Proving that Triangles are Similar
Lesson 4 of 9
Objective: Students will learn to do similar triangle proofs using the AA Similarity Postulate.
I begin by asking the students to draw on their graph paper two triangles:
- A right triangle with legs of 4 units and 5 units
- A triangle with angles of 30 degrees and 90 degrees
When all have completed this, I ask them to compare their triangles with their neighbor's.
- What do you observe about the triangles with legs 4 units and 5 units?
- What do you observe about the triangles with angles 30 degrees and 90 degrees?
Hopefully, the students will respond that their right triangles with legs 4 and 5 are congruent. We will generally spend a couple of minutes in discussion of the question: Why are all of these right triangles congruent?
Then, we will focus on the triangles with angles of 30 degrees and 90 degrees. I'll ask, "Are all of the triangles congruent in this case?" This can get interesting because there will have been some congruent triangles as well as some that are similar, but not congruent. After all of the students begin to realize that not all of the triangles are congruent, I will ask, " If they are not congruent, then what can we say about the triangles that were created in this case?"
Hopefully, the students will remember their recent work with similar polygons and they will respond that everyone’s triangles are similar. Either way, I will follow this with a discussion of the Angle-Angle Similarity Postulate, including a discussion of why just two angles are required and not three.
To close out this activity, I will directing the students back to their graph paper. Then, I will ask them to draw two more triangles:
- A right triangle with legs 3 and 4
- A right triangle with legs 6 and 8
Since my students are expected to know the Pythagorean Theorem at this point, I will also ask them to find and label the lengths of the hypotenuse of each triangle. Once the students complete their drawings, I will ask, "What is true about these two triangles? What can we say about the relationship between the sides of these triangles?" My hope is that it will not be a big leap for my students to conclude that the corresponding sides of these similar triangles are all in the same ratio.
In this section we will work more formally with the idea of similar figures. I begin by handing outthe worksheet entitled Introduction to Similar Triangle Proofs. We will briefly discuss the rules written at the top on the page. The Cross-Product Property was used by the students in previous lessons, and should need little discussion. Someone usually asks at this point why, in the second and third statements that we look at, line segment symbols are not used and why there are equal signs, rather than congruent symbols. These are great questions, and provide an opportune moment to revisit the subtle differences in a segment's name and its measure.
I then ask the students to work on the first proof, filling in the missing statements and reasons. As always, I remind them to mark their diagrams using colored pencil as they write in their ‘givens’ and find congruent angles or segments! This really helps students to see the information that they are working with, and is a huge aid in naming the triangles correctly. When the majority of the students have completed the proof, we discuss the students’ arguments, and continue this same process with each of the remaining proofs.
The last proof asks the students to prove a product. I have found that the step in which they must set up a proportion is particularly difficult for students. I encourage them to work backward from the products that they are asked to prove equivalent - make sure that the first two factors will multiply together in their proportion (one in the numerator of the first fraction, the other in the denominator of the second fraction, and then the last two factors can go in either of the remaining places). This seems to help.
Ticket Out the Door
A proof, with no hints included in it, is handed out. If there is more than a minute or two, I may decide to give students the proof as a Ticket out the Door. I'd prefer this, since it will leave me with a good sense of student understanding of the day's lesson. If time is limited, I will give the students the proof for homework and I will collect it upon their arrival the next day. In this case, I remind the students to refer back to the day’s worksheet on similar triangle proofs as they do the homework.
The students are also working on their Create Your Own Similar Figures Projects for homework.