FOUNDATIONS OF MATHEMATICS

UNIT 2

Properties of Angles and Triangles

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Email: jmckenzie@sd79. bc.ca Website; https://msmckenzle. weebly.com/fom-ll. html

Table of ContentsUnit 2: Properties of Angles and Triangles....................................................................................................3

CURRICULAR COMPETENCIES COVERED IN THIS UNIT .............................................................................3

CONTENT LEARNING OUTCOMES COVERED IN THIS UNIT .......................................................................3

J. McKENZIE FOUNDATIONS OF MATHEMATICS 11

Unit 2: Properties of Angles and Triangles

CURRICULAR COMPETENCIES COVERED IN THIS UNITReasoning and Modeling (RM)

. Develop thinking strategies to solve problems and play games

. Explore, analyze and apply mathematical ideas using reason, technology and other tools

. Model with mathematics in situational contexts

. Think creatively and with curiosity and wonder when exploring problemsUnderstanding and Solving (US)

. Develop, demonstrate and apply mathematical understanding through play, story, inquiryand problem solving

. Visualize to explore and illustrate mathematical concepts and relationships

. Solve problems with persistence and a positive disposition

. Engage in problem-solving experiences connected with place, story, cultural practices andperspective relevant to local First Peoples communities, the local community and othercultures.

Communicating and Representing (CmRp). Explain and justify mathematical ideas and decisions in many ways

. Represent mathematical ideas in concrete, pictorial, and symbolic forms

. Use mathematical vocabulary and language to contribute to discussions in the classroom

. Take risks when offering ideas in classroom discourse

Connecting and reflecting (CnRf)

. Reflect on mathematical thinking

. Connect mathematical concepts with each other, other areas, and personal interests

. Use mistakes as opportunities to advance learning

. Incorporate First Peoples worldviews, perspectives, knowledge, and practices to makeconnections with mathematical concepts

CONTENT LEARNING OUTCOMES COVERED IN THIS UNIT2A: Identify and describe situations that involve parallel lines cut by transversals. (2. 1)2B: Derive Proofs that involve the properties of angles formed by transversals and parallel lines. (2. 1)2C: Verify with examples, that if lines are not parallel the angle properties do not apply. (2. 1)2D: Prove using deductive reasoning properties of angles formed by. transversals and parallel lines. (2. 2)2E: Solve problems that involve angles, parallel lines and transversals. (2. 2)2F: Identify and correct errors in a given proof involving angles, parallel lines and transversals. (2. 2)2G: Develop and demonstrate strategies for constructing parallel lines. (2. 2)2H: Prove and apply the relationship relating the sum of the angles in a triangle. (2. 3)21: Solve problems that involve angles in triangles (2. 3)2J: Generalize, using inductive reasoning, a rule for the relationship between the sum of the interior

angles and the number of sides (n) in a polygon. (2.4)2K: Solve problems involving angles in polygons. (2. 4)

Email: jmckenzie@sd79. bc.ca Website: https://msmckenzie. weebly.com/f6[n-ll. html

UNIT REFLECTION

RED I still have difficulty understanding the learning outcome.

YELLOW I am somewhat ok in my understanding of the work.

GREEN I am strongly confident In my understanding of work.

Formative: DURING

2A: Identify and describe situations thatinvolve parallel lines cut by transversals.

(2. 1)2B: Derive Proofs that involve the

properties of angles formed bytransversals and parallel lines. (2. 1)

2C: Verify with examples, that if lines arenot parallel the angle properties do not

apply. (2. 1)2D: Prove using deductive reasoningproperties of angles formed bytransversals and parallel lines. (2. 2)

2E: Solve problems that involve angles,parallel lines and transversals. (2. 2)

2F: Identify and correct errors in a givenproof involving angles, parallel lines andtransversals. (2. 2)

2G: Develop and demonstrate strategiesfor constructing parallel lines. (2. 2)

2H: Prove and apply the relationshiprelating the sum of the angles in atriangle. (2. 3)

21: Solve problems that involve angles intriangles (2. 3)

2J: Generalize, using inductive reasoning,a rule for the relationship between thesum of the interior angles and thenumber of sides (n) in a polygon. 2.42K: Solve problems involving angles inpolygons. (2. 4)

Formative: REVIEW

(If red/yellow, explain what needs to be done)

FOM11

Angle Properties

Acute

Geometric Properties Review

Right Complimentaiy

Obtuse Straight Supplementaiy

Angles on a line Reflex Angles at a point

Triangle Properties

Vertically opposite angles

Ll

Quadrilateral Properties

Trapezoid

0Parallelogram

Rectangle Rhombus n

Square

ssignment: Geometric Properties Worksheet

eo e ri rope lesWoName:

A. Classify each of the following angles as Acute, Right, Obtuse, Straight, or Reflex.

1). 2). 3). 4.)

5). 21° 6). 90° 7). 270° 8). 93° 9). 172° 10). 180°

B. Identify the following shapes:

.A 2).

i):. 7).

3). 4). 5).

8). s'). <-» 10). i -tt-

Z^ "V '"£7"). p-+-a U)

h . ri

13). 14). 15).

C. Find the measure of each missing angle

i Remember Angles on a Straight Line add up to 180e ;|0 i

I).

a' 60"

2). 3).

b" 140' 30" c'

4). 5). 6).115"

130" d"r 35'

©

Remember Vertically Opposite angles are Equal

7). 8).

45-Xa-

9). 10). 11).

120'Xb- 135'

c"

25'

d" p'S 115'

12).

Remember Angles at a Point add up to 360°

13). 14).

72' b' .. 76'

154' iir135'

sr

Remember Interior Angles in a Triangle add up to 180°

15). 16). 17). 18).

^^ ^^19).

/^.-rss" 4r\m"

22). _. . 23).72' e"

24).

Remember Interior Angles In a Quadrilateral add up to 360°

25). 26)75"

130"

85"

31).

27). 28). 29)."' - 31 Xl:- f

43T " e5'154

d' ef

30).B7"

73-h"

52-

32).52°

FOM11 2.1 Exploring Parallel Lines

Parallel Lines and Transversals

A fa'ansversal is a line that intersects two or more other lines at distinct points.

Parallel lines are lines with the same slope but different y-intercepts. Parallel lines will neverintersect each other.

If two parallel lines are cut by a transversal, eight angles are created.

-a bc d

e

g h

Corresponding angles are on the same side of the transversal, and on the same side of theparallel Unes. (They are in the same position)

.a

d

/g

Interior angles lie inside the parallel lines.

Co-Interior Angles: Interior angles on the sameside of the transversal.

-c d

e f

Alternate Interior Angles: Interior Angles on oppositesides of the transversal.

Exterior angles lie outside the parallel lines.

Co-Exterior Angles: Exterior angles on the sameside of the transversal.

-a .b

g h

Alternate Exterior Angles: Exterior angles on oppositesides of the transversal.

***Iftwo parallel lines are cut by a transversal then Corresponding Angles,Alternate Interior Angles, & Alternate Exterior Angles are equal. ***

***Likewise, if two lines are cut by a transversal and the Corresponding Angles, or^Alternate Interior Angles, or the Alternate Exterior Angles are equal then the lines

[3J areparaUel. ***

Example 1: Find each indicated angle:

a.

60°

b.1

70°

c.

80° i

d.50°

e.

75°50

Assignment: Pg. 72 #2-6 ©

.. arallel Lines WorksheetName:

Find the measure of each required angle aad give the reason for your answer.

1.

42'

Zl=

2.37

62° 2

Z2=

Z3=

3.

73'

48" 101° 5

5.

Z9=

Zl=

Z.1=

Z3=

Z4=

6.

63°3 49

50"

872° ~ 6

2 1

^4=

Z5=

Z6=

4.

^.1=

Z8=

108'

Z5=

Z6=

2:7=

zs=

Z9=

7.

Zl=

2:2=

2 1

56'

®

8.

Z3=

^4=

Z5=

Z6=

9.

41° 4 3

117'

20'

Name all the pairs of parallel segments in eachfigure. State the reason for your answer.(Remember if any of the angles created by thelines satisfy the parallel line/angle rules it provesthe lines are parallel).

T

n- "00s

00

u

N110' 20

40

R

^1=

Z8=

Z9=

Zl=

20'

12. A B

0

D

10.

40'2 5

30

6

13.

50'

R

Z2=

2;3=

2^4=

Z5=

'. 6=

M N

115'

FOM11 2.2 Angles Formed by Parallel Lines

From last day we know that when a transversal crosses parallel lines, the corresponding anglesare equal. There are two other sets of angles that have a relationship when a transversal crossesparallel lines.

Alternate Interior AnglesWhen a transversal intersects a pair of parallel lines, the alternate Interior angles are equal.

Proof:

®

Co-Interior Angles:When a transversal intersects a pair of parallel lines, fhe co-lnterior angles are siq)plementaiy.

Proof:

Example 1: Determine the measures of a, b and c.

b

90°

100°

30°

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PROBLEMS M HANDLED W OURAurorwreo SADISTICPHONE SWTEM.

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TOR TECH SUPPORT,PRESS THE EXACTVALUE OF 22 DIVIDEDBY 1.

\/S^'S'^S'SBSk.

L.&». k-

eunini,

Example 2: Find the measure of Zl

lxl-25x)°{xr

Example 3; Detennine the measures of a, b, c and d.

60°

c ab

30°d

Assignment: pg. 78 #1-4, 10, 12, 13, 15, 16, 20

(fe)

FOM11 23 Angle Properties In Triangles

The sum of the angles in a triangle is 180°.

We can use our knowledge of parallel lines to prove (deductively) this theorem.

Example 1: Given AABC, prove Zl + Z2+Z3 = 180

Example 2: Determine the measures of Zl and Z2.

140"

The measure of an ertertor angle of a triangle ii equal to the sum of the measures of the two non-adjacent interior angles.

EmmpleS: Prove Ze= Za+Zb .

d ce

Example 4: Determine Zl, Z2, Z3, and Z4

60°

90°

5 3

Example 5: Given AS || CD

A

1

C 2

Prove

3 B 4

z

z

1=

1=Z4

Z2

Assignment: pg. 90 #2, 3, 5-9, 12, 15, 16, 18

FOM11 2.4 Angle Properties in Polygons

A polygon is a closed geometric figure made up of n straight sides.

A convex polygon has all interior angles less than 1 80°.

A concave polygon has at least one interior angle greater than 180'

#ofsidesina

polygon

3

4

5

6

7

8

9

10

11

12

sketch

A

# of trianglesformed

1

Sum of interior anglesof the polygon

1X180° =180°

2x180° =360°

n

In any polygon with n sides, the sum ofthe interior angles is 180°(n-2). Aregular polygon has equal sides andequal angles.

Example 1: Determine the measure of each interior angle of a regular 17-sided polygon.

The sum of the exterior angles of any convex polygon Is 360°.k0

Each exterior angle of a regular polygon isn

Example 2: Show that the sum of the exterior angles of a pentagon is 360'

Example 3: What type of regular polygon has an interior angle 3 times the exterior angle?

Awignment: Pg. 99 #1-4, 6-11, 14, 18

Name:

«-r

Ob ectlves:

1. Students will demonstrate their knowledge of parallel lines with a transversal.

2. Students will show when angles are congruent or supplementary given parallel lines anda transversal.

Materials needed:

. Pencil

. Colored pencils or markers

. Ruler

. Paper (graph paper, if desired)

Overview:

For this project, each student will make his or her own street map for a fictional city (you must

name your city). This city will consist of:

1. Five (5) streets that are parallel to each other. Each street should be named for reference.

2. Two (2) transversal streets. (Two or more streets that intersect aHjive of the above streets).

These should be named as well. Do not make the transversals parallel to each other! !!

3. Traffic lights or stop signs at four (4) different intersections.

4. The following buildings, represented in your city:

a. Post office

b. Bank

c. Fire Department

d. Police Station

e. Gas Station

f. School

g. Restaurant

h. Grocery Store

i. Courthouse

j. Your own house

Instructions on the back of this page- 1->

Instructions:

The point of this project is not to place these buildings anywhere, but to demonstrate yourunderstanding of different angles, as well as to understand when they are supplementary orcongruent. You can still be creative in doing so, but please place the buildings in the followinglocations.

1. Your house and the school should be located on alternate Interior angle corners.

2. The post office and the bank are positioned such that they are co-interior angles.

3. The fire department and the police station are on corners that are supplementary.

4. The restaurant and courthouse are gt alternate exterior angles.(Alternate Exterior Angles are like Alternate Interior Angles, but they lay on outside theparallel lines) x

5. The gas station and grocery store are at congruent corresponding angles.(Congruent is a fancy word for things that are equal)

Remember to be creative. You may be as artistic as you would like in drawing thebuildlngs/roads, but be sure to label each one. (Creativity will earn extra points).

Name:

Name of City:

N

W-- ^Evs

sA

vN

M

L

:aiuEN

:<lDjoauiEN

Marks

2

5

2

4

2

2

2

2

5

6

6

Rubric for Project: Parallel lines (Turn in with Project)

Name:

Category Score Comments

Name of City

Name of Each Street

Two Transversal Streets (named)

Traffic lights or stop signs at four (4)different Intersections

Your house and the school on

alternate interior angle corners

The post office and the bank are

co-interlor angles

The fire department and the policestation are on corners that are

supplementary

The restaurant and courthouse are at

alternate exterior angles.

The gas station and grocery store arecorresponding angles

Neatness

Drawing your own streets

Creativity

Total:(Out of 40)

^

^

FOM11 Ch.2 Practice TestProperties of Angles and Triangles

Name:

Block:

Multiple ChoiceIdentify the choice that best completes the statement or answers the question.

1. Which statement about the angles in this diagram is false?

a.

b.c.

d.

a

e 130°c

Z.b = 50°Zc-50°Ze=130"Z/=62°

62°

d

f

b

2. Which statement about the angles in this diagram is false?

144°

a. Zg=36°b. ^a=36"c. Zc=36°d. Zd=36°

s

3. Which are the correct measures of the indicated angles?

w

y 103'

a. Zw=77°, Zx=77°, Z)'=1030b. Zw= 77°, A =103°, ̂ =103°c. Zw=103°, A=77°, ^=77°d. Zw=l03°, Zx=lOy, Zy=77°

4. Which are the con-ect measures for ZBSZ and Z-tZr?

r

36-

117-

w

a. ZHZ= 63°, ZAZY= 91°b. ZESZ=53°,ZAZy=9I°c. ZKSZ=63°, ZXZy=81°d. ZEiZ=53°, ZAZy==81°

5. Which are the correct measures of the interior angles of ACD£?

12T1ST

a. ZDC£=92°, ZCD£=49°, andZC£D=39°b. ZDCE=52°. Z.CDE=69°, aadZCED=59°c. ZOCE=62°, ZCD£=49°, andZC£D=69°d. ZDC£=72°, ZCD£=59°, andZC£D=49°

y\

6. Determine the sum of the measures of the interior angles of this polygon.

a. 1080°b. 1440°c. 720°d. 540°

7. Each interior angle of a regular convex polygon measures 144°.How many sides does the polygon have?

a. 10b. 11c. 8d. 9

8. Determine the value of a.

a. 30°b. 35°c. 45°d. 25°

Sb

Short Answer

9. Detennme the measure of ZDBF.

A

114* B66'

10. Determine (he values of a, b, and c.

'? 3d 7o

b

7-c ^

11. Determine the measure of ZMfO.

L M75'

2-N

45' 0

3^

12. Determine the unknown angles.

B C

.

A

82'

£

13. Determine the value of ̂ .

2x+23'

x+31- 3x

14. Determine the sum of the measures of the angles in a 13-sided convex polygon.Show your calculation.

S2

Problem

15. Describe four different methods to prove EF || GH.

F

H

B

16. Prove: FG\\ HI

G

I

94- 73-F H /

17. Each interior angle of a regukr polygon is eight times as kuge as its coiresponding exterior angle. How manysides does the polygon have? Explain your answe

35

Ch.2 Practice Test - Properties of Angles and TrianglesAnswer Section

SHORT ANSWER

MULTIPLE CHOICE

1. A2. A3. C4. C5. D6. A7. A8. A

PROBLEM

9. ZDBF=U4°

10. Za-l 8°, Zt, - 54°, Zc = 27°

11. ZNMO=82°

12. ZEAD = 47°, 2^5C = 47°, ZAD£ =51°,ZBCD =51°, ACDA = 129°

13. ;c=21°

14. 180°(13-2)=1980°

15. 1) equal alternate interior angles, such as ZECD = Z.CDHU) equal corresponding angles, such as ZACE = Z.CDGiU) equal alternate exterior angles, such as ZAC£ = ZHDBIv) interior angles on same side of transversal are aapp\emmfsay, such as ZECD + ZCDG = 180'

16. ZFHG+^GHI+ZIHJ=1SO''94°+ZGff+73° =180°

ZGH7=180°-94°-73°^.GHI = 13°ZGffi = ZFGH

Therefore, FG II HI

Sum of angles in triangle is 180'Substitute known values.Determine Z. GHI.

alternate interior angles are equal

17. Let* represent the measure of the exterior angle. Then &c represents the measure of the mterior angle.The interior angle and its corresponding exterior angle are supplementary, so,

^+&c=180°9x = 180°x-20°

OR

Therefore, the measure of the interior angle is8(20°) = 160°. Now determine the number of sides, n,of a regular polygon with 160°-angle8:

(a-2)180°B

=160°

180°B-360°-ll50°B

B(180°-1(0°)-3150°

20°n - 3(0°

B=18

The measure of the interior angles in a regular18-sided polygon is 160° .The polygon has Igsides.

We can use the formula thateach Exterior angle of a regular

360Is - so,

20= 360n

_ 360"=^5-B-18

The polygon has 18 sides.

^