# Engaging students: Proving that two triangles are congruent using SAS

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Phuong Trinh. Her topic, from Geometry: proving that two triangles are congruent using SAS.

How does this topic extend what your students should have learned in previous courses?

Before learning how to prove that two triangles are congruent, the students learned about parts of a triangle, congruent segments, congruent angles, angle bisectors, midpoints, perpendicular bisectors, etc. These are some of the tools, if not all, that will aid them in proving two triangles are congruent. The basis of proving two triangles are congruent using SAS is to be able to identify the congruent sides and the congruent angles. That is where their knowledge of congruent segments and angles provide them the information they need. On other hands, not all problems of proving two triangles are congruent are straightforward with all the sides and angles needed are given to us. For example: Given a quadrilateral ABCD as shown below with AB = AD, and AC is the angle bisector of angle BAD. Prove that triangle ABC is congruent to triangle ADC. In this example, the problem did not clearly state what the congruent angles are. However, since the students have learned about what the angle bisector does to an angle, they can easily identify the congruent angles in this problem. Therefore, in order to successfully approach an exercise of proving two triangles are congruent using SAS, the students must first learn and understand the basics, which are parts of a triangle, angle bisectors, midpoints, etc.

How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic?

There are many resources that provide great aid to students in learning about proving two triangles are congruent using SAS. One of them is from ck12.org. The layout of the website is simple and straightforward. The site provides readings and color coded study guide to help the students understand the material of the lesson, such as definitions and properties of congruent triangles. It also provides videos that work out and explain example problems. The videos could potentially be a great resource and aid for students that are visual and/or auditory learners.  On other hands, the site also gives other practices and activities that help the student estimate how well they understand the material. It is a great resource for not only the students but also the teacher. Under the activity tab, the teacher can find student submitted questions. These questions can be brought up in class for discussion to help the students further understand the topic. Besides materials on SAS triangle congruence, the site also has materials on other cases of triangle congruence. Hence, ck12.org can be used as an aid for students to prepare for the lesson, and/or review on the materials of the lesson.

https://www.ck12.org/geometry/sas-triangle-congruence/

How could you as a teacher create an activity or project that involves your topic?

A three-part activity:

Part 1: At the beginning of the class, I will give the students some cut-out triangles and ask them to find the congruent pair. During this part, the students can easily find the pair by putting the triangles on top of each other to compare the shape and sizes. This is to introduce the students to congruence triangles.

Part 2: The next part will be after I introduce proving triangle congruence by SAS. I will give the students a guide sheet with congruence triangle pairs placed at random places, with side lengths and angles provided. Just like at the beginning, the students must match up the pairs. However, since this time the students cannot move the triangles around, they must utilize the clues provided to them, which are the side lengths and angles, to get the correct answers. Example: Match the congruent pairs by SAS.

Part 3: The last part will be before the end of the lesson. The students will be given a figure and asked to prove the congruent triangles using SAS. However, one of the components necessary for SAS is missing and the students will need to use other provided information to solve the problem. Example:

Given a quadrilateral ABCD as shown below with AB = AD, and AC is the angle bisector of angle BAD. Prove that triangle ABC is congruent to triangle ADC.

Reference:

CK-12 Foundation. CK-12 Foundation, CK-12 Foundation, www.ck12.org/geometry/sas-triangle-congruence/.

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