they are not congruent. In this activity, you will choose any three parts of a triangle and investigate whether those three parts create eactly one triangle or if those three parts can make more than one different triangle. Name the parts taken and circle if only one triangle can be formed or if more than one triangle can be made: 3 Parts Taken Number of Possible Triangles Eactly One Triangle can be formed More than one Triangle can be formed Eactly One Triangle can be formed More than one Triangle can be formed Eactly One Triangle can be formed More than one Triangle can be formed Eactly One Triangle can be formed More than one Triangle can be formed Eactly One Triangle can be formed More than one Triangle can be formed Eactly One Triangle can be formed More than one Triangle can be formed Eactly One Triangle can be formed More than one Triangle can be formed 4) If only one triangle can be formed given three parts of a triangle, then that is sufficient evidence needed to prove two triangles are congruent. I the three parts of one triangle are congruent to three corresponding parts of another triangle, the two triangles are congruent. Show the tick marks that would indicate the triangles are congruent in five different ways: 4

5 5 5) Indicate if the two triangles are congruent or not congruent given the marks. If they are congruent, justify this by indicating the three corresponding parts as listed on the previous page: 5

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7 7 CONGRUENCE, CONSTRUCTION AND PROOF Congruent Triangles to the Rescue A Practice Understanding Task Part 1 Zac and Sione are eploring isosceles triangles triangles in which two sides are congruent: Zac: I think every isosceles triangle has a line of symmetry that passes through the verte point of the angle made up by the two congruent sides, and the midpoint of the third side. Sione: That s a pretty big claim to say you know something about every isosceles triangle. Maybe you just haven t thought about the ones for which it isn t true. Zac: But I ve folded lots of isosceles triangles in half, and it always seems to work. Sione: Lots of isosceles triangles are not all isosceles triangles, so I m still not sure. 1. What do you think about Zac s claim Do you think every isosceles triangle has a line of symmetry If so, what convinces you this is true If not, what concerns do you have about his statement 2. What else would Zac need to know about the crease line through in order to know that it is a line of symmetry (Hint: Think about the definition of a line of reflection.) 3. Sione thinks Zac s crease line (the line formed by folding the isosceles triangle in half) creates two congruent triangles inside the isosceles triangle. Which criteria ASA, SAS or SSS could he use to support this claim Describe the sides and/or angles you think are congruent, and eplain how you know they are congruent. 4. If the two triangles created by folding an isosceles triangle in half are congruent, what does that imply about the base angles of an isosceles triangle (the two angles that are not formed by the two congruent sides) Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org 7 SECONDARY MATH I // MODULE 7 CC BY Anders Sandberg 24

8 8 25 SECONDARY MATH I // MODULE 7 CONGRUENCE, CONSTRUCTION AND PROOF If the two triangles created by folding an isosceles triangle in half are congruent, what does that imply about the crease line (You might be able to make a couple of claims about this line one claim comes from focusing on the line where it meets the third, non-congruent side of the triangle; a second claim comes from focusing on where the line intersects the verte angle formed by the two congruent sides.) Part 2 Like Zac, you have done some eperimenting with lines of symmetry, as well as rotational symmetry. In the tasks Symmetries of Quadrilaterals and Quadrilaterals Beyond Definition you made some observations about sides, angles, and diagonals of various types of quadrilaterals based on your eperiments and knowledge about transformations. Many of these observations can be further justified based on looking for congruent triangles and their corresponding parts, just as Zac and Sione did in their work with isosceles triangles. Pick one of the following quadrilaterals to eplore: A rectangle is a quadrilateral that contains four right angles. A rhombus is a quadrilateral in which all sides are congruent. A square is both a rectangle and a rhombus, that is, it contains four right angles and all sides are congruent 1. Draw an eample of your selected quadrilateral, with its diagonals. Label the vertices of the quadrilateral A, B, C, and D, and label the point of intersection of the two diagonals as point N. 2. Based on (1) your drawing, (2) the given definition of your quadrilateral, and (3) information about sides and angles that you can gather based on lines of reflection and rotational symmetry, list as many pairs of congruent triangles as you can find. 3. For each pair of congruent triangles you list, state the criteria you used ASA, SAS or SSS to determine that the two triangles are congruent, and eplain how you know that the angles and/or sides required by the criteria are congruent (see the following chart). Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org 8

9 26 9 SECONDARY MATH I // MODULE 7 CONGRUENCE, CONSTRUCTION AND PROOF- 7.5 Congruent Triangles Criteria Used (ASA, SAS, SSS) How I know the sides and/or angles required by the criteria are congruent If I say ΔRST ΔXYZ based on SSS then I need to eplain: how I know that RS XY, and how I know that ST YZ, and how I know that TR ZX so I can use SSS criteria to say ΔRST ΔXYZ 4. Now that you have identified some congruent triangles in your diagram, can you use the congruent triangles to justify something else about the quadrilateral, such as: the diagonals bisect each other the diagonals are congruent the diagonals are perpendicular to each other the diagonals bisect the angles of the quadrilateral Pick one of the bulleted statements you think is true about your quadrilateral and try to write an argument that would convince Zac and Sione that the statement is true. Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org 9

10 28 10 SECONDARY MATH I // MODULE 7 CONGRUENCE, CONSTRUCTION AND PROOF After working with these equations and seeing the transformations on the coordinate graph it is good timing to consider similar work with tables. 6. Match the table of values below with the proper function rule. I II III IV V f() f() f() f() f() A.!! =!!! +! D.!! =!! +! +! B.!! =!!! +!" E.!! =!! +! +!" C.!! =!!! +! SET Topic: Use Triangle Congruence Criteria to justify conjectures. In each problem below there are some true statements listed. From these statements a conjecture (a guess) about what might be true has been made. Using the given statements and conjecture statement create an argument that justifies the conjecture. 7. True statements: Point M is the midpoint of!"!"#!"#!"!" Conjecture: A C a. Is the conjecture correct b. Argument to prove you are right: Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org 10

11 11 29 SECONDARY MATH I // MODULE 7 CONGRUENCE, CONSTRUCTION AND PROOF True statements!"#!"#!"!" Conjecture:!" bisects!"# a. Is the conjecture correct b. Argument to prove you are right: 9. True statements!"# is a 180 rotation of!"# Conjecture:!"#!"# a. Is the conjecture correct b. Argument to prove you are right: GO Topic: Constructions with compass and straight edge. 10. Why do we use a geometric compass when doing constructions in geometry Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org 11

12 12 Etra Practice / Homework 12

13 -1- e Z2t0h1Z1k qkuuntras ZSZoHfRtqwnacr6e5 elklscz.c U 3ASl1lL Qr3iRguhNt2sE srieys0eirxvyepd2.7 Z MkakdJeO lwaiitwh9 tinvf9icnigtnes LGKecoTmTeZthr yg.0 Worksheet by Kuta Software LLC 13 Kuta Software - Infinite Geometry Parallel Lines and Transversals Name Date Period Identify each pair of angles as corresponding, alternate interior, alternate eterior, or consecutive interior. 1) 2) y y 3) 4) y y 5) 6) y y 7) 8) y y 13

14 -2- i L2c0j1e12 wkcuxtwan NSmoFfftDwaTr1eS OLwLXCZ.4 b GAKlzl 7rHi1gVhntisE 5r1eJsge5rwvteOd3.o Z TMTaOd0eF DwQihtohB TInnAfViYnRi5tveR 4G1eTo7mUeitOrGyL.k Worksheet by Kuta Software LLC 9) 14 10) y y Find the measure of each angle indicated. 11) 12) ) 14) ) 16) ) 18)

15 -3- F z2o0a1h1n EKTuttpaK YS1o3fQt4wIahryeN gl1lfcl.s W ta1lql4 krriwgbhztjsh krne6smevrevvebdi.1 Q ymna9djel ew4iptpht ki NnAf2iQnjiQtmeq egjeiopmye8tvrjym.o Worksheet by Kuta Software LLC 15 Solve for. 19) 20) ) 22) ) 24) Find the measure of the angle indicated in bold. 25) 26) ) 28)

16 -1- J K2L0H181m ek4untuan 9STo5fBtMwXa2ree9 slylrcp.h g CAHlGlA PrCiAgRhatqsv rrevsve3rrvceod8.k 5 CMoaBd9e8 5wAiAtghM viqnef6i7nni8tre7 QGYeXoEm9eatyrQyf.Q Worksheet by Kuta Software LLC 16 Kuta Software - Infinite Geometry Angles in a Triangle Find the measure of each angle indicated. Name Date Period 1) 65 2) ) 4) ) 6) ) 8)

17 -2- s m2a0q1m1s MKWuGtfad 6SmozfotkwEaWrDe1 GLbLfCl.I h yaklklm kr4ingch7tes7 BrqeVsIe9rXv4e7dC.r S pmpaadjed WwAiXtjhN GIsndfGiznFiXtoeq 8GlemoymfeUtgr1yh.c Worksheet by Kuta Software LLC 9) ) ) 86 12) ) 14) ) 16)

18 -3- k 22X0n1v1k 3KjurtSad ysmonfutbwzagrve3 hlvlcch.f N 4AJlilo greiugnhvtjst Xr8eCsBenrGvAezdn.Y V 6MlaEdhe2 AwwiDtUhX fiunef1iknuittwe8 wg7ehosmueotursyh.b Worksheet by Kuta Software LLC 18 Solve for. 17) ) ) 20) Find the measure of angle A. 21) 22) + 59 A A 23) A ) A

19 h0w1b26 8Kduftrak jsxo0fmthwuairseu klxl0cs.h B QAVl7la grn ig1hqtws5 kr5efsteorsvheqdn.a l nmeapdqeg lwdirtihn LIynzftiSnXirtAen OGyeHofm8eqt9rQy0.X Worksheet by Kuta Software LLC 19 Kuta Software - Infinite Geometry ASA and AAS Congruence State if the two triangles are congruent. If they are, state how you know. 1) 2) Name Date Period 3) 4) 5) 6) 7) 8) 9) 10) 19

20 -1- f M20F1S1l YK0uMtwav ospopfytpwnasruez qlulsc7.5 p ja5ljls ordi2gdhctiss trsemsqevrbvcefdw.c f IMMandSeQ Gw8i3tShv uionjf2irnqiztmey vgmelogmqectzrpyl.x Worksheet by Kuta Software LLC 21 Kuta Software - Infinite Geometry SSS and SAS Congruence State if the two triangles are congruent. If they are, state how you know. 1) 2) Name Date Period 3) 4) 5) 6) 7) 8) 9) 10) 21

21 -1- Q g20031z1q bkyuetpal jswoifntwwvarrzej rlklzce.f a camlylb grciogihntos6 vrlessbe2rfvnendn.c U RM8aedfeF ewcitthh QIinGftiFnuirtseV ygjejokm0e1tfrgyu.k Worksheet by Kuta Software LLC 23 Kuta Software - Infinite Geometry Congruence and Triangles Complete each congruence statement by naming the corresponding angle or side. Name Date Period 1) DEF KJI 2) BAC LMN E K I A M L D F J B C N FD A 3) TUV GFE 4) WVU GHI U F G W V I U G H T V E W U 5) ZXY ZXC Y 6) DEF DSR E F X Z D R S C Y F Write a statement that indicates that the triangles in each pair are congruent. 7) J T R 8) D H I I K S C B G 23

22 -2- Q T2m0L111J okmuft0a8 YSFovfut1wAaprzeH YLlL4CN.a 7 qajlhl8 srmi1g5h7tpse YrleisYeDrIvyeydr.C w 4Mfaadmem pwjiptqhe ZIOnGfSi0nuiqtceu sgde1obmvertbrhyo.q Worksheet by Kuta Software LLC 9) 24 R 10) I S Q P R T D 11) X C E 12) U D C W V D T S E Mark the angles and sides of each pair of triangles to indicate that they are congruent. 13) BDC MLK 14) GFE LKM D M K E G L B C 15) MKL STL L F 16) HIJ JTS M K K S I T M L 17) CDB CDL T H 18) JIK JCD J S C I K L J B D C D 24

23 -1- W R2R0Y1p1Y akfucttan 5SnogfStww2asr1eA L4LeCs.A b bazlel6 orwijgdhot6sm OrnezsqeqrbvMeadr.U b IMhaldVeb BwCiVthhG zianifzi2n3iutyeu ogye1ormoertor vyp.q Worksheet by Kuta Software LLC 25 Kuta Software - Infinite Geometry Isosceles and Equilateral Triangles Find the value of. Name Date Period 1) 2) 7 6 3) 6 4) 4 5) 6) ) 8) ) 65 10) 28 25

24 -2- K r250b1a19 4KmuBtraE ts9o7fotcwsanrred ylal1cw.g k sa3l7lt UrBiXghhytvsv rreeasbesrzvpegdh.x k nmfafdrej vwei4tthw oihnrfri8n5iwteel ug5eo8mie6trrqyh.a Worksheet by Kuta Software LLC 11) 26 12) ) 8 14) ) m 2 = ) m 2 = ) m 2 = ) m 2 =

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