Embedded Files
Materials
Materials
Sum of Angles in a Triangle flipbook (one per group)
Paper and something to write with to draw a triangle (one per person or pair)
Straight edge to draw triangle (one per person)
Protractor to measure each angle in the triangle (one per person)
Activity 3: 'Round About the Triangle
Activity 3: 'Round About the Triangle
1. Draw a triangle as big as you can on a piece of paper BUT extend two sides at a single vertex beyond the triangle.
2. Place the base of your flag at the vertex of your triangle with the flag pointing along its side.
3. Rotate the flag through the angle.
4. Without rotating, slide the flag to the next vertex of the triangle.
5. Rotate the flag through the "outside" angle.
6. Without rotating, slide the flag to the next vertex of the triangle.
7. Rotate the flag through the angle.
8. Without rotating, slide the flag to the next (initial) vertex of the triangle.
Compare the direction of the flag at the starting and ending position.
Reflection
Reflection
In the 5th step (Step "C" in the flipbook) the flag rotates through an angle vertical to the interior angle in the triangle. Vertical angles are formed by the intersection of straight lines and are congruent and proven in Euclid's Proposition 15.
Students in early grades can probably be convinced that the angles simply look equal. As students get into the seventh and eighth grades they should be able to reason in the following manner:
The two angles above shaded in orange combine to form a straight angle.
The two angles above shaded in blue also combine to form a straight angle.
The shaded angles share the angle in the middle so if it is removed, the remaining part of the shaded angles must be the same.
When students have some background in algebra they can represent the quantities algebraically and by setting both the straight angles equal (sum of alpha and beta and sum of alpha and delta) and then subtracting the common angle (alpha) we can see the vertical angles (beta and delta) must be congruent.
Activity Adaptation
Activity Adaptation
Note: A fun adaptation of this activity is to create a large triangle with tape on a floor and have students walk the triangle using their arm to swipe out the angles in place of a flag.
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