Don’t you just love it when things come together? That’s how I feel about transformations and congruent triangle proofs. They were made for each other. I love to follow up our transformations unit with our triangles unit. In this unit, we start with congruence and proofs, and then lead into triangle properties. Keep reading to learn all about this triangles geometry interactive notebook unit.
Corresponding Parts
This lesson is largely a repeat of the transformations lesson on congruence. To start, we define corresponding parts and then start to analyze congruence statements. Using two triangles, we complete congruence statements to practice. For these examples, we prioritize the order of corresponding vertices in our congruence statements. However, when it comes time to do proofs, I de-emphasize the order. I will not take credit from students for identifying correct parts of a triangle, but with the vertices not in the correct order. Proofs are difficult enough for students without adding more rules. However, I like to explain to students that it’s important to know about how order effects congruence statements when they are given information. It can help decode some tricky questions.
Students grasp congruence statements quickly, so we move on to practice marking diagrams. These practice questions are mini-proofs. Students are given information about what is congruent, or sometimes they are given information about parts that bisect and midpoints. Then, students look for vertical angles and reflexive sides, while writing out congruence statements for everything that they find along with its reason. After this practice, I tell students that they are one step short of a completed proof!
Triangle Congruence
Our next lesson is on the postulates and theorems that prove triangle congruence. In the past, I tried a few different activities for students to discover these relationships. None of them were worth the effort. Students ended up confused, and we actually wasted at least two class periods.
Instead, we complete a foldable and practice identifying these relationships. Students learn to count on the number of sides and angles that are congruent to identify which postulate or theorem proves triangles congruent. But every year the biggest sticking point for students is the difference between Angle-Side-Angle and Angle-Angle-Side. Since there’s two angles and one side for both theorems, students struggle to differentiate between them. The trick that I teach my students for this is to determine if the angles and side are all on one side of the triangle. If they are, then it’s Angle-Side-Angle. If they are spread out, then it’s Angle-Angle-Side.
Congruent Triangles Geometry Givens Cheat Sheet
After the triangle congruence foldable, I have my students add in a cheat sheet for interpreting givens. Even after our practice on corresponding parts, students struggle to come up with the steps for their proofs. So, I gave students a cheat sheet that details what is congruent when they’re given bisected sides/angles, midpoints, parallel and perpendicular lines. Students frequently reference this cheat sheet as they complete proofs.
Congruent Triangle Proofs
Our congruent triangle proofs lesson is where we finally put everything together. By now, students have completed proofs before. First we did algebraic proofs as we reviewed solving equations, and again in our lines and angles unit. However, the triangles unit is where I really start to assess students on their proofs.
As we complete 3 proofs in our notes together as a class, we discuss the formats that students can choose for their proofs. For all previous proofs, we used the two-column format. Here, students are introduced to paragraph and flow-chart proofs. From this point on, I like to give students a choice in what format they use for their proofs. (Though by now, most students are comfortable with two-column proofs, so they stick with those.) After we finish our notes, we spend the next two days completing Proof Cards. Students each complete a packet of 8 proofs, and have hints placed on cards for when they get stuck. This practice activity has been a game-changer and fostered much more independence in completing proofs than any of the traditional practice methods. (For more information about proof cards, check out this post!)
Triangles in Geometry: CPCTC Proofs
Once we get the hang of congruent triangle proofs, we incorporate CPCTC proofs. This lesson is very straightforward. We complete 3 proofs together in class, and then we begin practicing with a new set of proof cards.
In the past, I’ve had students come up with ways to remember the letters of CPCTC. My students’ favorite is always “Country People Cut The Corn.” But my favorite is “Concentrated People Come To Conclusions” because that’s what proofs are all about.
Triangle Sum Theorem
In college, I remember my geometry professor would have a party when they got to this theorem. Everyone seems to know that the interior angles of a triangle are supplementary, but until it was proven in her course, students were not allowed to use it.
That’s how I feel when we reach this lesson: Time to party! We’re out of the woods with proofs – for now, and we can enjoy some nice, relaxing algebra again.
To begin our lesson, I show students a triangle I cut out of paper, with angles A, B, and C labeled. (I always use a corner of the paper for one of the angles, so I can just cut across the paper diagonally.) After tearing off each angle, I tape them to the board so that every vertex touches, and show how they form a straight line – 180 degrees. After our practice problems, we also do a two-column proof of the theorem. (See? They weren’t gone for very long.)
In this lesson, I combine the triangle sum theorem with the exterior angle theorem. I do a similar paper triangle tearing demonstration. With a line extended from the triangle, I show how the two non-adjacent angles fill the angle formed by the line and the triangle. Naturally, this entire lesson includes algebraic practice examples.
Triangle Inequality Theorem
For our triangle inequality theorem lesson, we start by filling out a table to discover which side relationships will form a triangle. In groups, students use a set of Exploragons (Amazon affiliate link) to test out the different possible side lengths set up from the table. Once the table is complete, students are able to deduct that any time the 2 shorter sides were greater than the third, larger side, a triangle was formed. After summarizing the theorem, students practice determining if triangles can be formed from 3 given side lengths, and a range of possible third side lengths given two sides.
Triangles Geometry: Side-Angle Relationship
When demonstrating the side-angle relationship of triangles, analyzing a triangle is the easiest route. Although, I also like to show how the classroom door and doorway create an angle. By metnally connecting the non-hinge end of the door to the non-hinge end of the doorway, a triangle could be created. When the door is open wide, that imaginary side is longer, and when the door is open narrowly, the imaginary side is shorter. This gives a dynamic visual to make a connection between an angle’s size and the size of the side opposite it.
Again, a diagram of a triangle also works just fine, especially when you use two very different examples. We start our notes with some examples where students identify the largest and smallest angles and sides in triangles to help deduce the relationship. Then, students summarize the theorem and complete practice examples. This lesson is very short, and I’m usually able to cover it the same day that I cover the triangle inequality theorem.
Isosceles & Equilateral Triangles Geometry
Prior to geometry, students typically have an understanding of isosceles and equilateral triangles. We start this lesson by defining isosceles, and analyzing the different parts of an isosceles triangle – including the legs and base angles. Also, we extend our understanding of the angle-side relationship to come up with the base angle theorem. Then, we complete practice examples. Next, we define equilateral triangles, and figure out the corollary to the base angles theorem. Last, we complete a proof for the base angles theorem.
Midsegments of Triangles
Our next topic is midsegments of triangles. We keep this lesson simple. We go over the definition of midsegment, the midsegment theorem, and then we complete some practice examples. For this lesson, the formula we derive includes a fraction – one half. To deal with the fraction, I explain to my students that they have the choice of distributing the fraction (when necessary). Or, that they could multiply both sides 2 to avoid the fraction.
Medians of Triangles
In this lesson, before we talk about the vocabulary and theorem, I do a little demonstration. I cut a triangle out of a piece of paper, and draw medians on it. As I hold the paper up to show my students, I ask them what they notice about the triangle. Usually, they say that it is divided into six triangles. I ask them what their thoughts are on the point where the medians intersect. Usually, they have blank stares until someone notices that all three lines intersect at the exact same point. Then I ask, “would you say that point is the middle of the triangle?” Then, students have an entertaining debate about where the middle of the triangle is. This is my favorite part: to their delight and amazement (okay, I’m exaggerating), I hold up a pencil, and balance the paper triangle by placing that point on the eraser.
The demonstration is just for fun, and to build interest. Once we define the vocabulary, and explain and analyze the theorem, we complete our practice examples.
Points of Concurrency
Total disclosure: I have a love-hate relationship with points of concurrency. I love the concepts, I think they are so cool. However, I’m not a fan of teaching them. There’s a lot there, and students often get confused. Furthermore, it’s not something that even gets representation on our state exam, so I don’t spend much time on this topic. This lesson can easily go before medians of triangles. It would be fun to have a debate about which point of concurrency best indicates the middle of a triangle.
When I teach this lesson, I like to focus on the vocabulary. So, I teach this lesson with a foldable. It has four doors – one each for the incenter, circumcenter, centroid, and orthocenter. Behind the door, each term is defined, analyzed for where it falls in relation to the triangle, and the theorem is explained with a diagram. Since this isn’t my favorite lesson to teach, I usually save it for the very end of the triangles geometry unit.
Once upon a time, these lessons made up 2 separate units. It seems to be a serious change of pace from proofs to triangle sum theorem. However, when they were separate, I did not like assessing just triangles proofs (and the limited spiral review possible). Students didn’t like it either. Combining it all into one unit does make it longer, but the unit assessment is much more balanced and fair.
Getting ready to teach triangles in geometry? You can save time and get both a paper and digital version of my complete geometry interactive notebook triangles unit. Whether you are teaching online, or in person, I got you covered!
