1.4: Triangles and Congruence
Pacing
Day 1 | Day 2 | Day 3 | Day 4 | Day 5 |
---|---|---|---|---|
Triangle Sums Investigation 4-1 |
Congruent Figures |
Quiz 1 Start Triangle Congruence using SSS and SAS |
Finish Triangle Congruence using SSS and SAS Investigation 4-2 Investigation 4-3 |
Triangle Congruence using ASA, AAS, and HL Investigation 4-4 |
Day 6 | Day 7 | Day 8 | Day 9 | Day 10 |
More Triangle Congruence using ASA, AAS, and HL | Quiz 2 |
Isosceles and Equilateral Triangles Investigation 4-5 Investigation 4-6 |
Quiz 3 | Review Chapter 4 |
Day 11 | Day 12 | |||
Finish Review of Chapter 4 | Chapter 4 Test |
Triangle Sums
Goal
First, this lesson reviews the types of triangles. The Triangle Sum Theorem will be introduced and proven followed by the Exterior Angle Theorem.
Notation Note
A new symbol, \begin{align*}\Delta\end{align*}, is introduced to label a triangle. The order of the vertices do not matter for a triangle (unlike when labeling an angle). Usually the vertices are written in alphabetical order.
Teaching Strategies
To review finding angle measures, give them the six triangles at the beginning of the section and have them use their protractors to measure all the angles. Then, discuss their results. Students should notice that all the angles add up to \begin{align*}180^\circ\end{align*}, all the angles in an equilateral triangle are \begin{align*}60^\circ\end{align*}, and two of the angles in an isosceles triangle are equal.
Investigation 4-1 is a version of a proof of the Triangle Sum Theorem. One approach to this investigation is to demonstrate for the students. You can do the activity on the overhead and have students discover the sum of all the angles. You may need to remind students that a straight angle is \begin{align*}180^\circ\end{align*}. You could also have the students perform this investigation in pairs. After completing this investigation, go over the traditional proof in the text. Ask students how the traditional proof is similar to the investigation. This will make the traditional proof easier to understand.
Guide students through Example 5 before showing them the answer. Once all the exterior angles are found, ask students to find their sum. This will lead into the Exterior Angle Sum Theorem. Students might need a little clarification with this theorem. Explain that each set of exterior angles add up to \begin{align*}360^\circ\end{align*}. This theorem will be addressed again in Chapter 6.
The Exterior Angle Theorem can be hard for students to remember. Present it like a shortcut. If students forget the shortcut, they can still use the Triangle Sum Theorem and the Linear Pair Postulate. See Examples 7 and 8.
Congruent Figures
Goal
The goal of this lesson is to prepare students for the five triangle congruency theorems and the definition of congruent triangles.
Notation Note
Revisit congruence notation from earlier lessons. This is the first time students will apply congruence to a shape. Remind them that figures are congruent and measurements are equal. So, two triangles can be congruent and the measurements of their sides would be equal. Stress the importance of labeling each congruency statement such that the congruent vertices match.
Stress the tic mark notation in relation to the congruency statement. Simply because the letters used are in alphabetical order does not necessarily mean they will line up this way in a congruency statement. Students must follow the tic marks around the figure when writing congruency statements.
Teaching Strategies
When writing congruence statements, have students put the first triangle’s vertices in alphabetical order. Then, match up the second triangle’s vertices so that the congruent angles are lined up. Remember that it is very common to use letters in alphabetical order, however they might not always line up so that the congruent triangles vertices will be in alphabetical order. For example, \begin{align*}\Delta ABC\end{align*} might not be congruent to \begin{align*}\Delta DEF\end{align*}, but it could be \begin{align*}\Delta ABC \cong \Delta FDE\end{align*}.
Rather than needing to know all three pairs of angles and sides are congruent, the Third Angle Theorem eliminates one set of angles. Now, students need to know that two sets of angles and three sets of sides are congruent to show that two triangles are congruent. Ask students if they think there are any other shortcuts to finding out if two triangles are congruent. Can they show that two triangles are congruent using 4 pieces of information? 3 pieces? This could be a discussion for the end of the lesson and lead into the next.
Prove Move
In this lesson, we introduce CPCTC (corresponding parts of congruent triangles are congruent). Even though this is not a theorem, it will be used in proving that parts of triangles are congruent. CPCTC can only be used after two triangles are stated and proven congruent in a proof.
The Reflexive Property of Congruence is commonly used in proofs to say that a shared side or angle is congruent to itself. We will discuss this more in the next section.
Triangle Congruence using SSS and SAS
Goal
This lesson introduces students to the formal concept of triangle congruency through the SSS and SAS Congruence Theorems.
Teaching Strategies
When introducing SSS Congruence Postulate let students do Investigation 4-2 individually. Walk through the classroom and assist students with the steps. Once they reach Step 5, ask if they can make another triangle with these three measurements. Every student should have a 3-4-5 right triangle and have them show each other their constructions. Have students rotate and flip their triangles, but demonstrate that they still have the same shape.
You can also use the Distance Formula to show that two triangles are congruent using SSS (Examples 5 and 6). In Example 5 the two triangles are congruent. Show students that they are in different places, flipped and rotated. Put this example on a transparency and cut out \begin{align*}\Delta ABC\end{align*}. Then, place it over \begin{align*}\Delta DEF\end{align*} so that they are lined up. This also shows that the two triangles are congruent.
To introduce the SAS Congruence Theorem, you can either let students do Investigation 4-3 individually or in pairs. Like with the previous investigation, as students to compare their triangles to the triangles drawn by other students in the class. Again, they will see that all the triangles have the same shape and are congruent.
The concept of an included angle can be confusing for some students. Draw triangles to show students the difference. See picture.
Reinforce that the angle must be between the two sides to be a valid congruence theorem. The way the letters are written, SAS, also should remind students that the angle is between the two sides. SSA (or ASS) implies that the angle is not between the two sides.
Students may ask if SSA is a valid congruence theorem; it is not. There is an explanation of this in the next section. Also, students will realize that SSA and ASS are the same thing. You can address this however you seem fit. Some teachers approach it straight on while others may choose to avoid it and refer to this combination as SSA only.
Prove Move
When using SSS and SAS in a proof, students must present each piece as a step. For SSS, there needs to be three steps, one for each set of congruent sides. For SAS, there needs to be two steps for the two sets of congruent sides and one step for the included angles. Then, students can list the congruence statement and reason.
The Reflexive Property of Congruence can be used in triangle proofs. If two triangles share a side or an angle, the Reflexive Property is the reason this piece is congruent to itself. Students might feel as though this is an unnecessary step, but just remind them that they must right all three sets of congruent sides/angles in order to state that two triangles are congruent.
Triangle Congruence using ASA, AAS, and HL
Goal
Students will learn the ASA, AAS and HL Congruence Theorems and how to complete proofs using all five of the congruence theorems.
Teaching Strategies
Investigation 4-4 should be done individually and then students can compare their triangles with the students around them. Like with Investigations 4-2 and 4-3, students should realize that no other triangle can be drawn. Rotation and reflection do not change the shape of the triangle (Chapter 12).
ASA and AAS can be hard for students to distinguish between. Draw the two theorems side by side and compare. See picture.
Here we introduce the concept of an included side. The definition of an included side is very similar to that of an included angle. Ask students to compare the differences and similarities.
The proof of the AAS Congruence Theorem may help students better understand the difference between it and ASA. Explain that because of the Third Angle Theorem, AAS is also a congruence theorem. Example 3 shows which sides or angles are needed to show that the same two triangles are congruent using SAS, ASA, and AAS. Go over this example thoroughly with students.
Hypotenuse-Leg (HL) is the only congruence theorem that is triangle-specific. This theorem can only be used with right triangles. So, in order to use this congruence theorem in a proof, the student must know that the two triangles are right triangles (or be able to show it in the proof).
AAA and SSA are introduced at the end of the section as the other possible side-angle relationships that we have yet to explore. Neither of these relationships can prove that two triangles are congruent, but it is useful to show students why they do not work. AAA shows that two triangles are similar, as in Chapter 7 and SSA can actually create two different triangles.
Have students copy the Recap Chart into their notes. This chart will be a very helpful study guide for the chapter test.
Example 7 is the only example that touches on CPCTC, even though there are proofs that use it in the homework. Explain to students that they can only use CPCTC after they have proven two triangles are congruent.
This is a very challenging lesson for students. If you feel as though not everyone is grasping the concept of proofs or all the different triangle congruence theorems, slow down and go back over this lesson.
Additional Example: Put the reasons to the proof below in the correct order.
Solution: The correct order is B, D, C, G, F, A, E
Isosceles and Equilateral Triangles
Goal
This lesson illustrates the special properties that arise from isosceles and equilateral triangles.
Teaching Strategies
Investigation 4-5 can be done individually or teacher-led. As a teacher-led activity, this investigation should be done as a group discovery. You should ask questions of the students to keep them engaged while you are performing the construction (on the overhead or whiteboard). Then use a protractor to measure the angles. Or, you could have a student come up and measure the angles. Ask them to generalize this construction into the Base Angles Theorem.
This investigation also leads into the Isosceles Triangle Theorem. Have students duplicate \begin{align*}\Delta DEF\end{align*} (just before Example 1) in their notes. They should write down all markings and all corresponding congruence statements (for angles and sides) and any perpendicular statements. Stress that this theorem is only true at the vertex angle.
Investigation 4-6 can also be done individually or teacher-led. This investigation allows students to come to their own conclusion about equilateral triangles. They should discover that an equilateral triangle is also an equiangular triangle in Step 4.
Additional Example: Algebra Connection Solve for \begin{align*}x\end{align*} and \begin{align*}y\end{align*}.
Solution: It does not matter which variable you solve for first.
\begin{align*}(6y - 7)^\circ & = 65^\circ && (4x + 2)^\circ + 65^\circ + 65^\circ = 180^\circ\\ 6y & = 72^\circ && \qquad \qquad \quad 4x + 132^\circ = 180^\circ\\ y & = 12^\circ && \qquad \qquad \qquad \qquad \ 4x = 48^\circ\\ & && \qquad \qquad \qquad \qquad \quad x = 12^\circ\end{align*}