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)
Activity 6: Euclid for the Win!
Activity 6: Euclid for the Win!
1a. Draw a triangle that is about half the size of your paper.
1b. Extend all the sides beyond the triangle.
2. Choose a base for your triangle that has the opposite vertex across from it. Draw a line through the vertex at A that is parallel to segment BC.
3. Identify an angle in the triangle that is congruent to the angles at 1, 2, and 3 respectively.
The angle at 1 is congruent to the angle at C.
The angle at 2 is congruent to the angle at A.
The angle at 3 is congruent to the angle at B.
Reflection
Reflection
The combined angles at 1, 2, and 3 form a straight angle. Since angles 1, 2, and 3 are congruent to angles C, A, and B respectively, the sum of interior angles in the triangle is also 180 degrees.
For students who have been introduced to the parallel postulate, they can use the following reason to explain the above conclusions regarding congruence:
Since the line l is parallel to the line through B and C, the corresponding angles at C and 1 formed by the transversal through A and C are congruent.
The angles at A and 2 are vertical angles so they are congruent. Proof that vertical angles are congruent was given in Activity 3.
Since the line l is parallel to the line through B and C, the corresponding angles at B and 3 formed by the transversal through A and B are congruent.
This reasoning above constitutes a mathematical proof. Since we know that for every triangle we can always draw a line through a vertex opposite a base that is parallel to a base, the parallel postulate, along with vertical angles, allows us to deductively reason that the sum of interior angles for every triangle is 180 degrees.
Euclid gives proof that the sum of interior angles in a triangle is 180 degrees in Proposition 32 of The Elements. The formulation and proof is slightly different than the setup we explored in our activity. Euclid's statement of Proposition 32 is
In any triangle, if one of the sides is produced, then the exterior angle equals the sum of the two interior and opposite angles, and the sum of the three interior angles of the triangle equals two right angles.
This statement corresponds to the picture on the right and on the cover of the activity flipbook.
In Euclid's proof, one side of a triangle is extended and the exterior angle it forms is split in two by a line parallel to the base. In the image, we see that the two green angles are congruent because they are alternate interior angles. The blue and orange angle are congruent because they are corresponding angles. Since the three angles (red, green, and orange) on the extended side form a straight angle, we can conclude the sum of measures of the interior angles in the triangle is 180 degrees.
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