GEOMETRY  CURRICULUM  –  ST.  IGNATIUS  COLLEGE  PREP    

Part  1:    Course  Enduring  Understandings  Part  2:    Unit  Essential  Questions  and  Learning  Outcomes  Part  3:    Suggested  Pacing  Part  4:    Two  Sample  Common  Assessments  

 Part  1:  

Geometry  Enduring  Understandings    1.    Geometry  is  omnipresent  in  the  physical  world;  it  can  be  used  to  solve  problems  in  real  life.    2.    Geometry  knowledge  is  used  in  many  branches  of  mathematics.    3.    Geometry  uses  standard  vocabulary  and  symbols  to  communicate  facts  and  relationships  about  geometric  figures.      4.  Geometric  figures  are  ruled  by  known  relationships  of  measures,  often  expressed  as  theorems  and/or  algebraic  formulas.        5.    Proofs,  constructions  and  visual  observations  demonstrate  why  geometric  relationships  are  true.    6.      Logic,  in  combination  with  facts,  theorems  and  formulas  can  be  used  to  draw  conclusions  about  geometric  figures.      7.    A  proof  is  a  formal  argument  supported  by  postulates,  theorems  and  definitions;  it  uses  logical  reasoning  to  come  to  its  conclusion.      Part  2:  

Geometry  Essential  Questions  &  Performance  Objectives  by  Unit      Basic Skills 1. What  symbols,  formulas  and  vocabulary  are  conventional  for  communicating  within  the  context  of  Geometry?  

Students will explore the general concepts of patterns, points, lines, planes and angles. They will review midpoints and distances from Algebra, and will begin constructions. By the end of this unit, students will be able to: • Identify inductive reasoning • Find and describe visual and numeric patterns • Understand basic undefined terms of geom. • Identify intersections of lines and planes • Classify angles • Know and apply distance formula • Know and apply Segment Addition Postulate • Know and apply Angle Addition Postulate • Understand Geometry vocabulary • Know and apply Midpoint Formula or equivalent method • Perform constructions: bisect an angle, bisect a segment • Use a protractor to draw & measure angles • Identify vertical & linear pairs of angles, and use their relationships algebraically • Find areas & perimeters of common plane figures • Use area & perimeter formulas to solve problems

Reasoning 1.    How  and  why  is  deductive  reasoning  used  in  geometric  proof?  2.    How  can  traditional  constructions  deepen  understanding  and  illustrate  geometric  relationships?  

Students will begin an exploration of the deductive system in Geometry by exploring patterns, the geometric structure of theorems and postulates, conditional statements, and algebraic properties. Students will also explore segment and angle relationships. By the end of this unit, students will be able to: • Analyze & write conditional & biconditional statements • Understand basic point, line and plane postulates • Use symbolic notation for conditional statements • Form conclusions by using laws of logic • Use properties of length and measure to justify segment/angle relationships • Recognize algebraic properties of equality/properties of congruence • Write reasons for steps in a proof • Use deductive reasoning to prove statements about segments and angles • Perform constructions: copy a segment, copy an angle

Perpendicular & Parallel Lines 1.    What  algebraic  and  geometric  conditions  are  sufficient  and  necessary  to  prove  lines  parallel  

or  perpendicular?  2.    What  are  the  angle  relationships  when  parallel  lines  are  cut  by  a  transversal?  3.    What  are  the  conventional  forms  of  proof?  

Students will explore lines and planes, using algebraic connections and logic to lead them into different proof styles. Properties of parallel and perpendicular lines are introduced. By the end of this unit, students will be able to: • Construct lines parallel or perpendicular to a given line • Identify relationships between lines • Identify/use relationships of angle pairs formed by lines and a transversal • Prove & use angle relationships involving parallel lines and a transversal • Prove lines are parallel, given angle relationships • Use slopes to identify parallel/perpendicular lines in the coordinate plane • Write equations of lines, given point and/or slope information • Graph lines from equations • Become familiar with different types of proof • Use properties of lines to prove statements • Complete Geometer’s Sketchpad tutorial unit

   Congruent Triangles 1. What are the different classifications for triangles? 2. How can triangles be proven congruent? 3. How can congruent triangles be used to solve problems?

Triangles are explored extensively, with emphasis on congruent triangles and proof. Students will use congruent triangles and will extend their work with constructions. By the end of this unit, students will be able to: • Classify triangles by sides and angles • Solve problems based on interior and exterior angle relationships • Identify congruent figures and corresponding parts • Understand and apply postulates and theorems proving triangles congruent • (SSS, SAS, AAS, ASA, HL) • Prove triangles congruent with given information • Use congruent triangles to solve problems • Copy a triangle by construction • Use congruent triangles to prove segment or angle relationships • Understand & use properties of isosceles and equilateral triangles

Properties of Triangles 1. What segments have special purposes in understanding triangles and solving problems? 2. What are some traditional constructions involving special segments in triangles? 3. What is indirect proof and how is it different from direct proof?

Students will explore special segments in triangles as well as triangle inequalities. They will further explore constructions. By the end of this unit, students will be able to: • Use properties of perpendicular bisectors & angle bisectors to solve problems • Identify special segments in a triangle • Construct circle circumscribing a triangle • Construct centroid of a triangle

• Solve problems using properties of a centroid • Solve problems using properties of medians & altitudes of a triangle • Solve problems using properties using the midsegments of a triangle • Write and solve inequalities using properties of sides and angles of one triangle • Write and solve inequalities comparing sides/angles of two triangles • Write a paragraph-style indirect proof

   Quadrilaterals    1.    By  what  characteristics  can  one  classify  quadrilaterals?  2.    What  are  necessary  and  sufficient  conditions  for  proving  a  quadrilateral  is  a  parallelogram?  3.    How  can  algebra  be  used  to  classify  quadrilaterals?    

Students will explore polygons, with a special emphasis on quadrilaterals. By the end of this unit, students will be able to:

 • Identify & describe polygons • Solve problems using the sum of interior angles of a quadrilateral • Know and use properties of parallelograms, rectangles, rhombi & squares • Prove that a quadrilateral is a parallelogram or special parallelogram • Know and use properties of trapezoids & kites • Find areas of quadrilaterals • Use area formulas of quadrilaterals to solve problems • Use coordinate geometry in conjunction with quadrilaterals to solve problems

Transformations

1. What  are  the  types  and  characteristics  of  geometric  transformations?  2.    What  are  the  traditional  ways  to  represent  vectors?  

Students will explore rigid motion in a plane. By the end of this unit, students will be able to: • Identify the three basic rigid transformations – reflection, translation, rotation • Use transformations to solve problems • Identify the two types of symmetry – line and rotational • Use a vector to describe a translation • Write a vector in component form • Use vectors to solve problems

   Similarity    1.    How  are  ratio  and  proportion  related  to  geometric  figures?  2.    What  information  is  needed  to  prove  triangles  similar?  3.    How  is  knowledge  of  similar  figures  applicable  to  real-­‐world  problems?    

Students will explore and use ratio and proportion. Students will explore similar polygons with an emphasis on similar triangles. By the end of the unit, students will be able to:

• Compute the ratio of two numbers • Use proportions to solve problems • Use properties of proportions • Identify and Define similar polygons and find their scale factor • Use similar polygons to solve problems • Identify similar triangles • Use similar triangles in coordinate geometry • Use the AA, SSS, and SAS similarity theorems to prove two triangles are similar • Use similar triangles to solve real-life problems • Use proportionality theorems to solve problems • Identify a dilation and write the scale factor of a dilation • Use dilations to solve problems

   Right  Triangles    1.    What  theorems  and  other  rules  apply  specifically  to  right  triangles?  2.    What  information  is  needed  in  order  to  apply  these  rules  and  theorems?  3.    How  are  vectors  and  trigonometry  used  to  solve  real-­‐world  problems?    

Students will explore right triangles, special right triangles, trigonometric ratios in right triangles, and vectors in right triangles. They will explore models involving right triangles. By the end of the unit, students will be able to:

• Prove right triangles are congruent using HL congruence theorem • Use properties of right triangles • Compare similar right triangles using the altitude drawn to the hypotenuse • Use the Pythagorean Theorem and its converse to solve problems • Find the lengths of sides of special right triangles • Use special right triangles to solve problems • Find the sine, cosine, tangent, secant, cosecant and cotangent of any angle measured

in degrees (using a calculator) • Use trigonometric ratios to solve problems • Use coordinate geometry in the exploration of vector problems • Find the magnitude and direction of a vector • Add vectors and sketch vectors • Solve a right triangle • Use right triangles to solve problems

   Circles    1.    What  vocabulary  is  used  to  describe  circles  as  they  relate  to  lines  and  angles?  2.    How  can  circles  give  us  information  about  angle  measures  and  segment  lengths?  3.    How  are  the  equation  of  a  circle  and  its  graph  on  the  Cartesian  Plane  related?    

Students will explore circles, their parts and their equations. By the end of the unit, students will be able to:

• Use all vocabulary associated with circles • Use the properties of circles in problems • Use properties of tangents to solve problems in geometry • Name minor and major arcs of a circle • Find measures of central angles and arcs of circles • Use the measures of central angles and their arcs to solve problems • Use properties of chords and arcs to solve problems • Find the lengths of segments and chords in a circle • Use the properties of inscribed angles to solve problems • Use properties of the inscribed angles of a quadrilateral to solve problems • Calculate angles formed by tangents, chords, and secants • Use angle measures to solve real-life problems • Write the equation of a circle and use it to solve real-life problems • Sketch a circle on the coordinate plane given its equation

   

Planar Measurements  

1.    How  are  area  formulas  for  plane  figures  derived?  2.    How  are  area  and  perimeter  used  in  real-­‐world  applications?  

 Students will explore area and perimeter of polygons and area and circumference of circles. By the end of the unit, students will be able to: • Find the perimeter of a polygon • Find the area of a square, a rectangle, a parallelogram, a triangle, and a trapezoid • Find the area of an un-classified quadrilateral whose diagonals are perpendicular • Find the area of a regular polygon • Use areas to solve problems • Find the measures of and the sum of the interior and exterior angles of a polygon • Find the area and circumference of a circle • Find the length of an arc of a circle • Find the area of a sector of a circle (regions of a circle) • Compare the areas and perimeters of similar polygons • Calculate probabilities based on area

   

Spatial Measurements 1. How are surface area formulas for 3-D figures derived? 2. How  are  volume  formulas  for  3-D figures derived? 3. How  are  surface  area  and  volume  formulas  used  in  real-­‐world  applications?  

Students will explore surface area and volume of three-dimensional solids. By the end of the unit, students will be able to: • Identify solids that are polyhedrons • Use polyhedrons to solve real-life problems

• Find the surface area of a prism, cylinder, pyramid, cone and sphere • Find the volume of a prism, cylinder, pyramid, cone and sphere • Use volume and surface areas to solve real-life problems

   Part  3:    Suggested  Geometry  Pacing  (Regular)   (Based  on  Larson  Geometry  text,  2004)    Chapter         #  of  class  days,     Emphasis       Counting  assessments       &  Review  days    

 1     10       Notation,  segments,  angles    2     12       Bisectors,  two-­‐column  proof,  reasoning    3     9       Parallel  Lines,  two-­‐column  proof    (Sometimes  we  can  do  that  much  in  the  first  quarter,  and  sometimes  we  need  to  finish  Ch  3  in  the  second  quarter.    A  common  cumulative  quiz  may  be  given  at  midterm  time  or  in  the  second  quarter  as  agreed-­‐upon.    Traditionally,  we  have  had  the  students  learn  Geometer’s  Sketchpad  during  two  class  days  in  the  first  or  second  quarter.)    4     10       Congruent  triangles  with  proof    5     10       Special  segments  in  triangles;  some  construction    (In  past  years,  we  have  gotten  through  some  of  Chapter  6  before  December  final  exams.    In  ’09-­‐’10,  we  got  through  Ch.  5,  and  some  classes  did  some  of  Ch.  6.)    6     12       parallelograms,  area    7     5     Rotations,  reflections,  translations,  vectors  (7.1-­‐7.4)    8     11       Similar  triangles,  proportion    9     15     Pythagorean  Theorem,  Special  Rt.  Triangles,  Basic  Trig    Trig  Supp.     6       Radians,  Unit  Circle,  Exact  trig  values      (Some  years,  some  teachers  have  included  9.6  &  9.7  with  the  trig  supplement  and  assessed  them  together,  so  Ch.  9  was  13  days,  and  the  trig  supp  was  8  days.    Also,  we  have  typically  given  a  common  cumulative  quiz  on  the  midterm  day  that  ends  the  third  quarter.)    10     7       De-­‐emphasize  segments/geom.  mean  in  circle    

11     7       Time  crunch?    Keep  problems  fairly  simple    12     7     Give  all  formulas  for  test  but  emphasize  understanding    (In  recent  years,  some  have  combined  Ch.  11  and  Ch.  12  for  assessments.)      Part  4:        

Geometry Fall Final Exam 2011 Name________________________ 70 points A.M.D.G. December 2011 Multiple Choice Portion – 1 point each

1. True or False: Skew lines can be perpendicular. [A] true [B] false

2. True or False: If two lines are perpendicular to a third line, they are perpendicular to each other. [A] true [B] false

4. Classify ΔQVB by its side lengths. 3. Find the slope of the line given the points

6, −7( ) and

9, −9( ) .

[A] 0 [B] undefined

[C] −

23

[D] −

32

[A] equilateral [B] scalene [C] isosceles [D] obtuse

5. Which of the statements is FALSE, given ΔABC ≅ ΔMNO ? [A] AC ≅ MO [B] CB ≅ ON [C] ∠A ≅ ∠M [D] CA ≅ NM

6. Identify the property: If m∠ABC=m∠DOG and m∠DOG= m∠MAP, then m∠ABC= m∠MAP. [A] reflexive [B] symmetric [C] transitive [D] distributive

8. Given: CD is the perpendicular bisector

of HJ. Which statement is false?

7. What additional information do you need to prove ΔABC ≅ ΔADC by the SAS Postulate?  

                       

[A] AB ≅ AD [B] ∠ABC ≅ ∠ADC

[C] BC ≅ DC [C] ∠ACB ≅ ∠ACD

[A] CH ≅ CJ [B] DH ≅ IJ [C] m∠CIJ = 90° [D] ΔCIH ≅ ΔCIJ

9. Which side lengths allow you to 10. Which statement is a true biconditional?

8

8B

V

Q

JI

H

D

C

construct a triangle? [A] 2, 3 and 7 [B] 5, 4 and 9

[C] 8, 2 and 4 [D] 6, 4 and 7

[A] Angles are congruent if and only if they are vertical angles. [B] Lines intersect if and only if they are not parallel. [C] Lines are perpendicular if and only if they intersect to form right angles. [D] Geometry is enjoyable if and only if you are a math teacher.

11. In the diagram, use the given information to find the value of x. CU = QT; UQ = QT CU = 10x – 6 UQ = 7x + 2

[A] 8/3 [B] 4/3 [C] 10.8 [D] none of these 12. Refer to figure below.

Given: AF ≅ FC , ∠ABE ≅ ∠EBC. Which segment is the median?

[A] BD [B] BE [C] BF [D] GF 13. Which statement is FALSE based on the diagram?

A [A] Lines h and i are parallel. [B] Points D and G are collinear. [C] Points A, D, and K are coplanar. [D] FK

intersects plane J at point K.

14. Which equation is accurate based on the given information and diagram. A, B, and C are collinear.

[A] AB = 7x +1 [B] AC = 10x − 3 [C] BD = 10x − 2 [D] Both B & C

T

Q

U

C

ED FC

GB

A

Jh

K F

E, B F

BDG KAG

!"AB A,C B

A,C,D E

AB BA

##"CD

!##DC

15. Which of the following is an angle bisector in the diagram?

[A] EF

[B] AH

[C] EB

[D] EH

FREE RESPONSE PORTION. POINTS AS NOTED. SHOW WORK. 2 points each 1. Describe the pattern below and find the next two terms. −7,−1,5,11,... 2. Find the area and perimeter of ΔABC .

3. This rectangle has a perimeter of 52 feet. Find CD and the area of the rectangle.

4. Write the third sentence using the Law of Syllogism:

If  I  get  detention,  then  I  won’t  go  to  practice.  If  I  don’t  go  to  practice,  then  I  won’t  play  in  the  game.  

5. Given that m ⊥ n, find the values of x and y.

25°y°x°n

m

6. One side of an equilateral triangle measure 2y + 3 units. If the perimeter of the triangle is 33 units, what is the value of y? 7. C is the centroid of ΔGHJ and CM = 8. Find HM and CH .

6 points each 8. Solve for x, y, and z in the following diagram. Explain your reasoning using definitions/postulates/theorems.

x = _______ because of ________________ y = _______ because of ________________ z = _______ because of ________________

9. Write each form of the following conditional statement. For each form, determine if it is true or false. If the answer is false, provide a counterexample. Original: If I’m sick, then I don’t go to school. TRUE Converse:______________________________________________ _______________________________________________________

TRUE / FALSE Counerexample?

Inverse:______________________________________________ _______________________________________________________

TRUE / FALSE Counerexample?

Contrapositive:___________________________________________ _______________________________________________________

TRUE / FALSE Counerexample?

2 points each 10. Given that m || n, find the value of x. 11. What value of p ensures that m || n?

5 points each 12. G, E, and F are the midpoints of the sides of ΔABC . If EF = 6 and FG = 5, find AB.

If EG = 52x − 3 and AC = 3x + 8 find x.

If m∠GCF = 36° , find m∠BGE . EXPLAIN. 13. Complete the proof using congruent triangles.

Given: P is the midpoint of RS TR SQ Prove: TR ≅ SQ

14. Write an indirect proof. Given: m∠1 = 64° and m∠115° Prove: l is not to m

3 points each 15. Find m∠K . 16. Find m∠Q .

n

m

(5x+13)°

(8x–2)°

(2p–98)°

(p+20)°

n

m

2

1 l

j

m

17. Mark the figure with the given information and find the following. Given: FGHJ is a parallelogram, m∠JHG = 68°, JH = 34, FK = 19.

A. Find m∠FJH. B. Find FH. C. Find FG.

1 point 18. The perpendicular bisectors of ΔXYZ intersect at W .

If WX = 10 and AX = 8 , find WZ.

Extra Credit (2 points): Use your compass and straightedge to bisect this angle.

     

x°(2x+15)°

Q

R

P

F G

HJ

K

Z

B

WX

A

Y

Spring 2010 Final Exam Name: Geometry 2°, 7° Date: Total Points: 76 Part I: Multiple Choice. 16 pts, 1 point each. Answer on the Scantron form. Good luck everyone! 1. Consecutive angles in a parallelogram are always ________. [A] complementary angles [B] vertical angles [C] congruent angles [D] supplementary angles 2. Which statement is true? [A] All quadrilaterals are squares. [B] All rectangles are squares. [C] All rectangles are quadrilaterals. [D] All quadrilaterals are rectangles. 3. Which type of quadrilateral has no parallel sides? [A] rhombus [B] trapezoid [C] kite [D] rectangle 4. Mr. Jones has taken a survey of college students and found that 70 out of 76 students are liberal arts majors. If a college has 7613 students, what is the best estimate of the number of students who are liberal arts majors? [A] 70,120 [B] 8266 [C] 41 [D] 7012 5. One way to show that two triangles are similar is to show that ______. [A] two angles of one are congruent to two angles of the other [B] two sides of one are proportional to two sides of the other [C] a side of one is congruent to a side of the other [D] an angle of one is congruent to an angle of the other 6. If the side lengths of a triangle are 7, 6, and 9, the triangle _____. [A] is obtuse [B] is right [C] is acute [D] cannot be formed 7.  Find  the  equation  of  a  circle  with  center  (1,  3)  and  passes  through  (1,  6).  

  [A]    (x −1)2 + (y − 3)2 = 3                                [B]    (x +1)2 + (y + 3)2 = 9     [C]     (x +1)2 + (y + 3)2 = 3  [D]       (x −1)

2 + (y − 3)2 = 9   8.  Find  the  area  of  an  equilateral  triangle  that  has  side  length  8  in.     [A]    8   [B]       [C]  16     [D]      9.  The  surface  area  of  the  right  cone  shown  is  _____.  

[A] [B] [C] [D]

8 3 16 3

�

44π in.2

�

36π in.2

�

16 33π in.2

�

112π in.2

C

B

A

     10.  If   ,  find   .       [A]  24º       [B]  48º             [C]  96º     [D]12º      11.  Find  the  value  of  x.       [A]  31       [B]  32       [C]  33       [D]  34      12. An aquarium in a restaurant is a rectangular prism and measures 3.5 feet by 4 feet by 4 feet. What is the volume of the aquarium? [A] 56 cubic feet [B] 19.5 cubic feet [C] 11.5 cubic feet [D] 48 cubic feet 13. Find the surface area of a sphere with a diameter of 12 cm. Express your answer in terms of π . [A] 288π [B] 144π [C] 576π [D] 36π

14. Which diagram shows a rotation of approximately –200° in standard position? [A] [B] [C] [D] 15. Which of the following shows a triangle and its reflection image in the x-axis? [A] [B] [C] [D]

16. is a right triangle. AB = _____.

m∠ACB = 48° mAB

�

cm2

�

cm2

�

cm2

x

y

x

y

x

y

x

y

ΔABC

�

cm2

x°76°

123°112

80°

[A] [B] [C] 117 [D]

Part II: Free Response – 60 points Centers of circle are shown with a point. 3 points each unless otherwise stated. Use radical form for special triangles; otherwise round to hundredths. 1. Refer to the figure below. Given: UVWX is a parallelogram, UY = 15, UX = 24, XW = 21. A. Find m WVU. B. Find m XUV. C. Find UW. 2. Write a two column proof. Given: and Prove: is a parallelogram

Statement Reason

3. Trapezoid ABCD has midsegment . If and , find the length of . If 82m EFB∠ = ° then m DCF∠ =

3 5 3 13

3 6

EF

AB = 6 EF = 8 DC

D C

A B

E F

4. Find in the diagram, if 120m R∠ = ° and 80m S∠ = °

5. Find the value of x. 6. Graph PQR with and Graph after the translation

described by the vector . 7. Solve for x and y.

m T∠

T

S

R

U

8. Solve for the unknown side lengths and angles in the figure below. AC = ______, CB = _______, = ________ 9. Find the following values.

a)    sinA  =    

b)    tanA  =      

c)    Find  the   .  

        10. Ruby wants to find the height of the tallest building in her city. She stands 480 feet away from the building. There is a tree 32 feet in front of her, which she knows is 17 feet tall. How tall is the building?

11. The length of the diagonal of a square is 22 cm. What is the length of each side? Draw and label a sketch first. 12.    are  tangent  to  the  circle  and      Find  the  value  of  x.    

y

35

x

40

5

12

E

CB

D

A

m∠B

PR and PQ PR = 2x +15 and PQ = 5x − 6.

C B

A

12

33°

R

Q

P

x°

M L

K

J93°

47°

               13.    If   ,  find  the  length  of  the  diameter.              14.    Find  the  length  of   .    Express  your  answer  as  a  simplified  fraction  in  terms  of  π.  

                     15.    Find  the  volume  of  the  right  prism  below.            

     16.    If   ,  find  the  value  of  x.                    17.    Find  the  volume  of  the  right  cone.    Leave  π  in  your  answer.        

18. Find the surface area of the right cylinder. Leave π in your answer.

EF = 6 and DF = 8

JK

KJ 120° 5

mJK = 93° and mML = 47°

F

E

D

                                                                                                                                                          19.

a) Convert to degrees.

    b)  Convert  60º  to  radians.       c)  Find    from  the  diagram.  

                  20. Make a sketch of a 135° angle in standard position for trigonometry and then find the exact value of sec135°.  

5π2

R

cotA