1. Let’s see why the angles in a triangle add to 180 degrees.
1-16: Learning Goals
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Let’s see why the angles in a triangle add to 180 degrees.

2. 1-16-1: Computational Relationships
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1-16-1: Computational Relationships
Is each equation true or false?
62−28=60−30
3⋅-8=(2⋅-8)−8
Launch - 5 minutes
Display one problem at a time. Tell students to give a signal when they have an answer and a strategy. After each problem, give students 1 minute of quiet think time and follow with a whole-class discussion.
Anticipated Misconceptions
In the first question, students may think you can round or adjust numbers in a subtraction problem in the same way as in addition problems. For example, when adding 62+28, taking 2 from the 62 and adding it to the 28 does not change the sum. However, using that same strategy when subtracting, the distance between the numbers on the number line changes and the difference does not remain the same.
Activity Synthesis
Ask students to share their strategies for each problem. Record and display their explanations for all to see. Ask students how they decided upon a strategy. To involve more students in the conversation, consider asking:
Do you agree or disagree? Why?
Who can restate ___’s reasoning in a different way?
Does anyone want to add on to _____’s reasoning?
After each true equation, ask students if they could rely on the reasoning used on the given problem to think about or solve other problems that are similar in type. After each false equation, ask students how we could make the equation true.

3. 1-16-2: Angle Plus Two
Launch - digital access for students - 15 minutes
Arrange students in groups of 2–3. Provide access to geometry toolkits. Give 5 minutes work time building the diagram and measuring angles. Then allow for a short whole-class check-in about angle measurement error, and then provide time for students to complete the task.
Important note for classes using the digital activity: The applet measures angles in the standard direction, counterclockwise. Students need to select points in order. For example, to measure angle ACB in this triangle, students would select the angle measure tool and then click on A, then C, and then B. If they click on B, then C, and then A, they get the reflex angle.
Anticipated Misconceptions
Some students may have trouble with the rotations. If they struggle, remind them of similar work they did in a previous lesson. Help them with the first rotation, and allow them to do the second rotation on their own.
Activity Synthesis
Ask students how the grid lines helped to show that the sum of the angles in this triangle is 180 degrees. Important ideas to bring out include:
A 180 rotation of a line BC (with center the midpoint of AB or the midpoint of AC) is parallel to line BC and lays upon the horizontal grid line that point A is on.
The grid lines are parallel so the rotated angles lie on the (same) grid line.
Consider asking students, "Is it always true that the sum of the angles in a triangle is 180˚?" (Make sure students understand that the argument here does not apply to most triangles, since it relies heavily on the fact that BC lies on a grid line which means we know the images of BC, AE and DA, also lie on grid line that point A is on.)
It turns out that the key to showing the more general result lies in studying the rotations that were used to generate the three triangle picture. This investigation is the subject of the next activity.

4. 1-16-2: Angle Plus Two
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Download for free at openupresources.org.
Rotate triangle ABC 180˚ around the midpoint of side AC. Label the new vertex D.
Rotate triangle ABC 180˚ around the midpoint of side AB. Label the new vertex E.
Look at angles EAB, BAC, and CAD. Without measuring, write what you think is the sum of the measures of these angles. Explain or show your reasoning.
Is the measure of angle EAB equal to the measure of any angle in triangle ABC? If so, which one? If not, how do you know?
Is the measure of angle CAD equal to the measure of any angle in triangle ABC? If so, which one? If not, how do you know?
What is the sum of the measures of angles ABC, BAC, and ACB?
Launch - digital access for students - 15 minutes
Arrange students in groups of 2–3. Provide access to geometry toolkits. Give 5 minutes work time building the diagram and measuring angles. Then allow for a short whole-class check-in about angle measurement error, and then provide time for students to complete the task.
Important note for classes using the digital activity: The applet measures angles in the standard direction, counterclockwise. Students need to select points in order. For example, to measure angle ACB in this triangle, students would select the angle measure tool and then click on A, then C, and then B. If they click on B, then C, and then A, they get the reflex angle.
Anticipated Misconceptions
Some students may have trouble with the rotations. If they struggle, remind them of similar work they did in a previous lesson. Help them with the first rotation, and allow them to do the second rotation on their own.
Activity Synthesis
Ask students how the grid lines helped to show that the sum of the angles in this triangle is 180 degrees. Important ideas to bring out include:
A 180 rotation of a line BC (with center the midpoint of AB or the midpoint of AC) is parallel to line BC and lays upon the horizontal grid line that point A is on.
The grid lines are parallel so the rotated angles lie on the (same) grid line.
Consider asking students, "Is it always true that the sum of the angles in a triangle is 180˚?" (Make sure students understand that the argument here does not apply to most triangles, since it relies heavily on the fact that BC lies on a grid line which means we know the images of BC, AE and DA, also lie on grid line that point A is on.)
It turns out that the key to showing the more general result lies in studying the rotations that were used to generate the three triangle picture. This investigation is the subject of the next activity.

5. 1-16-2: Angle Plus Two
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Launch - digital access for students - 15 minutes - solution!!
Arrange students in groups of 2–3. Provide access to geometry toolkits. Give 5 minutes work time building the diagram and measuring angles. Then allow for a short whole-class check-in about angle measurement error, and then provide time for students to complete the task.
Important note for classes using the digital activity: The applet measures angles in the standard direction, counterclockwise. Students need to select points in order. For example, to measure angle ACB in this triangle, students would select the angle measure tool and then click on A, then C, and then B. If they click on B, then C, and then A, they get the reflex angle.
Anticipated Misconceptions
Some students may have trouble with the rotations. If they struggle, remind them of similar work they did in a previous lesson. Help them with the first rotation, and allow them to do the second rotation on their own.
Activity Synthesis
Ask students how the grid lines helped to show that the sum of the angles in this triangle is 180 degrees. Important ideas to bring out include:
A 180 rotation of a line BC (with center the midpoint of AB or the midpoint of AC) is parallel to line BC and lays upon the horizontal grid line that point A is on.
The grid lines are parallel so the rotated angles lie on the (same) grid line.
Consider asking students, "Is it always true that the sum of the angles in a triangle is 180˚?" (Make sure students understand that the argument here does not apply to most triangles, since it relies heavily on the fact that BC lies on a grid line which means we know the images of BC, AE and DA, also lie on grid line that point A is on.)
It turns out that the key to showing the more general result lies in studying the rotations that were used to generate the three triangle picture. This investigation is the subject of the next activity.

6. 1-16-3: Every Triangle in the World
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1-16-3: Every Triangle in the World
Launch - slide is animated - 15 minutes
Keep students in the same groups. Tell students they’ll be working on this activity without the geometry toolkit.
In case students have not seen this notation before, explain that m∠DBA is shorthand for "the measure of angle DBA."
Begin with 5 minutes of quiet work time. Give groups time to compare their arguments, then have a whole-class discussion.
Anticipated Misconceptions
Some students may say that a,b, and c are the three angles in a triangle, so they add up to 180. Make sure that these students understand that the goal of this activity is to explain why this must be true. Encourage them to use their answer to the first question and think about what they know about different angles in the diagram.
For the last question students may not understand why their work in the previous question only shows a+b+c=180 for one particular triangle. Consider drawing a different triangle (without the parallel line to one of the bases), labeling the three angle measures a,b,c, and asking the student why a+b+c=180 for this triangle.
Activity Synthesis
Ask students how this activity differs from the previous one, where △ABChad a horizontal side lying on a grid line. Emphasize that:
This argument applies to any triangle ABC.
The prior argument relies on having grid lines and having the base of the triangle lie on a grid line.
This argument relies heavily on having the parallel line to AC through B drawn, something we can always add to a triangle.
The key inspiration in this activity is putting in the line DE through B parallel to AC. Once this line is drawn, previous results about parallel lines cut by a transversal allow us to see why the sum of the angles in a triangle is 180∘. Tell students that the line DE is often called an ‘auxiliary construction’ because we are trying to show something about △ABC and this line happens to be very helpful in achieving that goal. It often takes experience and creativity to hit upon the right auxiliary construction when trying to prove things in mathematics.
Here is △ABC. Line DE is parallel to line AC.
What is m∠DBA+b+m∠CBE? Explain how you know.
Use your answer to explain why a+b+c=180.
Explain why your argument will work for any triangle: that is, explain why the sum of the angle measures in any triangle is 180˚.

7. 1-16-4: Four Triangles Revisited
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This diagram shows a square BDFH that has been made by images of triangle ABC under rigid transformations.
Given that angle BAC measures 53 degrees, find as many other angle measures as you can.
Launch - 5 minutes - optional activity
Provide 3 minutes of work time followed by a whole-class discussion.
Activity Synthesis
Display the image of the 4 triangles for all to see. Invite students to share how they calculated one of the other unknown angles in the image, adding to the image until all the unknown angles are filled in.
In wrapping up, note that angles ACE, CEG, EGA, and GAC are all right angles. It’s not necessary to show or tell students that ACEG is a square yet. This diagram will be used in establishing the Pythagorean Theorem later in the course, so it’s a good place to end.

8. 1-16: Lesson Synthesis
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Revisit the basic steps in the proof that shows that the sum of the angles in a triangle is 180˚. Consider asking a student to make a triangle that you can display for all to see and add onto it showing each step.
We had a triangle, and a line through one vertex parallel to the opposite side.
We knew that the three angles with their vertices on the line summed to 180˚.
We knew that two of these angles were congruent to corresponding angles in the triangle, and the third one was inside the triangle.
Therefore, the three angles in the triangle must also sum to 180˚.
Tell students that this is one of the most useful results in geometry and they will get to use it again and again in the future.

9. 1-16: Learning Targets
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I can explain using pictures why the sum of the angles in any triangle is 180 degrees.

10. 1-16-5: Angles Sizes
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In an equilateral triangle, all side lengths are equal and all angle measures are equal. Sketch an equilateral triangle. What are the measures of its angles?
In an isosceles triangle, which is not equilateral, two side lengths are equal and two angle measures are equal. Sketch three different isosceles triangles.
List two different possibilities for the angle measures of an isosceles triangle.
Cool-down - 5 minutes