CCSS-M Congruence G-CO 10
Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.
I’m not sure where the idea of inequalities in triangles has fallen with CCSS-M. Maybe it is one of those topics that hasn’t surfaced in our efforts to teach concepts more deeply. Do students need to know that they shortest side is opposite the smallest angle? It is at least helpful to know when solving triangles and evaluating whether solutions are reasonable. So we did a brief exploration with dynamic geometry software and a chart to ensure students know that the smallest angle is opposite the shortest side.
We didn’t prove the Triangle Inequality Theorem – students have some exposure to it in middle school. But most haven’t thought about why the sum of two sides of the triangle must be larger than the third. So we used dynamic geometry software to show why some side lengths can make a triangle and others can’t and gave student a visual for the “collapsing triangle” that happens when the sum of two side lengths equal the 3rd side.
And a brief making sense of what values the 3rd side of the triangle can have when two of the side lengths are fixed. (In the diagram, AB=6 cm, and AB=5 cm.)
My students know that the sum of the measures of the angles of a triangle sum to 180°, but they have not thought about why.
So how could we prove that the sum of the measures of the angles of a triangle sum to 180°?
What do you know is true in the diagram?
The measures of the angles sum to 180°. Yes – we know that because our teachers told us. But we haven’t proven it yet. If that is what our goal is, how can we get there?
Student struggled for a couple of minutes before a hint. So I am gathering that you are not seeing much in the diagram as it is given that we can use in our proof. Last time, we talked specifically about the practice of look for and make use of structure. You drew auxiliary lines to solve some problems. Are there any auxiliary lines that you can draw for this diagram that might be helpful in proving The Triangle Sum Theorem?
I set the timer again. This time for 3 minutes. And I walked around to look at the auxiliary lines that students were adding to their diagrams. When the timer went off, we shared some of the diagrams with the whole class. Could any of these be helpful in our proof?
Of course it turns out that drawing in a line parallel to a base of the triangle is helpful. But my students figured that out. I didn’t have to tell them. And even though they all settled on the diagram in the top right for their proof, they didn’t all use exactly the same steps or angles for their argument as to why the sum of the measures of the angles of a triangle is 180°.
During the next few minutes, I heard cries of success from different groups in the class. They had proven The Triangle Sum Theorem using our deductive system. They were proud of themselves. They understand why the sum of the measures of the angles of a triangle is 180° – not just that it is.
We had a short conversation about the implications of our proof. On what does our proof rely? Ultimately, it relies on our Corresponding Angles Postulate, our version of Euclid’s 5th Postulate. Without it, our geometry would be different. Without it, we could be developing spherical geometry or elliptical geometry or hyperbolic geometry. Without it, the sum of the measures of the angles of the triangle could be more than 180° or less than 180°.
We ended the lesson with a Quick Poll.
Which most students got. How did you get it? I found the sum of the angles and subtracted from 180. Then I subtracted that result from 180. How long did it take you to get that?
What do you notice in the diagram? Write down your observations.
m∠1 + m∠2 + m∠3 = 180
m∠3 + m∠4 = 180
As we began to list student observations, the students recognized that the exterior angle is equal to the sum of the two remote interior angles. Of course they didn’t use those words exactly. But they concluded that m∠1 + m∠2 = m∠4.
Is this a waste of our time? I don’t think it is. The students are beginning to realize that we have choices in our deductive system, and that those choices affect what is true. The students are beginning to develop arguments as to why one result has to be true and why another result cannot be true.
And so the journey continues …