Interior vs. Exterior Angles – Students frequently have trouble keeping interior and exterior angles straight. They may fail to identify to which category a specific angle belongs and include an exterior angle in a sum with two interior. They also sometimes use the wrong total, verses . Encourage the students to draw the figure on their papers and color code it. They can highlight or use a specific color of pencil to label all the exterior angle measures and another color for the interior angle measures. Then it is easy to do some checks on their work. Each interior/exterior pair should have a sum of , all of the interior angles should add to , and the measures of the exterior angles total .
Find All the Angles You Can – When a student is asked to find a specific angle in a complex figure and they do not immediately see how they can do it, they can become stuck, and don’t know how to proceed. A good strategy is to find any angle they can, even if it is not the one they are after. Finding other angles keeps their brains active and working, they practice using angle relationships, and the new information will often help they find the target angle. Many exercises are not designed to do in one step. It is important that the students know this.
Congruent Angles in a Triangle – In later sections students will study different ways of determining if two or more angles in a triangle are congruent, and will then have to use this information to find missing angles in a triangle. To start them on this process it is good to have them work with triangles in which two angles are stated to be congruent.
1. An acute triangle has two congruent angles each measuring . What is the measure of the third angle?
2. An obtuse triangle has two congruent angles. One angle of the triangle measures . What are the measures of the other two angles?
Answer: The two remaining angles must be congruent since a triangle can not have more than one obtuse angle.
Rotation Difficulties – When congruent triangles are shown with different orientations, many students find it difficult to rotate the figures in their head to align corresponding sides and angles. One recommendation is to redraw the figures on paper so that they have the same orientation. It may be necessary for students to physically rotate the paper at first. After students have had some time to practice this skill, most will be able to skip this step.
Stress the Definition – The definition of congruent triangles requires six congruencies, three pairs of angles and three pairs of sides. If students understand what a large requirement this is, they will be more motivated to develop the congruence shortcuts in subsequent lessons.
The Language of Math – Many students fail to see that math is a language, a form of communication, which is extremely dense. Just a few symbols hold great amounts of information. The congruence statements for example, not only tell the reader which triangles are congruent, but which parts of the triangle correspond. When put in terms of communication students have an easier time understanding why they must put the corresponding vertices in the same order when writing the congruence statement.
Third Angle Theorem by Proof – In the text an example is given to demonstrate the Third Angle Theorem, this is inductive reasoning. A deeper understanding of the theorem, and different types of reasoning, can be gained by using deductive reasoning to write a proof. It will also reinforce the idea that theorems must be proved, and shows how inductive and deductive reasoning work together.
Key Exercise: Prove the Third Angle Theorem.
Answer: Refer to the figures on the top of page 213, where the example of the Third Angle Theorem is given.
Triangle Sum Theorem
Triangle Sum Theorem
Substitution Property of Equality
Subtraction Property of Equality
Substitution Property of Equality
Triangle Congruence Using SSS
One Triangle or Two – In previous chapters, students learned to classify a single triangle by its sides. Now students are comparing two triangles by looking for corresponding pairs of congruent sides. Evaluating the same triangle in both of these ways helps the students remember the difference, and is a good way to review previous material. For instance, students could be asked to draw a pair of isosceles triangles that are not congruent, and a pair of scalene triangles that can be shown to be congruent with the SSS postulate.
Correct Congruence Statements – Determining which vertices of congruent triangles correspond is more difficult when no congruent angles are marked. Once the students have determined that the triangles are in fact congruent using the SSS Congruence postulate, it is advisable for them to mark congruent angles before writing the congruence statement. Corresponding congruent angles are found by matching up side markings. The angle made by the sides marked with one and two tick marks corresponds to the angle made by the corresponding sides in the other triangle, and so on.
Translation Rotation – Translating a triangle on a coordinate plane in order to see if it fits exactly over another triangle is a good way to demonstrate that two triangles are congruent. The notation used to describe these translations can sometimes be confusing. The text writes out the movement in words “ is to the right and below ”. If students use other materials for reference, they may see this same translation as . This could be confused with the point located at . It may be helpful to alert students to this difference.
1. Use the congruence statement and given information to find the indicated measurements.
Find and .
Answer: and .
Triangle Congruence Using ASA and AAS
An Important Distinction – At first students may not see why it is important to identify whether ASA or AAS is the correct tool to use for a specific set of triangles. They both lead to congruent triangles, right? Yes, but this will not always be the case, as they will see in the next lesson. Sometimes the configuration of the corresponding congruent sides and angles in the triangles determines if the triangles can be proved to be congruent or not. Knowing this will motivate students to study the difference between ASA and AAS.
Flowchart Proofs – Flowchart proofs do a much better job of showing implication than two-column proofs. In a two-column proof one statement following another does not necessarily mean that the previous statement implies the next. Sometimes all the given information is listed at the beginning or another parallel argument needs to be developed before the implication is made. This can be confusing for students without much experience with proofs, or who have trouble understanding the argument. In a flowchart proof the implications are clearly indicated with arrows, and when parallel arguments are being developed, they are arranged vertically. The flowchart holds much more information.
Different Folks – People think and learn in different ways. When teaching, it is best to provide a few different explanations and have a variety of ways to present content. Some students, the linear thinkers, will understand two-column proof perfectly, and others, the special thinkers, will find flowchart proofs clearer. It is best to use both so that all students understand and develop their reasoning skills. One option to introduce the flowchart format is to have the students go back to key two-column proofs provided in the text and convert them to flowchart proofs.
Patterns and Structure – All of the shortcuts to triangle congruence require three pieces of information, therefore the box of the flowchart proof that states that two triangles are congruent will have three boxes leading into it. These kinds of structural relationships help students write and understand flowchart proofs and should be noted. It is also helpful to give students incomplete flowchart proofs and have them fill in the missing information. Subsequent proofs can be given with less information provided each time, until the boxes are all empty, and then with no help at all. The only problem is that sometimes there is more than one way to write a proof and a different chart may be required for the proof that the student wants to write. In that case, students can start from scratch if they like.
Proof Using SAS and HL
AAA – Students sometimes have to think for a bit to realize that AAA does not prove triangle congruence. Ask them to think back to the definition of triangle. Congruent triangles have the same size and shape. Most students intuitively see that AAA guarantees that the triangles will have the same shape. To see that triangles can have AAA and be different sizes ask them to consider a triangle they are familiar with, the equiangular triangle. They can draw an equiangular triangle on their paper, and you can draw an equiangular triangle on the board. The triangles have AAA, but are definitely different in size. This is a counterexample to AAA congruence. Have the students note that the triangles are the same shape; this relationship is called similar and will be studied in later chapters.
SSA – Student will have a hard time seeing the two possible triangles with SSA. The best way to describe it when the congruent parts are set up, is to tell them to take that the last congruent side can bend in, so that the third side is short to make one triangle, and bent out, so that the third side is long, to make the other triangle. Some students will see it right away and others will really have to play around with their triangles for awhile in order to understand.
Why Not LL? – Some students may wonder why there is not a LL shortcut for the congruence of right triangles. It also leads to SSS when the Pythagorean theorem is applied. Have the students explore the situation with a drawing. They can draw out two congruent right triangles and mark sides so that the triangles have LL. There is already a congruence guarantee for this, SAS. What would the non-right triangle congruence be for HL? Is this a guarantee? (It would be SSA, and no, this does not work in triangles that are right.)
Importance of Right Triangles – When using math to model situations that occur in the world around us the right triangle is used frequently. Have the students think of right angles that they see every day: walls with the ceilings and the floors, widows, desks, and many more constructed objects. Right triangles are also important in trigonometry which they will be studying soon. Stressing the usefulness of right triangles will motivate them to think about why HL guarantees triangle congruence but SSA, in general, does not.
Using Congruent Triangles
The Process – When students first start examining pairs of triangles to determine congruence it is difficult for them to sort out all the sides and angles.
The first step is for them to copy the figure onto their paper. It is helpful to color code the sides and angles, congruent sides marked in one color and the congruent angles in another. Some congruent parts will not be marked in the original figure that is given to the students in the text. For example, there could be an overlapping side that is congruent to itself, due to the reflexive property; mark it as well. Then they should do a final check to ensure that the congruent parts do correspond.
The next step is for them to count how many pairs of congruent corresponding sides and how many pairs of congruent corresponding angles there are. With this information they can eliminate some possibilities from the list of way to prove triangles congruent. If there is no right angle they can eliminate HL, or if they only have one set of corresponding congruent angles, they can eliminate both ASA and AAS.
If at this point there is still more than one possibility, they are going to need to decide if an angle is between two sides or if a side is between two angles. Remind them that both ASA and AAS can be used to guarantee triangle congruence, and that SAS works, but that SSA can not be used to prove two triangles are congruent.
If all postulates and theorems have been eliminated, then it is not possible to determine if the triangles are congruent.
AAS or SAA – Sometimes students try to list the congruent sides and angles in a circle as they move around the triangle. This could result in AAS or SAA when there are two pairs of congruent angles and one pair of congruent sides that is not between the angles. They know AAS proves congruence and want to know if SAA does as well. When this occurs it is best to redirect their thinking process. With two sets of angles and one set of sides there are only two possibilities, the side is between the angles or it is another side. When it is between the angles we have ASA, if it is either of the other two sides we use SAA. This same situation occurs with SSA, but is even more important since SSA is not a test for congruence. A good way for the students to remember this is that when the order of SSA is reversed it makes an inappropriate word. This word should not be used in class or in proofs, even if it is spelled backwards.
Isosceles and Equilateral Triangles
The Useful Definition of Congruent Triangles – The arguments used in the proof of the Base Angle Theorem apply what the students have learned about triangles and congruent figures in this chapter, and what they learned about reasoning and implication in the second chapter. It is a lot of information to bring together and students may need to review before they can fully understand the proof.
They have been practicing with proofs throughout the chapter, so they should be adept with the logic at this point. If they are having trouble, a review of the Deductive Reasoning section in Chapter Two: Reasoning and Proof will help. It could be assigned as reading the night before the current lesson will be done in class.
This is a good point to summarize what the students have learned in this chapter about congruent triangles and demonstrate how it can be put to use. To understand this proof, students need to remember that the definition of congruent triangles requires three pairs of congruent sides and three pairs of congruent angles, but realize that not all six pieces of information need to be verified before it is certain that the triangles are congruent. There are shortcuts. The proof of the Base Angle Theorem uses one of these shortcuts and jumps to congruence which implies that the base angles, a pair of corresponding angles of congruent triangles, are congruent.
To a student new to geometry this argument is not as straightforward as it may seem to an instructor experienced in mathematical proofs. Plan to take some time explaining this important proof.
A Proved Theorem Can Be Used – Now that the students have the proof of the Base Angle Theorem they can use it as opportunities present themselves. They should be on the lookout for isosceles triangles in the proofs of other theorems, in complex figures, and in all other situations. When they spot them, they need to immediately apply the Base Angle Theorem and mark those base angles congruent. This is true for the converse as well. When they spot a triangle with congruent angles, they should mark the appropriate sides congruent. Students sometimes do not realize what a powerful tool this theorem is and that they will be using it extensively throughout this class, and in math classes they will take in the future.
1. What are the measures of the angles of an equilateral triangle? What postulates or theorems did you use to obtain your answer?
Answer: , Base Angle Theorem, Triangle Sum Theorem
Reflection or Rotation – When looking a two triangle, where one is a transformation of the other, students sometimes have trouble distinguishing between a reflection and a rotation. This is particularly true when the triangles are almost equilateral. When demonstrating these transformations, it is best to use an obviously scalene triangle. Good use of labels is also helpful. The prime notation clearly indicates the new location of each vertex under the transformation. A rotation preserves the order of the vertices, and a reflection reverses the order of the vertices. If the students are unsure of what transformation has been applied to the figure, have them choose one vertex and then move counterclockwise around the polygon listing off the vertices as they occur. If they start with the image of that first vertex in the new figure, and again move counterclockwise, they will get the images of the vertices in the same order for a rotation, and in reverse order for a reflection.
Don’t Just Memorize, Reason – This section contains many ordered pair rules for different transformation. Students will try to memorize them without really thinking about them or looking for patterns. This is challenging, if not impossible for most students. Have the students discuss similarities and differences between the rules. Ask them if the rule surprises them, or seems logical. Why? If they really get stuck, they can do a test. Graph a scalene triangle and apply different rules to it until the desired transformation occurs. Students will be motivated to use reason to shorten the guess and check process. In geometry problems are written so that students will have to think about them for awhile, and figure out an answer. Once students realize that they are not supposed to know the answer immediately, they are much more willing to spend time thinking about an exercise.
1. Graph a scalene triangle in the first quadrant of the coordinate axis. Reflect the triangle over the axis. Take this new triangle and reflect it over the axis. What single transformation would take the first triangle to the final triangle? How can this be predicted by the ordered pair rules?
Answer: A single rotation of about the origin would result in the final triangle. The reflection over the axis takes the opposite of the coordinate and the reflection over the axis takes the opposite of the coordinate. If opposite of both coordinates are taken, the result is a rotation of about the origin.