Mathematics 20-2 - Activating Students in High School Math Web viewMathematics 20-2 Properties of...

MATHEMATICS 20-2

Properties of Angles and Triangles

Lessons 3 and 4 not completed

High School collaborative venture withHarry Ainlay, McNally, M. E LaZerte, Ross Sheppard, Scona, and

W.P. Wagner

Harry Ainlay: Colin VeldkampHarry Ainlay: Debby SumantryHarry Ainlay: Mathias StewartHarry Ainlay: Meriel HughesMcNally: Enchantra GramlichM. E. LaZerte: Monique MerchantRoss Sheppard: Jeremy KlassenRoss Sheppard: Tim GartkeScona: Joe JohnstonW. P. Wagner: Kiki Brisebois

Facilitator: John Scammell (Consulting Services)Editor: Jim Reed (Contracted)

2010 – 2011

Mathematics 20-2 Properties of Angles and Triangles of 47

TABLE OF CONTENTS

STAGE 1 DESIRED RESULTS PAGE

Big Idea

Enduring Understandings

Essential Questions

4

4

4

Knowledge

Skills

5

6

STAGE 2 ASSESSMENT EVIDENCE

Transfer Task (on a separate page which could be photocopied & handed out to students)

Carpentry Short CutsTeacher Notes for Transfer TaskTransfer TaskRubricPossible Solution

79

1517

STAGE 3 LEARNING PLANS

Lesson #1 Parallel Lines 21

Lesson #2 Triangles and Polygons 27

Lesson #3 Congruent Triangles 31

Lesson #4 Geometric Proofs 34

Appendix – Worksheets/Keys 37


Implementation note:Post the BIG IDEA in a prominentplace in your classroom and refer to it often.

Implementation note: Ask students to consider one of the essential questions every lesson or two.Has their thinking changed or evolved?

Mathematics 20-2 Properties of Angles and Triangles

STAGE 1 Desired Results

Big Idea:

Angles and lines are encountered in many places in everyday life. The capacity to describe and relate angles and lines allows us to design and describe objects in many contexts. The angle and line theorems are a context in which students can apply their knowledge of logic and reasoning.

Enduring Understandings:

Students will understand …

When a transversal intersects parallel lines, there are numerous pairs of equivalent angles.

A proof follows a logical set of linked steps. The sum of the angles in a triangle is 180 degrees. Congruent means same size and shape. It is possible to use logic to determine whether two triangles are congruent based on

incomplete information.

Essential Questions:

How can you construct parallel lines using only a compass and straight edge? What conditions are necessary to prove that two triangles are congruent? How can you tell if an argument is invalid? Is it possible to draw a triangle whose angles do not have a sum of 180 degrees? What is so important about proofs?


Knowledge:

EnduringUnderstandingList enduring understandings (the fewer the better)

SpecificOutcomesList the reference # from the Alberta Program of Studies

Description ofKnowledgeThe paraphrased outcome that the group is targeting

Students will understand…


*G1, G2

Students will know …

Which angle pairs are equal when a transversal intersects parallel lines.

Which pairs are supplementary.


A proof follows a logical set of linked steps.

G1, G2


How to construct a proof.


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

G2


That the sum of the angles in a triangle is 180 degrees.

The sum of the interior angles of a polygon is only dependent on the number of sides.


Congruent means same size and shape.

It is possible to use logic to determine whether two triangles are congruent based on incomplete information.

G1, G2


The necessary and sufficient conditions for congruency.

Congruent triangles have the same size and shape.

888888I*G = Geometry


Implementation note:Teachers need to continually askthemselves, if their students are acquiring the knowledge and skills needed for the unit.

Skills: EnduringUnderstandingList enduring understandings (the fewer the better)

SpecificOutcomesList the reference # from the Alberta Program of Studies

Description ofSkillsThe paraphrased outcome that the group is targeting



*G1,G2

Students will be able to…

Determine if lines are parallel, given the measure of an angle at each intersection formed by the lines and a transversal.

Determine particular values of unknown angles in a diagram of parallel lines intersected by a transversal.

Construct parallel lines given a compass or protractor.

Solve a contextual problem involving angles.Students will understand…

A proof follows a logical set of linked steps.

G1, G2


Construct a valid proof. Identify and correct errors in a proof.


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

G2


Identify the necessary pieces of information from a diagram.

Generalize a rule for the relationship between teh sum of the interior angles and the number of sides of a polygon.

Determine the sum of the angles in an n-sided polygon.



It is possible to use logic to determine whether two triangles are congruent based on incomplete information.

G1, G2


Prove that two triangles are congruent. Solve a contextual problem that involves

congruent triangles.


*G = Geometry


Implementation note:Students must be given the transfer task & rubric at the beginning of the unit. They need to know how they will be assessed and what they are working toward.

STAGE 2 Assessment Evidence

1 Desired Results Desired Results

Carpentry Short Cuts

Teacher Notes

There is one transfer task to evaluate student understanding of the concepts relating to properties of angles and triangles. A photocopy-ready version of the transfer task is included in this section.

Each student will:

Use properties of angles and triangles to prove that a drawer is divided into thirds. Use properties of angles and triangles to find the error in a proof showing two

triangles are congruent. Use properties of angles and triangles to show 4 triangles within a larger triangle are

congruent.


Implementation note:Teachers need to consider what performances and products will reveal evidence of understanding?What other evidence will be collected to reflectthe desired results?

Teacher Notes for Carpentry Short Cuts Transfer Task

Glossary

adjacent angles – Angles with a common vertex and a common arm

alternate exterior angles – Angles that are in opposite positions relative to a transversal intersecting two lines. If the alternate angles are outside the two lines intersected by the transversal, they are called alternate exterior angles.

alternate exterior angles – Angles that are in opposite positions relative to a transversal intersecting two lines. If the alternate angles are inside the two lines intersected by the transversal, they are called alternate interior angles.

congruent – Have the same shape and size

converse – A conditional statement formed by interchanging the if and then clauses of another conditional statement

convex polygon – A polygon with all interior angles smaller in measure than a straight angle (180°).

corresponding angles – Angles that are in the same position relative to lines intersected by a transversal

equilateral triangle – A triangle with three congruent sides (and three congruent angles)

exterior angle of a polygon – An angle at a vertex of the polygon, outside the polygon, formed by one side and the extension of an adjacent side

isosceles triangle - A triangle with two congruent sides (and two congruent angles)

non-adjacent interior angles – The two angles in a triangle that do not have the same vertex as the exterior angle [Math 20-2 (Nelson: page 516)]

scalene triangle – A triangle with no congruent sides (and no congruent angles)

transversal – A line that intersects two (or more) lines


Glossary hyperlinks redirect you to the Learn Alberta Mathematics Glossary (http://www.learnalberta.ca/content/memg/index.html). Some terms can be found in more than one division. Some terms have animations to illustrate meanings.

http://www.learnalberta.ca/content/memg/index.html

http://www.learnalberta.ca/content/memg/Division03/Transversal/index.html

http://www.learnalberta.ca/content/memg/Division03/Scalene%20Triangle/index.html

http://www.learnalberta.ca/content/memg/Division03/Isosceles%20Triangle/index.html

http://www.learnalberta.ca/content/memg/Division03/Exterior%20Angle%20(of%20a%20Polygon)/index.html

http://www.learnalberta.ca/content/memg/Division03/Equilateral%20Triangle/index.html

http://www.learnalberta.ca/content/memg/Division03/Corresponding%20Angles%20(Transversal)/index.html

http://www.learnalberta.ca/content/memg/Division03/Polygon%20(Convex)/index.html

http://www.learnalberta.ca/content/memg/Division03/Congruent/index.html

http://www.learnalberta.ca/content/memg/Division03/Alternate%20Angles%20(Transversal)/index.html


http://www.learnalberta.ca/content/memg/Division03/Adjacent%20Angles/index.html

Carpentry Short Cuts - Student Assessment Task

1. Part A: Your friend is a carpenter and he is making a chest of drawers. He would like each drawer to have a handle centered in the face of the drawer. The problem he faces is that he has a twelve-inch ruler, but the drawer is only ten inches wide. What he needs to do is to divide the drawer perfectly into thirds horizontally so that he can put the handle on correctly, but he is concerned that dividing ten by three will mean that he is slightly off when he goes to measure. What he does instead is to rotate the ruler slightly so that he is measuring twelve inches from edge to edge as shown below. It is easy to mark off a third of 12: he simply makes his line from a corner to the opposite side and marks every 4 inches.

Source: http://www.smallscalechemistry.colostate.edu/equipment.html

http://www.smallscalechemistry.colostate.edu/equipment.html


He then slides the ruler down a few inches and repeats the steps above.

Now if he connects the marks he made, he will have cut the drawer into thirds.



As a math student you find this interesting, but you are concerned that what he is doing is actually correct. Prove that your friend has actually divided the drawer into thirds. Assume that the two vertical lines you drew are perpendicular to the top and bottom of the drawer, the corners of the drawer are square, and each oblique line segment is equal. What you are proving is that the horizontal line segments between the vertical lines and the edges are all equal.

A

BC D


Part B: Your friend is building a rafter in the shape of an isosceles triangle. He knows the rafter will be strongest when the two smaller triangles created by the support beam are congruent. He draws an isosceles triangle with a blue line segment representing the support beam, as shown in the diagram below. The following proof is used to show that no matter where the line segment representing the support beam meets the opposite side, that the two triangles formed are congruent.

Statement ReasonΔ ABC is isosceles Given

AC=AB Definition of Isosceles∠ ACB=∠ABC Definition of IsoscelesAD=AD Reflexive property / Shared SideΔ ADC≃Δ ADB SAS

Identify and explain the error in the above proof. Under what circumstances is the above proof valid? In other words what other condition(s) would be necessary for the two triangles to be congruent?

s



Part C: In the shaded space provided, draw points A, B and C. Construct a parallel line to line segment AB, through C. Construct another parallel line to line segment BC though A, and finally repeat this process drawing a line parallel to AC through B. These lines should intersect to form a large triangle. Now connect points A, B and C. Prove that all four of the resulting triangles are congruent.


Put your proof here.

Statement Reason

Glossary



























Assessment

Mathematics 20-2

Properties of Angles and Triangles

Rubric

Level Excellent Proficient Adequate Limited InsufficientCriteria 4 3 2 1 BlankMath ContentPart A

All required elements are present and correct

All required elements are present but may contain minor errors

Some required elements are missing, or contain major errors

Most required elements are missing or incorrect

No score is awarded as there is no evidence given

Math ContentPart B






Math Content Part C






Presents Data

Presentation of data is clear, precise and accurate

Presentation of data is complete and unambiguous

Presentation of data is simplistic and plausible

Presentation of data is vague and inaccurate

Presentation of data is incomprehensible

Explains Choices

Provides insightful explanations

Provides logical explanations

Provides explanations that are complete but vague

Provides explanations that are incomplete or confusing.

No explanation is provided

When work is judged to be limited or insufficient, the teacher makes decisions about appropriate intervention to help the student improve.

Possible Solution to Carpentry Short Cuts

As a math student you find this interesting, but you are concerned that what he is doing is actually correct. Prove that your friend has actually divided the drawer into thirds. Assume that the two vertical lines you drew are perpendicular to the top and bottom of the drawer, the corners of the drawer are square, and each oblique line segment is equal. What you are proving is that the horizontal line segments between the vertical lines and the edges are all equal.

Use corresponding angles to show that all of the A1’s and B1’s are equal. Since the sum of interior angles in a triangle is 180o, the missing angle in the lightest grey triangles is 90 – B1. This establishes that Δ ABC≃Δ DEF (ASA). Within each of these large triangles there are 2 similar triangles (lightest grey triangle and the triangle formed by the two lightest grey areas. Since the original diagonal line was divided into 3 equal regions, the corresponding lines in each similar triangle are also equal.

A

BC D


Part B: Your friend is building a rafter in the shape of an isosceles triangle. He knows the rafter will be strongest when the two smaller triangles created by the support beam are congruent. He draws an isosceles triangle with a blue line segment representing the support beam, as shown in the diagram below. The following proof is used to show that no matter where the line segment representing the support beam meets the opposite side, that the two triangles formed are congruent.

Statement ReasonΔ ABC is isosceles Given

AC=AB Definition of Isosceles∠ ACB=∠ABC Definition of IsoscelesAD=AD Reflexive property / Shared SideΔ ADC≃Δ ADB SAS

Identify and explain the error in the above proof. Under what circumstances is the above proof valid? In other words what other condition(s) would be necessary for the two triangles to be congruent?


The error is in the property used to show the congruence of the triangles. The reasons given actually demonstrate the SSA property, which cannot be used to show congruence of triangles, because under certain conditions two triangles can be constructed that are not congruent.

The proof would be valid if the support beam is a perpendicular bisector. This would make DC=DB and now we can say Δ ADC≃Δ ADB (SAS).

We could also establish congruence by ASA if we establish ∠ ADC =∠ ADB .

s


Part C: In the shaded space provided, draw points A, B and C. Construct a parallel line to line segment AB, through C. Construct another parallel line to line segment BC though A, and finally repeat this process drawing a line parallel to AC through B. These lines should intersect to form a large triangle. Now connect points A, B and C. Prove that all four of the resulting triangles are congruent.


A1 and B1 are placed using corresponding angles. The third angle in the outer triangles must be 180 – (A1 + B1).

The innermost triangle angles can be determined because the sum of angles that form a line is 180o. This establishes the four smaller triangles are similar. Equal angles with a common side proves congruence (ASA).


Put your proof here.

Statement Reason

A1 angles are equal corresponding anglesB1 angles are equal corresponding anglesThird angle in outer triangles is sum of interior angles is 180o

180 – (A1 + B1)Adjacent triangles share a These triangles are congruent (ASA)common side.

Implementation note:

Each lesson is a conceptual unit and is not intended to be taught on a one lesson per block basis. Each represents a concept to be covered and can take anywhere from part of a class to several classes to complete.

STAGE 3 Learning Plans

Lesson 1

Parallel Lines

STAGE 1

BIG IDEA: Angles and lines are encountered in many places in everyday life. The capacity to describe and relate angles and lines allows us to design and describe objects in many contexts. The angle and line theorems are a context in which students can apply their knowledge of logic and reasoning.

ENDURING UNDERSTANDINGS:




ESSENTIAL QUESTIONS:

How can you construct parallel lines using only a compass and straight edge?

KNOWLEDGE:


Which angle pairs are equal when a transversal intersects parallel lines.

Which pairs are supplementary.

SKILLS:

Students will be able to …

Determine if lines are parallel, given the measure of an angle at each intersection formed by the lines and a transversal.

Determine particular values of unknown angles in a diagram of parallel lines intersected by a transversal.

Construct parallel lines given a compass or protractor.

Solve a contextual problem involving angles.


Lesson Summary

Introduce Terminology related to pairs of angles formed by transversals and parallel lines.

Use applets to introduce terminology. Have students generate notes. Explore properties of transversals and parallel lines. Provide access to Math Interactives – Exploring Parallel Lines (Explore It #1). Provide students with Quick Check. Practice properties involving two parallel lines and transversal(s). Provide access to Math Interactives – Exploring Parallel Lines (Use It).

Lesson Plan

Introduce Terminology related to pairs of angles formed by transversals and parallel lines.

Use the applets found at http://members.shaw.ca/jreed/math20-2/program2.htm to introduce the relationships and terminology related to transversals and parallel lines. Text links are provided for the LearnAlberta resources that are displayed on http://members.shaw.ca/jreed/math20-2/program2.htm. Screenshots on this page are hot linked to the original resources.

Discuss the relationships using the applets titled:


http://members.shaw.ca/jreed/math20-2/program2.htm


http://www.learnalberta.ca/content/mejhm/html/object_interactives/parallel_lines/use_it.html

http://www.learnalberta.ca/content/mejhm/html/object_interactives/parallel_lines/explore_it.html


Opposite Angles Alternate Angles (Transversal) Angles on the Same Side of the Transversal, Corresponding Angles (Transversal)

Have students generate their own notes individually or as a group to help them remember the properties discussed. Encourage students to draw pictures of the equal or supplementary angle pairs.


http://ronblond.com/MathGlossary/Division03/Opposite%20Angles/index.html

http://ronblond.com/MathGlossary/Division04/Alternate%20Angles%20(Transversal)/index.html

http://ronblond.com/MathGlossary/Division04/Angles%20on%20the%20Same%20Side%20of%20the%20Transversal/index.html

http://ronblond.com/MathGlossary/Division04/Corresponding%20Angles%20(Transversal)/index.html

Explore properties of transversals and parallel lines.

Math Interactives – Exploring Parallel Lines (Explore It #1) found on the same page as the previous applets (http://members.shaw.ca/jreed/math20-2/program2.htm)

Allow students to play and explore using this applet. This applet provides a great visual of the relationships of transversals and parallel lines and will help students reinforce their understanding. Encourage students to play with all the options within the applet (Angle Type, Reference Angle, Parallel Line Orientation)

Quick Check

This may be used to assess student’s knowledge. The next section of this lesson involves an interactive game and this quick check could be used as an entry pass to be able to play the game.

Quick Check copy was added to Appendix



http://www.learnalberta.ca/content/mejhm/html/object_interactives/parallel_lines/explore_it.html

Practice properties of transversals

Math Interactives – Exploring Parallel Lines (Use It) found on the same page as the previous applets (http://members.shaw.ca/jreed/math20-2/program2.htm)

This game is a great way for students to practice their skills. Encourage students to use the Hint button, the Explore It applet, or their notes to help them win the game.

This would also be a great place to discuss the Essential Question for the lesson and challenge students to come up with a strategy to draw two parallel lines using a ruler and a protractor.

Going Beyond

Resources

Math 20-2 (Nelson: sec 2.1 and 2.2, page(s) 70-82)

Supporting




Assessment

Glossary



alternate interior angles – Angles that are in opposite positions relative to a transversal intersecting two lines. If the alternate angles are inside the two lines intersected by the transversal, they are called alternate interior angles.




Other










Implementation note:

Each lesson is a conceptual unit and is not intended to be taught on a one lesson per block basis. Each represents a concept to be covered and can take anywhere from part of a class to several classes to complete.

Lesson 2

Triangles and Polygons

STAGE 1




A proof follows a logical set of linked steps. The sum of the angles in a triangle is 180

degrees.


Is it possible to draw a triangle whose angles do not have a sum of 180 degrees?

KNOWLEDGE:


That the sum of the angles in a triangle is 180 degrees.

The sum of the interior angles of a polygon is only dependent on the number of sides.

SKILLS:


Generalize a rule for the relationship between the sum of the interior angles and the number of sides of a polygon.

Determine the sum of the angles in an n-sided polygon.

Lesson Summary

The students will explore angles in polygons. The will see several ways of illustrating that the sum of the angles in a triangle is 180°, and will extend this to other closed polygons.


Lesson Plan

Lesson GoalBy the end of the lesson, students should be aware that the sum of the angles in a triangle is 180°, and the sum of the angles (°) in an n-sided polygon is 180(n-2).

Hook/Activate Prior KnowledgeGive students a piece of paper, a straight edge, and a protractor. Tell them that you are going to give a prize to the student who draws the triangle whose angles have the greatest sum, and another prize to the student who draws the triangle whose angles have the smallest sum. Let them draw and measure some triangles, and conclude that all triangles have angles with a sum of 180°.

Show them a rudimentary proof, by having them cut out one of their triangles, and piece it together as shown below.

Lesson

Define Exterior Angle

Have students draw a triangle and an exterior angle. Use the protractor to measure the exterior angle and compare it to the interior angles. Students should notice that the exterior angle is equal to the sum of the interior and opposite angles. They should also notice that the exterior angle and its adjacent angle are supplementary.

Give an example like Example 3 on Page 88 of the Nelson resource, and give students time to work through it in pairs.

Have students use their rulers to draw 4-, 5-, and 6-sided polygons. Ask them to measure the interior angles of each of them, and find the sum.

Ask students to look for a pattern and use that pattern to predict the sum of the angles in a 25-sided polygon. Compare answers. Students should discover that the sum of the angles (°) in an n-sided polygon have a sum of 180(n-2).



Give students an example like #16 on Page 102 of the Nelson resource, and have them work through it in pairs.

Going Beyond

Students could use Geogebra or Geometer’s Sketchpad to complete the investigations in this lesson.

Students could use a spreadsheet to determine the sum of the angles in an n-sided polygon.

Ask students whether the 180(n-2) formula works on all polygons. Does it work on convex, concave, regular, and irregular polygons?

Resources

Math 20-2 (Nelson: sec 2.3 and 2.4, page(s) 86 to 103)

Supporting

Sum of interior angles in a triangle applets: http://www.walter-fendt.de/m14e/anglesum.htmhttp://206.110.20.132/~dhabecker/geogebrahtml/triangle_sum/triangle_sum.htmlhttp://staff.argyll.epsb.ca/jreed/math9/strand3/triangle_angle_sum.htm

Supplementary angles in a triangle applet: http://206.110.20.132/~dhabecker/geogebrahtml/supplementary_defined/supplementary.html

(Really Nice) Sum of angles in a polygon applet: http://www.mathopenref.com/polygoninteriorangles.html

Exterior angles applets: http://www.mathopenref.com/polygonanglerelation.htmlhttp://www.mathopenref.com/triangleextangle.html

Interior/exterior polygon angles:http://staff.argyll.epsb.ca/jreed/math9/strand3/polygon_angles.htm


http://staff.argyll.epsb.ca/jreed/math9/strand3/polygon_angles.htm

http://www.mathopenref.com/triangleextangle.html

http://www.mathopenref.com/polygonanglerelation.html

http://www.mathopenref.com/polygoninteriorangles.html

http://206.110.20.132/~dhabecker/geogebrahtml/supplementary_defined/supplementary.html

http://staff.argyll.epsb.ca/jreed/math9/strand3/triangle_angle_sum.htm

http://206.110.20.132/~dhabecker/geogebrahtml/triangle_sum/triangle_sum.html

http://www.walter-fendt.de/m14e/anglesum.htm

Assessment

Students could be given an exit slip with a couple questions like:http://www.onlinemathlearning.com/quadrilateral-worksheets.html

a. b. c.

Questions could be assigned for homework from the Nelson resource sections 2.3 and 2.4.

Glossary




Other







http://www.onlinemathlearning.com/quadrilateral-worksheets.html

Lesson 3

Congruent Triangles

STAGE 1




A proof follows a logical set of linked steps. Congruent means same size and shape. It is possible to use logic to determine

whether two triangles are congruent based on incomplete information.


What conditions are necessary to prove that two triangles are congruent?

How can you tell if an argument is invalid? What is so important about proofs?

KNOWLEDGE:


How to construct a proof. The necessary and sufficient conditions for

congruency. Congruent triangles have the same size and

shape.

SKILLS:


Construct a valid proof. Identify the necessary pieces of information

from a diagram. Prove that two triangles are congruent. Solve a contextual problem that involves


Lesson Summary


Lesson Plan

Hook

Lesson Goal

Activate Prior Knowledge

Lesson

Going Beyond

Resources

Math 20-2 (Nelson: sec 2.5, page(s) 104-106)

Supporting

Congruent triangles: http://staff.argyll.epsb.ca/jreed/math9/strand3/3203.htm

Assessment

GlossaryMathematics 20-2 Properties of Angles and Triangles of 47

http://staff.argyll.epsb.ca/jreed/math9/strand3/3203.htm














Other















Lesson 4

Geometric Proofs

STAGE 1





A proof follows a logical set of linked steps. The sum of the angles in a triangle is 180

degrees. Congruent means same size and shape. It is possible to use logic to determine

whether two triangles are congruent based on incomplete information.


How can you tell if an argument is invalid? What is so important about proofs?

KNOWLEDGE:


How to construct a proof.

SKILLS:


Construct a valid proof.. Prove that two triangles are congruent. Solve a contextual problem that involves


Lesson Summary


Lesson Plan

Hook

Lesson Goal

Activate Prior Knowledge

Lesson

Going Beyond

Resources

Math 20-2 (Nelson: sec 2.6, page(s) 107-115)

Supporting

Assessment

Glossary















Other















Appendix

Appendix 1: Properties Quick Check


B

H

FE

C D

A

G

Appendix 1: Properties Quick Check

Alternate Interior Angles have equal measures. They are: ___ and ___ ___ and ___

Alternate Exterior Angles have equal measures.

They are: ___and____ ____and___

Corresponding Angles have equal measures.

They are:___ and ___ ___ and ______ and ___ ___ and ___

Vertically Opposite angles are equal.

They are:___ and ___ ___ and ______ and ___ ___ and ___

Interior angles on the same side of the transversal are supplementary (add to180°).

They are: ___ and ___ ___ and ___ Exterior angles on the same side of the transversal are supplementary (add to180°).

They are: ___ and ___ ___ and ___


B

H

FE

C D

A

G

Properties Quick Check Key

Alternate Interior Angles have equal measures. They are: _C_ and _F_ _D_ and _E_

Alternate Exterior Angles have equal measures.

They are: _A_and_H__ __B_and_G_

Corresponding Angles have equal measures.

They are:_A_ and _E_ _C_ and _G__B_ and _F_ _D_ and _H_

Vertically Opposite angles are equal.

They are:_A_ and _D_ _B_ and _C__E_ and _H_ _F_ and _G_

Interior angles on the same side of the transversal are supplementary (add to180°).

They are: _C_ and _E_ _D_ and _F_ Exterior angles on the same side of the transversal are supplementary (add to180°).

They are: _A_ and _G_ _B_ and _H_


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