Working with Similar Triangles, Part 2

This time is used for the students to refresh their memories of the work completed the day before on Triangle Diagrams. I ask the students to look over and discuss in their group their work from the previous lesson, so that we can begin to discuss the conclusions they have reached with the entire class.

Now, the discussion phase of this lesson starts! Beginning with the first problem in Triangle Diagrams, I choose a number from 1 to 4, and ask each student in the groups with this number to briefly explain his/her group’s findings on that problem. As students offer explanations I highlight different approaches, should they occur, and bring up for discussion among the students any uncertainty or potential misconceptions in students’ reasoning. 

I then move on to the next diagram, choosing another number from 1 to 4, asking students with this number to present, and continue in this fashion, through item 8.  With regard to these problems, here are some preliminary expectations based on my experiences with the lessons:

(See Discussion Notes Narrative Full for a slightly more detailed presentation.)

1. This problem is pretty straight forward, applies both an isosceles triangle theorem, parallel lines, and the sum of the interior angles of a triangle, and the triangles are clearly similar. 
2. In this problem, very similar to #1 in appearance, the triangles are clearly not similar and it applies reasoning much like that in #1.
3. This is an important diagram, as it leads to one of the theorems referenced specifically in the Common Core:  A line parallel to one side of a triangle divides the other two proportionally.  It is important that students become comfortable with the application of this theorem in problems involving proportionality.  I have found that it really helps to ask the students to use two different colored pencils to outline the two triangles.  This enables them to better visualize the triangles and to see exactly which segments constitute each triangle.  This problem also lends itself to revisiting the different sets of congruent angles formed by a pair of parallel lines. 
4. This problem is the first in which similarity cannot be determined.  I would ask the students what information could be given to result in similar triangles, hoping to lead a discussion of two cases. In this problem, it may be helpful for students to redraw the diagram. Colored pencils are helpful here.
5. In this problem, the triangles are clearly similar, and this problem can be used to again emphasize that only 2 sets of congruent angles are necessary.  Additionally, this problem recalls supplementary angles and sets of angles formed by parallel lines (e.g., alternate interior angles).
6. There are a lot of things going on in this diagram: an angle bisector, congruent alternate interior angles, and one isosceles triangle.  Again, I would ask the students to determine what information would be needed to guarantee similarity.  In this case, since the larger triangle is isosceles, the smaller triangle would need to be proven isosceles also.  Students need to be careful of making assumptions about this diagram. I think it’s important to stress that alternate interior angles are not always congruent – that they are only congruent when the lines cut by the transversal are parallel.  It is also worth noting that it is possible for the triangle to be isosceles in two different ways, because the diagram is not drawn to scale.  Again, I would encourage the students to sketch the two possibilities.
7. This is a pretty straight-forward diagram for similar triangles.  However, students can prove similarity using two sets of congruent alternate interior angles OR using one set of congruent alternate interior angles and the pair of vertical angles.  This is another opportunity to drive home that only two sets of congruent angles are required and that there is not always just one way to solve problems.
8. This problem is challenging in that there are three triangles in the diagram. The discussion of this problem is always interesting, as students want to claim similarity with the third triangle, but cannot. This is another problem in which colored pencils are a key tool.

Similar to the first part of this lesson, delivered the day before, I ask the students to return their attention to the board on which they have listed ways of proving angles congruent when working with similar triangles.  I ask several questions to summarize the investigation and followup discussion:

• Are there any more additions that we should make to our list of ways to prove angles congruent in similar triangles? 
• What is always true about the angles of similar triangles? 
• How many angles are required to be known in order to prove that two triangles are similar? 
• With respect to the sides of similar triangles, what is known? (If the students do not have a comprehensive explanation I will allude to the fact that we will be working on finding the lengths of sides in tomorrow's lesson.) 

As the students leave the class, I hand out Similar Triangles Homework Proof.  This proof recalls the work done on similar triangle proofs previously, and in particular applies the work that students did in this lesson.