1

Angles and Lines 10th Grade Geometry

5-Day Unit Plan Tools Used: Geometer’s Sketchpad, Protractors

By: Holly Kubicki

2

Objectives for the unit: Students must be able to:

• Recognize, define and discuss complimentary angles, supplementary angles, right angles, vertical angles, angle bisector, intersecting lines, perpendicular lines, oblique lines, parallel lines and transversals

• Draw and measure angles using a protractor • Practice drawing and measuring angles and lines by using Geometer’s Sketchpad • Discover relationships among the angles formed by intersecting lines,

perpendicular lines and two parallel lines cut by a transversal • Observe and locate interior angles and exterior angles • Locate and explain why same side interior angles are supplementary • Locate and explain why alternate interior angles, alternate exterior angles,

corresponding and vertical angles are congruent NCTM Standards for Grades 9-12: Geometry Standard Measurement Standard Communication Standard New York State Standards MST Standard 3 Key Idea 4- Modeling/Multiple Representation Key Idea 5-Measurement Resources Textbook- D.C. Heath and Company, Geometry: An Integrated Approach, Roland E. Larson, Laurie Boswell, Lee Stiff, Chapter 2&3: pg. 71,90-91,101,105,134-140, Copyright 1998 Web Site- http://library.thinkquest.org/2647/geometry/angle/angle.htm Angles and Lines by ThinkQuest Computer Applications- Geometer’s Sketchpad Materials Computers with Geometer’s Sketchpad Overhead Projector Protractors Rulers Pencils Worksheets

3

♦ Unit Overview ♦

This unit is based on students discovering, exploring and understanding the angle

relationships formed by lines. Students will be learning the definitions and properties of angles and lines with the assistance of technology and manipulatives. Through teacher guided lessons and worksheets, the students will have the opportunity to make connections and have a solid understanding of the unit. Day 1: Introduction and Review of Terms

Students and teacher will fill in and discuss angle and line vocabulary on a graphic organizer.

Day 2: Measuring Angles with Protractors and GSP Teacher will demonstrate and show students how to measure and draw angles using a protractor. Students will compete in group games on measuring and drawing angles. Teacher will then introduce GSP and show students how to construct and measure angles. Students will then practice using GSP to construct and measure angles. Day 3: Intersecting Lines and Perpendicular Lines with GSP

Students will use GSP to complete a worksheet on discovering relationships among the angles formed by intersecting and perpendicular lines.

Day 4: Parallel Lines Cut by a Transversal with GSP Students will use GSP to complete a worksheet on discovering relationships among the angles formed by parallel lines cut by a transversal. Day 5: Review of Definitions and Concepts Students will work in pairs to complete a review worksheet. Teacher and students will go over the review sheet.

4

Day 1

Topic: Introduction and review of terms Learning Objectives: Students must be able to recognize, define and discuss complimentary angles, supplementary angles, right angles, vertical angles, angle bisector, intersecting lines, perpendicular lines, oblique lines, parallel lines and transversals. Motivational Transitioning: Relate: Does anyone know any types of lines or angles? Actively involve: Does anyone know how many degrees are in a straight line? What makes an angle? Do you see any angles around the room? Focus: We need to know the vocabulary of angles and lines in order to explain and prove ourselves when discussing angles and lines. Sharing the goals: Today we are going to learn some useful vocabulary about angles and lines. Instructional Presentation: Teaching/Modeling: 1) Pass out graphic organizer worksheets to the students and display the graphic organizer transparency on the overhead projector. 2) The teacher will state that we will go over, discuss and fill in all the vocabulary terms that are on the graphic organizer and draw in the pictures that represent each term. 3) The teacher will state the first vocabulary term, discuss and write the meaning of the word in the definition part of the graphic organizer on the overhead. The teacher will then think aloud and explain his/her thoughts of drawing the illustration of the word in the picture part of the graphic organizer. To aid in the understanding of the first term- complimentary angles, the teacher could draw a circle on the board which has 360 degrees, then break it into 4 equal parts to show how a straight line has 180 degrees and how complimentary angles have 90 degrees. Guided Practice: 1) Students will then get into groups of about three. 2) In their groups, have the students discuss the meaning of the rest of the words and try to come up with an illustration of the word. The teacher will walk around and ask some questions to get the students to communicate their thoughts about the terms. By thinking about and discussing the terms, this will hopefully stir up some previous knowledge. Closure: 1) After the groups have thought about the meaning of the vocabulary, the teacher will have the groups explain what their group thought the words meant. The teacher will fill in the rest of the graphic organizer with the definitions and illustrations while getting input from the students. Independent Practice/Evaluation: Students will be given a worksheet where they will have to answer questions based on the vocabulary.

5

Name:____________________________ Angle and Line Vocabulary

VocabularyVocabulary DefinitionDefinition PicturePicture

complimentary angles

supplementary angles

right angle

vertical angles

angle bisector

intersecting lines

perpendicular lines

oblique lines

parallel lines

transversal

6

Name:_________________________ 1)

Is ray BD the angle bisector? Why or why not. __________________________________ __________________________________ __________________________________

2) Are angles ABD and DBC complimentary to

each other? Why or why not. __________________________________ __________________________________ __________________________________ 3) Are these intersecting lines oblique or

perpendicular? Why. __________________________________ __________________________________ __________________________________

4)

Name all the sets of supplementary angles. ______________ ______________ ______________ ______________ ______________ ______________ ______________ ______________ ______________ ______________ ______________ ______________

Name all the sets of vertical angles.

______________ ______________ ______________ ______________ ______________ ______________ 5) Draw three parallel lines cut by a transversal.

m!D B C = 3 2

°

m!A B D = 3 3

°

B

A

C

D

m!D B C = 3 6

°

m!A B D = 5 4

°

B

A

C

D

m!D B E = 8 9

° m!C B E = 9 1

°

m!A B D = 9 1

° m!A B C = 8 9

°

B

A

E

D C

7

10

11

5

12

6

8

4 2

3

9

1

7

Name:____Answer Key___________ 1)

Is ray BD the angle bisector? Why or why not. Ray BD is not the angle bisector because_ angle ABD is not equal to angle DBC.___ __________________________________

2) Are angles ABD and DBC complimentary to

each other? Why or why not. Angle ABD is complimentary to angle DBC because when added they both equal 90____ degrees._____________________________ 3) Are these intersecting lines oblique or

perpendicular? Why. These intersecting lines are oblique because angle ABC, ABD, DBE, & CBE are not right angles that is they are not 90 degrees.______

4)

Name all the sets of supplementary angles. _angles 1&2____ _angles 3&4____ _angles 1&4_____ _angles 2&3____ _angles 5&6_____ _angles 7&8____ _angles 6&7_____ _angles 5&8____ _angles 9&10____ _angles 11&12__ _angles 10&11___ _angles 9&12___

Name all the sets of vertical angles.

_angles 1&3____ _angles 2&4_____ _angles 5&7____ _angles 6&8_____ _angles 9&11___ _angles 10&12___ 5) Draw three parallel lines cut by a transversal.

m!D B C = 3 2

°

m!A B D = 3 3

°

B

A

C

D

m!D B C = 3 6

°

m!A B D = 5 4

°

B

A

C

D

m!D B E = 8 9

° m!C B E = 9 1

°

m!A B D = 9 1

° m!A B C = 8 9

°

B

A

E

D C

7

10

11

5

12

6

8

4 2

3

9

1

8

Day 2

Topic: Measuring and drawing angles with protractors and GSP Learning Objectives: Students must be able to draw and measure angles using a protractor. Students must practice drawing and measuring angles using GSP. Motivational Transitioning: Relate: Does anyone know what a protractor is used for? Actively Involve: Does anyone know how to use a protractor or any other tool that would help measure and draw angles? Focus: We need to know how to use protractors and other tools so we can measure and draw angles. Sharing the Goals: Today we are going to learn how to measure and draw angles using protractors and a computer program, Geometer’s Sketchpad, so we can have a better understanding of the vocabulary we learned yesterday. Instructional Presentation: First part of Lesson: Teaching/Modeling: 1) Pass out protractors to every student. 2) The teacher will question students about the protractors. For example, ask questions such as, How many degrees does the protractor go up to? How do you know this? How many degrees does a circle have? Is the protractor half a circle? These questions should get the students thinking about what they already know. 3) The teacher will have large angles drawn on the overhead projector so the teacher can demonstrate and explain with his/her protractor how to measure an angle so all the students can see. The teacher will then ask students to explain to teacher how to measure the next few angles. 4) The teacher will then demonstrate and explain to students how to draw an angle for a given measurement on the overhead. The teacher will then ask students to explain to teacher how to draw an angle for a given measurement. Guided Practice: 1) Pass out worksheets of angles and blank sheets of paper to students and have them get into groups of 3. 2) Teacher will have students measure the angles for #1 & #2 on the worksheet to get a feel for using the protractor. The teacher will walk around and help students. 3) The students will then draw an angle for measurement on #3 & #4 on the blank sheet of paper. The teacher will continue to walk around to help students who are having difficulty. Closure: 1) The teacher will then have the groups compete to see which group can measure the angle for #5 the fastest and with the correct angle measurement. Every student in the group must measure the angle and make sure each member came up with the same angle measurement, then raise their hands as a group. The teacher will verify

9

the angle measurement and determine which group won. This game will continue for #6-#10. 2) The groups will compete again to see which group can draw angles for the measurement on #11 the fastest and with the correct angles drawn. Every student in the group must draw the angle for angle for a given measurement and make sure every member of the group has the correct angle drawn, then raise their hands as a group. The teacher will verify the angle measurement and determine which group won. The game will continue for #12-#16. 3) The teacher will also ask questions throughout the lesson pertaining to the vocabulary learned yesterday such as, What would the compliment of this angle be?, Have the students draw the compliment angle, Are these angles supplementary? Second part of lesson: Teaching/Modeling: 1) The teacher will display Geometer’s Sketchpad on an overhead screen and give students an introduction of how to use Geometer’s Sketchpad. The teacher will give students an overview of how to sketch points, lines, rays, segments, label points and so forth. The teacher will also tell students how to highlight, press Esc to unhighlight everything, use edit tab to undo a mistake, use file tab to open a new sketch and some other important things about the program. 2) The teacher will demonstrate and explain how to draw an angle using rays. Then to find an angle measurement, highlight all 3 points in order and use the measure tab to measure angle. Teacher will show students if you move a point the angle measure will change. Guided Practice: 1) The students will practice at their own computers for a few minutes to get a feel for how the program works. 2) The teacher will then have students draw an angle and get the measure of it while talking them through steps. The teacher will be walking around and helping students. The students can then move around the points on their angle to see how the angle measure changes. Closure: 1) Students will open a new sketch and sketch a right angle, and prove it’s a right angle by showing its measure. The teacher will check the students’ sketches. Independent Practice/Evaluation: Students will be given a worksheet where they will use protractors to measure and draw angles, and answer questions based on the vocabulary.

10

Name:_________________________ Measuring and Drawing Angles 1) 2) 3) 155° 4) 40° 5) 6) 7) 8) 9) 10) 11) 90° 12) 104° 13) 52° 14) 76° 15) 38° 16) 180°

11

Name:__________________________ 1) Give the angle measure for a & b a) b) 2) Draw angles for a & b a) 33° b) 161° 3) Are the two angles complimentary? Why or why not.

__________________________________________________________________________________________________________________

4) Are the two angles supplementary? Why or why not.

____________________________________________________________________________________________________________

5) Is ray BC the angle bisector? Why or why not. _____________________________ _____________________________ _____________________________ _____________________________

B

A

C

D

12

Name:____Answer Key__________ 1) Give the angle measure for a & b a) 44° b) 143° 2) Draw angles for a & b a) 33° b) 161° 3) Are the two angles complimentary? Why or why not.

Yes, the two angles are complimentary because the first angle is 65°__ and the second angle is 35° and both equal___ 90°________________

4) Are the two angles supplementary? Why or why not.

No, the two angles are not supplementary___ because the first angle is 96° and the second angle is 86° and both do not equal 180°___

5) Is ray BC the angle bisector? Why or why not. Yes ray BC is the angle bisector___ because angle ABC is 53° and____ angle CBD is 53°______________ _____________________________

B

A

C

D

13

Day 3

Topic: Intersecting lines and perpendicular lines with GSP Learning Objectives: Students must discover relationships among the angles formed by intersecting lines and perpendicular lines. Students must locate and explain why vertical angles are congruent. Motivational Transitioning: Relate: Can someone tell me what intersecting lines are? Actively Involve: Are the vertical angles in intersecting lines always congruent? Focus: We are going to further our understanding of intersecting lines to know why vertical angles are congruent. Sharing the Goals: Today we are going to discover why vertical angles are congruent using what we already know about intersecting lines, vertical angles and supplementary angles.

Instructional Presentation: Teaching/Modeling: 1) The teacher will display Geometer’s Sketchpad on an overhead screen and give students a review of the program. 2) The teacher will demonstrate and show students how they will discover angle relationships formed by intersecting lines and perpendicular lines through examples of complimentary and supplementary angles. 3) The teacher will construct complimentary angles and show the measure of each angle and the right angle measure as shown below. By moving the ray BD, the students will see the 2 angles will always equal 90˚ 4) The teacher will then construct supplementary angles and show the measure of each angle and the straight angle measure as shown below. By moving ray BD, the students will see the angles will always equal 180˚.

m!D B C = 4 5

°

m!A B D = 4 5

°

m!A B C = 9 0

°

B

A

D

C

m!D B C = 4 9

°

m!A B D = 1 3 1

°

m!A B C = 1 8 0

°

BA

D

C

14

Guided Practice: 1) The teacher will pass out worksheets and have students work at their own computers to complete the worksheet. The students will be able to help each other out and work along with the students sitting next to them. The teacher will also be circulating the room aiding students and asking questions. Closure: 1) The teacher will review the worksheet by asking students for their answers and explanations. Independent Practice/Evaluation: Students will be given a worksheet where they will have to exercise their understandings of intersecting lines, vertical angles and perpendicular lines.

15

Name:_____________________ Intersecting and Perpendicular Lines Directions: Follow instructions and answer questions. 1) GSP: Construct two lines that intersect. 2) GSP: Highlight both lines, go to Construct tab and construct intersection. 3) GSP: Label all 5 points. 4) GSP: Find the measure of all 4 angles created by the intersecting lines. What angles are supplementary? _______ & _______, _______ & _______, _______ & _______, _______ & _______

What angles are vertical angles? _______ & _______, _______ & _______ What angles are equal in measure? _______ & _______, _______ & _______

5) GSP: Move a point on one of the lines, move a point on both of the lines.

Are the angles still equal in measure? _______ 6) Look at the diagram of intersecting lines below.

<A + <B = 180˚ and <B + <C = 180˚ Is this true? ____ <A + <B = <B + <C Is this true? ____ <A + <B = <B + <C

___- <B -<B_____ <A = <C Explain why <A is congruent to <C. ____________________________________ ____________________________________ Show why <B is congruent to <D by using the above method. (Show on back of worksheet) 7) GSP: Construct intersecting lines that are perpendicular. Prove they are perpendicular by finding the angle measurements. Why are the intersecting lines you constructed perpendicular? ___________________________________________________________ ________________________________________________________________________

C

A

BD

E

A

CD

B

16

Name:_____________________ 1) If <A = 145˚, What are the measurements of <B = ______ <C = ______ <D = ______ If <D = 47˚, What are the measurements of <B = ______ <C = ______ <A = ______ 2) If <A = <B = <C = <D, What type of

lines formed these angles? ________________________

How do you know? ______________________________ ______________________________ ______________________________

A

C

D B

A

C

DB

17

Name:__Answer Key___________ 1) If <A = 145˚, What are the measurements of <B = _35˚___ <C = _145˚__ <D = _35˚___ If <D = 47˚, What are the measurements of <B = _47˚___ <C = _133˚__ <A = _133˚__ 2) If <A = <B = <C = <D, What type of

lines formed these angles? __Perpendicular Lines______

How do you know? _Since all the angles are equal, they_

_are each 90˚ because all the angles_ _added up must equal 360˚.________

A

C

D B

A

C

DB

18

Day 4

Topic: Parallel lines cut by a transversal with GSP Learning Objectives: Students must discover relationships among the angles formed by two parallel lines cut by a transversal. Students must observe and locate interior angles and exterior angles. Students must locate and explain why same side interior angles are supplementary and why alternate interior angles, alternate exterior angles and corresponding angles are congruent. Motivational Transitioning: Relate: Can someone tell me what parallel lines are? What is a transversal? Actively Involve: Are parallel lines cut by a transversal the same thing as two pairs of intersecting lines? Focus: We are going to discover which angles are congruent and supplementary in parallel lines cut by a transversal. Sharing the Goals: Today we are going to discover why alternate interior angles, alternate exterior angles and corresponding angles are congruent and why same side interior angles are supplementary using what we already know about vertical angles and supplementary angles. Instructional Presentation: Teaching/Modeling: 1) The teacher will display a Geometer’s Sketchpad drawing of parallel lines cut by a transversal on an overhead screen. 2) The teacher will explain and show students which angles are called the exterior and interior angles as well as alternate angles and corresponding angles on the GSP drawing. 3) The students will be given worksheets with parallel lines cut by a transversal on it. They will with assistance from teacher name all the exterior and interior angles, alternate exterior angles, alternate interior angles, corresponding angles and same side interior angles. The students will not yet know which or why the angles are congruent or supplementary.

Guided Practice: 1) The teacher will then pass out GSP worksheets and have students work on their own computers to complete the worksheet. This worksheet will guide students in discovering the angle relationships through using GSP. By drawing constructions, measuring angles and answering thought provoking questions, the students will begin to make internal connections between the angles and see the angle relationships. The students will be able to help each other out and work along with the students sitting next to them. The teacher will also be circulating the room aiding students and asking questions.

Closure: 1) The teacher will review the GSP worksheet by asking students for their answers and explanations. Students and teacher will return to the first worksheet and complete the summary.

19

Independent Practice/Evaluation: Students will be given a worksheet where they will have to exercise their understandings of the angles formed by parallel lines cut by a transversal.

20

Name: ________________________ Parallel Lines Cut by a Transversal Exterior Angles: _____,_____,_____,_____ Interior Angles: _____,_____,_____,_____ Alternate Exterior Angles: _____&_____, _____&_____ Alternate Interior Angles: _____&_____, _____&_____ Corresponding Angles: _____&_____, _____&_____, _____&_____, _____&_____ Same Side Interior Angles: _____&_____, _____&_____ Fill in after completing GSP worksheet Summary: Corresponding Angles are ________________ Alternate Exterior Angles are ________________ Alternate Interior Angles are ________________ Same Side Interior Angles are ________________

Angle 5

Angle 1

Angle 8 Angle 7

Angle 4 Angle 3

Angle 2

Angle 6

21

Name:_____________________ GSP-Parallel Lines cut by a Transversal Directions: Follow instructions and answer questions. 1) GSP: Construct two lines that are parallel. 2) GSP: Construct a transversal. 3) GSP: Find the measure of all the angles. (First construct the 2 points of intersection of parallel lines and transversal) Move the angle measurements to the angle for which its measure is. Then hide all the points. (Your measurements will not be the same as this one) 4) GSP: Re-label all the angles as the diagram below is by right clicking on each angle, going to properties, under label tab retype the label then click on ok. Describe everything you notice about your diagram of parallel lines cut by a transversal. ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ What do you notice about the angle measurements? ______________________________ ________________________________________________________________________ Which angles are congruent? ______=______=______=______ ______=______=______=______

m!H E G = 1 2 2

°m!

F E H = 5 8

°

m!B E G = 5 8

°m!

F E B = 1 2 2

°

m!D B E = 1 2 2

°m!C B E = 5 8

°

m!A B D = 5 8

°m!

A B C = 1 2 2

°

Angle 7 = 122°

Angle 5 = 122 ° Angle 6 = 58°

Angle 8 = 58°

Angle 4 = 58° Angle 3 = 122°

Angle 1 = 122° Angle 2 = 58°

22

Are the alternate interior angles congruent? ______ Are the alternate exterior angles congruent? ______ Are the corresponding angles congruent? ______ Are the same side interior angles supplementary? ______ 5) GSP: Move a point on the transversal to move the transversal, this will change the angle measurements. Are your answers from 4) the same after you moved the transversal? _______ 6) Look at the diagrams of parallel lines cut by a transversal below. If we cut a line along the dotted line through the transversal, this will leave two sets of intersecting lines in which we already know vertical angles are congruent.

If we slide the intersecting lines together so the lines match up, the two sets of intersecting lines match up exactly. Now using the diagram below and knowing that the intersecting lines are equal

and vertical angles are equal, name all the congruent angles. ______=______=______=______

______=______=______=______

Are the alternate interior angles congruent? ______ Are the alternate exterior angles congruent? ______ Are the corresponding angles congruent? ______

Are the same side interior angles supplementary? ______

Angle 5

Angle 1

Angle 8 Angle 7

Angle 4 Angle 3

Angle 2

Angle 6

23

Name: _______________________

If Angle 2= 65˚, what are the angle measurements of <1= _____ <3= _____ <4= _____ <5= _____ <6= _____ <7= _____

<8= _____ If Angle 5= 135˚, what are the angle measurements of <1= _____ <2= _____ <3= _____ <4= _____ <6= _____ <7= _____ <8= _____

If Angle 7= 90˚, what are the angle measurements of <1= _____ <2= _____ <3= _____ <4= _____ <5= _____

<6= _____ <8= _____

If Angle 1=115˚, what are the angle measurements of <2=_____ <3=_____ <4=_____

If Angle 3=64˚, what are the angle measurements of <1=_____ <2=_____ <4=_____

Angle 5

Angle 1

Angle 8 Angle 7

Angle 4 Angle 3

Angle 2

Angle 6

Angle 5

Angle 1

Angle 8 Angle 7

Angle 4 Angle 3

Angle 2

Angle 6

Angle 5

Angle 6

Angle 4

Angle 2

Angle 8

Angle 1Angle 3

Angle 7

Angle 1

Angle 4

Angle 2

Angle 3

Angle 1

Angle 4

Angle 2

Angle 3

24

Name: ___Answer Key_____________

If Angle 2= 65˚, what are the angle measurements of <1= _115˚ <3= _115˚ <4= _65˚_ <5= _115˚ <6= _65˚_ <7= _115˚

<8= _65˚_

If Angle 5= 135˚, what are the angle measurements of <1= _135˚ <2= _45˚_ <3= _135˚ <4= _45˚_ <6= _45˚_ <7= _135˚ <8= _45˚_

If Angle 7= 90˚, what are the angle measurements of <1= _90˚_ <2= _90˚_ <3= _90˚_ <4= _90˚_ <5= _90˚_ <6= _90˚_

<8= _90˚_ If Angle 1=115˚, what are the angle measurements of <2=_65˚_ <3=_115˚ <4=_65˚_

If Angle 3=64˚, what are the angle measurements of <1=_64˚_ <2=_116˚ <4=_116˚

Angle 5

Angle 1

Angle 8 Angle 7

Angle 4 Angle 3

Angle 2

Angle 6

Angle 5

Angle 1

Angle 8 Angle 7

Angle 4 Angle 3

Angle 2

Angle 6

Angle 5

Angle 6

Angle 4

Angle 2

Angle 8

Angle 1Angle 3

Angle 7

Angle 1

Angle 4

Angle 2

Angle 3

Angle 1

Angle 4

Angle 2

Angle 3

25

Day 5

Topic: Review of Definitions and Concepts Learning Objectives: Students will be able to review and consolidate newly learned material. Sharing the Goals: Today we are going to be reviewing the definitions and all the things we discovered about angles and lines. Instructional Presentation:

Guided Practice: 1) Students will be able to work in pairs to complete the review worksheet. The teacher will also be circulating the room aiding students. Closure: 1) The teacher will review the worksheet by asking the pairs of students to go up to the board and explain their answers.

26

Name: ____________________ Review Sheet Directions: Match the phrase with a figure. (Match each letter exactly once.) 1)Intersecting Lines ____ a) b) 2)Angle Bisector ____ 3)Vertical Angles ____ 4)Parallel Lines ____ 5)Complimentary Angles ____ 6)Perpendicular Lines ____ 7)Parallel Lines Cut by a Transversal ____ 8)Supplementary Angles ____ 9)Right Angle ____ c) d)

e) f) g)

h) i)

m!A B C = 9 0

°

B

A

C

m!A B D = 2 0

°

m!A B C = 2 0

°

B

D

C

A

Angle 2 = 35°

Angle 1 = 35°

m!A B C = 9 0

°

A

B

C

27

Directions: Use your protractor to measure and draw the given angles and answer the questions. 1) 2)

3) Are these angles complimentary? _______ Why or why not? ______________________ ____________________________________ 4) Are these angles supplementary? _______ Why or why not? ______________________ ____________________________________ 5) Draw perpendicular lines. Directions: Find the measure of x. 1) How do you know what x is equal to? Why?

_______________________________ _______________________________

_______________________________

2) How do you know what x is equal to? Why?

__________________________________ __________________________________ __________________________________

Angle 1 = 100°

x

Angle 1 = 56° x

28

Directions: Answer the questions. 1) Angle 1= 123˚

What is the measure of angle a? _____ How do you know this? ______________________ __________________________________________ What is the measure of angle b? _____ How do you know this? ______________________ __________________________________________

2) Angle 1=84˚ What is the measure of angle x? _____

Why? _____________________________________ 3)

Angle 1=124˚ What is the measure of Angle 2? _____

Angle 3? _____ Angle 4? _____ Why?________________________________ _____________________________________

4) Angle N=67˚ & Angle P=150

What is the measure of Angle m? _____ Why? _____________________________________ __________________________________________

5) Angle C=46˚ & Angle B=92˚

What is the measure of Angle A? _____ Why?________________________________ _____________________________________

6 ) If Angle X=151˚, what is the measure of Angle Y=_____ & Angle Z=______

Why?____________________________________ _________________________________________

Angle 1

b

a

Angle 1

Angle 4

Angle 2

Angle 3

Angle N

Angle Pm

Angle B

A

Angle C

Angle 1

x