interior and exterior angles of a triangle lesson planwebpages.charter.net/druby/papers/triangle sum...

Interior and Exterior Angles of a Triangle Lesson Plan

By: Douglas A. Ruby Date: 10/10/2002 Class: Geometry Grades: 9/10

INSTRUCTIONAL OBJECTIVES:

This is a two part lesson designed to be given over a several day time span. At the end of the first part of the lesson, the student will:

1. Be able to use Geometer’s Sketchpad Version 4.0x (GSP), its tool bar and menus to: a. construct and label points, segments, midpoints, parallel lines, angles, triangles,

and polygons b. measure the magnitude of angles in GSP constructions c. Use the calculator tool to add and subtract measurements as necessary d. shade the interior of triangles e. reflect, rotate, and translate a triangular construction

2. Measure and explore the relationships of the interior angles of a triangle. 3. Discover and use the “Triangle Sum”, i.e. the sum of the measures of the interior angles

of any arbitrary triangle. 4. Explore the relationship between adjacent and opposite angles created by a transversal of

parallel lines. 5. Be able to discover that the relationship between the triangle sum and the measures of

adjacent angles create by a transversal of parallel lines are similar (i.e. that both are 180o).

At the end of the second part of the lesson, the student will:

1. Have increased fluency with Geometer’s Sketch Pad Version 4.0.x by repeated use of similar tools and constructions as in Part 1.

2. Measure and explore the relationships of an exterior angle of a triangle with its remote interior angles.

3. Discover that the measure of an exterior angle is equal to the sum of the measures of its two remote interior angles.

4. Write a formal geometric two column proof that shows that:

Given: ABC with exterior angle BCD (as shown below) Prove: m BCD = m ABC + m CAB

Exploring Interior and Exterior angles of a Triangle with Geometer’s Sketchpad – Mr. Ruby

A

B

C D


Relevant Massachusetts Curriculum Framework

10.G.1 Identify figures using properties of sides, angles, and diagonals. Identify the figures’

type(s) of symmetry. 10.G.2 Draw congruent and similar figures using a compass, straightedge, protractor, and other

tools such as computer software. Make conjectures about methods of construction. Justify the conjectures by logical arguments.

10.G.3 Recognize and solve problems involving angles formed by transversals of coplanar lines. Identify and determine the measure of central and inscribed angles and their associated minor and major arcs. Recognize and solve problems associated with radii, chords, and arcs within or on the same circle.

10.G.4 Apply congruence and similarity correspondences (e.g., ABC

XYZ) and properties of the figures to find missing parts of geometric figures, and provide logical justification.

10.G.5 Solve simple triangle problems using the triangle angle sum property and/or the Pythagorean Theorem.

10.G.6 Using rectangular coordinates, calculate midpoints of segments, slopes of lines and segments, and distances between two points, and apply the results to the solutions of problems.

10.G.7 Draw the results, and interpret transformations on figures in the coordinate plane, e.g., translations, reflections, rotations, scale factors, and the results of successive transformations. Apply transformations to the solutions of problems.

INTENDED AUDIENCE

This lesson is targeted at a 9/10th grade honors geometry audience. These students already know how to use Windows 98/Me/XP based computers and are fluent with use of keyboard, mouse movement, left and right mouse click, and normal conventions and terms such as <esc>, <tab>, double-click, etc. Students have already taken Algebra (either in Middle school or high school) and have already been introduced in prior classes to the notions and conventions of formal proof, logic, and the basics of Theorems, postulates, definitions, and construction.


CLASS ACTIVITIES

Basics of using Geometer’s Sketchpad.

We are going to start this class by using the Geometer’s Sketchpad (GSP) version 4.0.x. It appears on your Windows desktop as the following icon:

If there are other earlier versions of GSP (i.e.version 3.x) on your machine, do not use these. Make sure you are using the 4.0.x version of GSP. Double-click the GSP 4.0.3 icon to start GSP. You should now see a startup screen that looks like this

Note that the version number of GSP appears in its “welcome banner” above. In this case we are using GSP 4.0.3, though any variety of GSP 4.0 is fine for this lesson.


In order to use Geometer’s Sketchpad (hereafter referred to as GSP) in this lesson, you will need to understand two major conventions. These are the use of the GSP menus and the GSP Tool Bar.

GSP Menus

At the top of the GSP window are the GSP Menus. There are 9 menus, in the GSP Menu Bar. These are File, Edit, Display, Construct, Transform, Measure, Graph, Window, and Help. The Menu bar looks like this:

Whenever you are asked to select a menu function, we will write this in italics separated by the “>” character. So for instance, if you need to save a GSP construction to a file, you will be asked to save it using the File>Save As menu. These are normal Windows™ “pull down” menus. So in our example of File>Save As, you would use the mouse pointer to select the File menu, hold left-click, and pull down to the Save As submenu item as you see in the following picture:

When you release the left click, you will have selected that function.

The Menu items will normally be used to operate on items that we have selected in our Geometer’s Sketchpad window. The File, Window, and Help items behave much like any Windows™ program. The other six menu items are unique to GSP. In this first lesson, we will use Display, Construct, Transform, and Measure.

GSP Tool Bar:

The second major aspect of GSP is the GSP Tool Bar. Normally this tool bar appears along the left edge of the window as pictured above. However, someone may have hidden the tool bar from you. If so, select the Display>Show Toolbar menu to cause it to reappear. If the Tool Bar is


floating in the middle of the screen, just select it by moving your mouse pointer over the GSP Tool Bar and holding left click. Now move it back over to the left side of the GSP window. It will “reattach” itself so that it appears like the picture above.

This is a picture of the GSP Tool Bar and its 6 tools:

Selection Tool

Point Tool

Circle tool

Segment Tool

Text Tool

Custom Tool

We will use the names above to refer to each tool. If you are asked to use a particular tool, use the mouse arrow and a single left click to select that tool.

Part 1A – Investigating the Triangle Sum

1. Use the File>New Sketch menu to create a new GSP window.

Let’s start by opening GSP 4.0.x and getting the startup window as described above. Put your floppy disk in the floppy drive of your computer. Now put your mouse pointer over the File> menu. Hit left click, pull down to the File>New Sketch and release the mouse button. This will clear any prior sketch that might have been in place. Now you will go back and pull down (using left click) the File>Save As menu. A pop-up window will appear that looks something like this:


Put your mouse pointer over the symbol and left click. This will move you up in the directory structure. Continue to do this until your pop-up window looks like this:

Now double-click on the “My Computer” icon and then on the “3 ½ Floppy A:” icon. You now should see a pop-up window that looks like:

Save your workspace to your floppy disk as A:>trianglesum.gsp by changing the file name from untitled.gsp to trianglesum.gsp. We have now created a named Geometer Sketchpad workspace. Anytime you want to save your work during this session, just use the File>Save pull-down menu and your current workspace will be saved as A:>trianglesum.gsp.


2. Start by using the Point Tool to create three points

Next we will use the point tool to create three points in our workspace. By default, GSP starts with the selection tool highlighted. You can see this above in the discussion regarding the various GSP tools. Put the mouse pointer over the GSP Point Tool and hit left click once. This

tool should now look like this: When your mouse is in the GSP workspace on the screen, you will be using the Point Tool. To create three points, just hit left click in three locations. You should see this:

Please note: From here on out, we will only show geometric figure rather than the whole GSP window.

If you end up with more than three points, move your mouse over to the GSP Selection Tool and click once. Then move the Selection tool to the excess points, select them by hitting left click once, and hit the <del> key on your keyboard to remove them.


3. Label the points A, B and C using the text tool

We want to label the points as A, B, and C using the Text tool. Select the Text Tool

just as

you selected the Point Tool or Selection Tool above. Move the Text Tool to the left-most point and hit left-click once. You should see an “A” appear. Go to the other two points and left-click each one. The letters B and C should also appear. When you have completed this, it will look like this:

4. Construct a figure using the Construct>Segment menu

We will now create a geometric figure. To do this, highlight the GSP Selection Tool. Now highlight each of the three points, A, B, and C. Make sure that all three points have a bright red spot over them like this:


Now pull down the Construct>Segments menu. This should look like:

Release the left click button once the Segments sub-menu has been highlighted.

What happens when we do this?: _Three segments connect the three points_______________.

____________________________________________________________________________

What kind of figure is created? ___A triangle________________________________________

And how do we name this figure? __ ABC__________________________________________

5. Use Measure>Angle menu to measure the angle outlined in the prior step.

Your triangle should now look like this:

To “deselect” the highlighted lines and points, just click with the GSP Selection Tool anywhere in the white “free space” outside of the triangle. Now sse the selection tool to highlight the three


vertices of ABC in any particular order. Pull down the Measure>Angle menu and release the mouse pointer.

What happened? ___We see that the measure of the angle is displayed in the workspace__

Try highlighting the three vertices in a different order and pulling down the Measure>Angle menu. How does changing the order in which you select the three points (vertices) of the triangle affect the measurement? __It changes the angle. The 2nd point we select becomes the vertex of the angle.

What happens if one of the line-segments is highlighted as well as three of the vertices when you do the Measure>Angle? _The Measure>Angle menu does not work.______________________

Do you think you can now measure all three angles of your ABC? If so, highlight the angles you measured previously (They might look like the picture to the right) and delete them by hitting the <del> key on your keyboard. Make sure you do not delete any other part of your construction. If you accidentally deleted part of your triangle, use the Edit>Undo pull down menu to back up a step or two.

6. Use Measure>Angle menu to measure ABC, BAC, and BCA of the triangle.

Once you have cleared out the prior measurements, you need to create measurements for the three angles of your triangle in the following order:

m ABC = __varies_________ m BAC = __varies_________ m BCA = __varies_________

7. Use Measure>Calculate to create a triangle sum

of the three measurements.

GSP has a built-in calculators that allows us to work with measurements or other numbers related to our constructions. Make sure you have the GSP Selection Tool selected. Now use the Measure>Calculate pull down menu. You should see a pop-up window that looks like this:

Using the selection tool, highlight the measure of m ABC . What happens in the calculator window? _that angle

appears in the window____


Now hit the “+” key on the calculator. What do you see? ___I see an addition or “+” operation

after the m ABC _______________________________________________________________

If you have successively highlighted each of the three angle measures and used the addition “+” key on the calculator, your pop-up window should look something like what you see to the left. When you hit “OK” at the bottom of the calculator window, what happens? _The sum of the three angles appears in the workspace.___

__________________________________

You should now see this in your workspace:

Based on what we have just done, what can we conjecture about the sum of the interior angles of a triangle? _that they will

Always be 180o______________________

What do we call that sum? _Triangle __

Sum_______________________________ ___________________________________

8. Change the shape of your triangle

Now, making sure that the GSP Selection Tool is highlighted, I want you to deselect all of your construction and its measurements by clicking in the white “free space” outside of the triangle and away from the measurement and calculations. Once you have done that, “grab” one of the vertices of the triangle by highlighting it with the selection tool and holding the left-click button down.

If you now move the mouse, what happens? _The triangle changes shape__________________

____________________________________________________________________________

Try changing the shape of your triangle from any of the three vertices (points A, B, and C). What happens to the measurements of the angles? _The measurements change__________________

What happens to the sum of the three angles? __It stays at 180o__________________________


We are nearing completion of Part 1A of this lesson. When you are done moving your triangle around, please try to make it look something like this in preparation for Part 1B:

Note: the angles do not have to be exact, but point B should be at the top and segment AC should be nearly horizontal

9. Completion of Part 1A of Lesson Plan

Please save your work by using the File>Save pull down menu. You should hear your floppy drive make a little noise while it is saving the file. You can now exit Geometer’s Sketch Pad if you choose. Once you have saved your work I want you to think about what you have just done and write a summary of current observations in the space provided below. You should be able to answer some of the following questions. Feel free to make up your own questions to answer:

How is “constructing” a triangle with GSP different from just drawing a triangle using a paint program with three lines? Did I do something that proved my triangle was a “construction” rather than a drawing?

What else did I learn about using GSP?

What is the most important conclusion about triangles from this lesson so far?

If I learned something about triangles, do I know enough to prove it using geometric Statement/Reason formal proof?

Answers will vary. We are looking for the student to observe the following:

Constructing a triangle is different than drawing one since when we construct the triangle, it is always a triangle. We can change its shape, biut its still retains the properties of a triangle. If I just drew three lines, moving one of them would cause the figure to no longer be a triangle.

We learned how to use the Select Tool, Text Tool, and the Construct and Measure menus

So far we learned that a triangle “has” 180o. Or that the sum of the measures its interior angles is 180o.

Student may believe that this is a proof; however we have not proven this. We have “demonstrated” empirically that it appears that triangle has 180o, but this is inductive rather than deductive.

Do not turn the page until you have completed your summary!


Part 1B – “Evaluating” the Triangle Sum

Note: You have now learned how to use the GSP Menus and GSP Tools. In part 1B, we do not specify steps in as much detail as we did in part 1A. If you have problems using GSP in this exercise, please see the teacher or ask one of your fellow students how to perform a specific task.

1. Starting where we left off

If you are in GSP 4.0.x and have your A:>trianglesum.gsp file loaded as your current work space, then you are ready to proceed. If not, start GSP 4.0.x and use the File>Open menu to load this file.

2. Create a parallel line through vertex B

We now are going to create a line parallel to segment AC through vertex B of ABC. To do this, use the selection tool to select vertex B of the triangle and its opposite side (segment AC). Now use the menu Construct>Parallel line to create a line through B parallel to AC. It should look like:

3. Create midpoints in segments AB and CB adjacent to point B

Deselect the line above by clicking the Selection tool in white space. Select segment AB. Use the Construct>Midpoint menu to create a mid-point. Do the same for of the other side CB adjacent to vertex B.

4. Construct an interior to the triangle

Deselect anything else in your triangle and select the three vertices using the Selection Tool. Then use the Construct>Triangle Interior menu to shade the interior of the triangle. This should now look like this:


5. Transforming the triangle by rotation.

We are now going to transform the triangle interior we created in Part 1B, Step 4 above by rotating it around the midpoints of AB and BC. To do this, select the midpoint of segment BC using the Selection Tool Mark that midpoint as the center of rotation using the Transform>Mark segment BC using the Selection Tool Mark that midpoint as the center of rotation using the Transform>Mark Center menu. You should see a little “bullseye” pattern appear momentarily on

the midpoint. Now we want you to select the interior of the triangle using the Selection Tool. When the interior is selected, you will see a “cross-hatch pattern like the picture the left. When the interior has not been selected, the cross hatch will

disappear look like the diagram to the right.

Now, use the Transform>Rotate menu to rotate the triangle 45o around the previously marked midpoint. You should see a pop-up window that allows you to change the rotation angle. It will look like this:

Put your cursor in the degrees window and change the number to 45o. When you do, you will see exactly what is above. Without clicking the Rotate button, change the number to 90o. What just happened? _The lightly shaded triangle appeared to rotate to the right by another 45o._____

Now change the number of degrees to 135o. Try also some negative numbers. What happened to the little “ghost triangle” that appears to be behind your original triangle? _It turned to the right

by 135o or 90o more than before.

_________________________________________________

Finally, change the value in the degree window to 180o. Now press the Rotate button in the Transform>Rotate window. Where is the second triangle located now? _It is aligned with the

parallel line that goes through vertex B of ABC____________________________________ ______________________________________________________________________________


What did we just do? Explain this in your own words? __We rotated an image

of ABC around

the midpoint of segment BC by 180 degrees this made a second triangle that is lined up with the parallel line through point B._____________________________________________________

Give the new triangle a different color. Select it and use the Display>Color menu to change its color from yellow to another color. Now mark the midpoint of segment AB as the center of rotation and rotate the interior of triangle ABC by 180o. Change the color of the new triangle to a third color. Finally, using the Point Tool and the Text Tool, put points E and F on the line parallel to segment AB such that they are on either side of point B. You should now have a construction that looks like this:

Are the second and third triangle just created by rotation congruent with our original ABC? Why? __Yes they are congruent because they are just rotated images of the original triangle.__

Measure angles DBA, EBC, DBE using the Measure>Angle tool as you did in part 1A. What are their values?

m DBA = __varies_________ m EBC = __varies_________ m DBE = __180o_________

Based on this, which are the corresponding angles between the new triangles and the original triangle? _ We know that the three triangles are congruent so that DBA corresponds to BAC and EBC corresponds to ACB. DBE is a straight angle____________________________

Try “grabbing” one of the vertices (points A, B, or C) of the original triangle using the Selection Tool and moving it around. Change the shape of the triangle any way you want. What happens? _The corresponding relationships do not change. All three triangles are still congruent. ______

What does the Angle Addition Postulate tell us about the sum of angles? _It would say that if point B lies in the interior of DBE, then DBE = DBA + ABC + EBC______________

What do we know about alternate interior

created a transversal that cuts two || lines? ___They are congruent.__________________________________________________________________

6. Save your workspace to your floppy disk using the File>Save menu.

Do not proceed until this is done and you have shut GSP down.


Part 1C – Formal Proof of the sum of interior angles of a triangle

Using what you have discovered in Part 1A and Part 1B, we will proceed with a formal proof of the sum of the interior angles of a triangle. Please use the page provided below to proceed. Fill in the blank parts of the proof.

Theorem: The sum of the measures of the interior angles of a triangle is 180o.

Given: ABC Line DE with points D-B-E Line DE || segment AC

Prove: m ABC + m BAC + m BCA = 180o

Statement Reason 1. ABC Given 2. Line DE with points D-B-E Given 3. Line DE || segment AC Given 4. DBE is a straight angle Definition of straight angle 5. m DBE = 180o A straight angle has a measure of 180o

6. m DBA + m ABC + m CBE = m DBE Angle Addition Postulate 7. m DBA + m ABC + m CBE = 180o Substitution 8. DBA

BAC Alternate Interior angles are

9. m DBA m BAC Congruent angles have equal measure 10. CBE

BCA Alternate Interior angles are

11. m CBE

m BCA 12. m BAC + m ABC + m CBE = 180o Substitution 13. m BAC + m ABC + m BCA = 180o Substitution 14. m ABC + m BAC + m BCA = 180o Commutative Property of Addition

We have now completed Part 1C of our lesson and a formal proof of the Triangle Sum. We may use this for Part 2 of our lesson.


Part 2A – Exterior Angles in a Triangle

In part 1 of this lesson, we were able to create and examine the features of a triangle. In doing this, we learned how to use Geometer’s Sketchpad and several of its tools. More importantly, we learned that the sum of the measures of the interior angles of a triangle is 180o and that this is called the Triangle Sum Theorem. Despite the fact that we had numerous cases where we could show the truth of this Theorem in parts 1A and 1B, it was not until part 1C that we proved the Triangle Sum Theorem using formal Statement/Reason proof. Our proof took no measurements, but instead proved by deductive argument that the sum of the measures of the interior angles is equal to the measure of a straight angle (180o) of a line drawn through one of the vertices. We will now use that prior knowledge to examine the exterior angles of a triangle.

1. Create a new workspace and save it as A:exterior.gsp

We begin by starting Geometer’s Sketchpad and using File>Save As to save the blank workspace as A:exterior.gsp.

2. Create Triangle ABC

Create a triangle ABC using the Point Tool, Text Tool, and Construct>Segments menu as you did in part 1.

3. Construct Ray AC to extend side AC

Use the Selection tool. Select Point A (first) and then point C. Construct Ray AC by using the Construct>Ray menu. If you select the point C before point A, you will end up creating Ray CA rather than Ray AC. How are these different? __A ray is a set of points that begins at an

endpoint A and continue infinitely through another point B. The first point we select is the endpoint

4. Construct Point D, segment BD, and hide ray AC

Now construct point D on ray BC to the right of point C using the Point Tool and Text Tool. Once this point is created on the ray, highlight point B and point D. Use the Construct>Segment Nomenu to create a line segment (rather than a ray) to connect the two point. Finally, deselect everything by clicking in white space and select just the ray AC by clicking on it with the Selection Tool to the right of point D. Use Display>Hide Ray to make the ray disappear. When you are done, your construction should look like this:


5. Definition of Terms.

We now have ABC and BCD. Since segment AC (a side of the triangle) land segment CD both lie on ray AC, we call BCD an exterior angle. In fact, BCD is the exterior angle to the interior angle ACB of ABC. In addition, ABC and CAB are called the remote interior angles to exterior angle BCD. If we constructed rays along the other two sides of ABC, we could have created two more exterior angles. However, for today’s lesson, the current diagram will suffice.

6. Angle Measurements (Use the Measure>Calculate menu if necessary)

Now let’s measure a few angles:

m ABC = ___varies________ m CAB = ___varies________ m BCA = ___varies________ m BCD = ___varies________ What is m ABC + m CAB =__same as m BCD __

What do we call ABC and CAB relative to the exterior angle BCD? _remote interior

angles_______________________________________________________________________

What does this tell us about the measure of an exterior angle? _The measure of an exterior angle is equal to the sum of the measures of the remote interior angles. _________________________

What is the relationship between BCD and its adjacent interior angle BCA? _They are supplementary. That is the sum of their measures is 180o. ______________________________

7. Is this always true?

Try grabbing one of the three vertices of ABC using the Selection Tool. Holding down the left click button, drag that vertex around and change the shape of the triangle. What happens to m ABC + m CAB relative to m BCD? __Stays the same. They are still equal.___________________________________________


8. Conclusions

Suppose you were to write a conjecture or theorem regarding the relationship between an exterior angle of a triangle and its remote interior angles. Word this Theorem below: __Theorem 2.4.5 - The measure of the exterior angle of a triangle is equal to the sum of the measures of the two non-adjacent (remote) interior angles.

______________________ _____________________________________________________________________ _____________________________________________________________________

Do not go on until you have completed this page


Part 2C – Examining the Exterior Angle.

We have created an exterior angle and have been able to observe that the measure of its exterior angle is equal to the sum of the measures of the two remote interior angles. This is still a conjecture, however since we have no deductive proof. Formal geometric proof is not the product of observation (inductive method), but rather must be by deducing the theorem using prior knowledge in the form of definitions, postulates, and theorems that have already been proven. To be able to do this we need to proceed a little further in our examination of the exterior angle of a triangle.

1. Construct and Translate Triangle Interior

Start by opening the file you saved in Part 2A. Select the three vertices of your triangle and then create an interior to the triangle by using the Construct>Triangle Interior menu. Our triangle should look like the triangle to the left.

Now we want to select points A and C of our triangle and create a vector around which the triangle will be translated. To do this, deselect

the triangle interior and select first point A and then point C. Then use the Transform>Mark Vector menu. You should see an animation in which a dotted line appears to move from point A to point C. This is the vector that establishes the translation direction for a triangle that we will create.

Now select the triangle interior and choose the Transform>Translate menu. You should see a pop-up window that looks like this:

Making sure that the box has the Marked Translation Vector checked off, hit the Translate button.


What just happened? Describe it in your own words? _We “slid” a copy of the triangle along segement AC. This “translated” the triangle by the width of its base. _____________________

As we did before, select the interior of the second triangle and use the Display>Color to make it a color different from the yellow of ABC. Before we go any further, I want you to construct a line that goes through point C and is parallel to segment AB. You should be able to do this using what you learned in part 1. Your construction should now look like this:

2. Fill the other triangle interior.

Ok, it appears that we have some “white space” that could be filled with another image from ABC. If we look at the diagram we did previously, we can see that we could do this by rotating ABC 180o around the midpoint of its side BC. Do this. (Go back and look at how you did it in

part 1 if necessary). Make this another color. Further, I want you to draw another line parallel to side AC of the triangle that passes through point B. Your construction should now look like this:

What do we know about the three triangles above? Why? _They are congruent by construction. Our transformations did not change the triangles’ properties.___________________________

3. Create and label a point at the intersection the two parallel lines

Now using the Selection Tool, deselect everything in your construction. Then select the two parallel lines you just created. Use Construct>Intersection to create a point exactly at the intersection of those two lines. Give it the label E using the Text Tool.


4. Measure all of the rest of the angles

Your construction should now look something like this(note that your measures will be different):

Since we may have changed the angles by dragging the original triangle around, I want to make sure that you write down the following measures:

m ABC = ___varies________ m CAB = ___varies________ m BCA = ___varies________ m BCD = ___180o - m BCA

m BCE = ___ same as m ABC

m DCE = ___ same as m CAB

m ABC + m CAB =___ same as m BCD

What can we conclude about the two new angles we created? __They correspond to (or are congruent with) the two remote interior angles to the exterior angle BCD

________________

5. Is this always true?

As we have done before, we want to see if our construction always has angles whose measures behave according to our theorem. To do this, we grab one of the vertices of ABC and change its shape. Do this.

What happens to the relationship between m ABC and m BCE? __stays the same________ What happens to the relationship between m CAB and m DCE? __stays the same________ What happens to the measures represented by our conjecture from part 2A? _The sum of the remote interior angles is still the same as the measure of the exterior angle. ______________


Do not go on until you have completed this page


Part 2C – Formal Proof of the Exterior Angle Theorem

Using what you have discovered in Part 2A and Part 2B, we will proceed with a formal proof. Please fill in the statement of the Exterior Angle theorem that you wrote in step 8 of part 2A of this lesson. Then proceed with the rest of the proof by providing as many statements and proofs of those statements as necessary. You may use whatever was proven in Part 1 of this lesson.

Theorem: Theorem 2.4.5 - The measure of the exterior angle of a triangle is equal to the

sum of the measures of the two non-adjacent (remote) interior angles.

Given: ABC Line Segment AC with points A-C-D Line BE || segment AC Line CE || segment AB

Prove: m ABC + m CAB = m BCD

Statement Reason 1. ABC Given 2. Line Segment AC with points A-C-D Given 3. Line BE || segment AC Given (note by construction) 4. Line CE || segment AB Given (note by construction) 5. BCE

ABC Alternate Interior angles from 3 6. CAB

BAC Alternate Interior angles from 4 7. m BCE m ABC

’s have = measures 8. m CAB m BAC

’s have = measures 9. m ABC + m CAB + m BCA= 180 Triangle Sum Theorem 10. m ABC + m CAB = 180o - m BCA Subtraction property of identity 11. BCA + BCD = ACD Angle Addition Postulate 12. ACD is a straight angle Definition from 2 13. m ACD = 180o Measure of a straight angle is 180o

14. m BCA + m BCD = 180o Substitution from 11 and 13 15. m BCD = 180o - m BCA Subtraction property of identity 16. 180o - m BCA = m BCD Reflexive Property of equality 17. m ABC + m CAB= m BCD Substitution from 10 and 16

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