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, 360∘, verses 180∘. 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 adjacent (linear) pair should have a sum of 180∘, all of the interior angles should add to 180∘, and the measures of the exterior angles total 360∘.
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 not see how to proceed. A good strategy is to find any angle they can and write the measures in the diagram, 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. Students often miss possible connections when they don’t write down all of the angles as they find them. It is important that students know that many exercises are not designed to do in one step.
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. A few example problems follow:
Example 1: An acute triangle has two congruent angles each measuring 70∘. What is the measure of the third angle?
Encourage students to make a triangle diagram and label two angles 70∘. This helps review making and marking diagrams and gives them a concrete visual to work with. Next, students should set up an equation showing that the two given angles plus the third, unknown angle, will add up to 180∘. Finally, students should solve and find that the third angle is 40∘.
Example 2: An obtuse triangle has two congruent angles. One angle of the triangle measures 130∘. What are the measures of the other two angles?
Again, drawing a diagram will help students organize their thoughts. They will recognize that the obtuse angle cannot be one of the congruent angles. They should arrive at the conclusion that each of the congruent angles should be 25∘.
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. Remind them that the angles and sides marked congruent are the corresponding pairs of angles and sides. Have students practice listing the corresponding pairs of congruent sides and angles. Another 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. Use review question 23 as a template for a proof of the Third Angle Theorem.
Triangle Congruence Using SSS and SAS
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.
Congruent Segments - This is a good time to remind students that overlapping (shared) segments will be congruent. Also, remind students that when they are given a midpoint, they can mark the two halves of the segment congruent. Practice this with them and illustrate marking the diagram appropriately.
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.
The Included Angle - Students often have a hard time differentiating between SAS and SSA. It helps to have students practice identifying the angle included between two particular sides in a triangle. This should be practiced using a diagram and using the name of the triangle. This will help students identify which triangle congruence theorem is being used are help them identify correct pairs of corresponding triangle parts.
Example: What is the included angle between sides AB¯¯¯¯¯¯¯¯ and BC¯¯¯¯¯¯¯¯ in ΔABC?
Answer: The included angle is ∠B. Note that it is the point that is an endpoint in both segments.
Tricky Problems - When looking to see which triangle congruence theorem is being used, students are liable to only look at one of the triangles and make a mistake as shown in the example below. This is problem #3 from the review questions for this section in the textbook.
If a student were to only look at ΔABC, he or she might say that these triangles are congruent by SAS. The second triangle, though it does have a pair of sides and an angle congruent to a pair of sides and an angle in the first triangle, does not have the angle included between the sides.
SSA - Students are apt to try to use this as a triangle congruence theorem. It is helpful to have them attempt to construct two different triangles with the same two side lengths and non-included angle. This will only be possible in certain cases (think back to the ambiguous case for the Law of Sines- this is the connection). Here are two side lengths and a non-included angle for which two distinct triangles can be formed. ΔABC with AB¯¯¯¯¯¯¯¯=12 cm, BC¯¯¯¯¯¯¯¯=5 cm and m∠A=35∘.
Student diagrams should look something like this:
This diagram is scaled down but should reflect the general “shape” of student work.
Using the Distance Formula to Prove Triangles Congruent - It may be necessary to review the distance formula with students again and remind them that they cannot “distribute” the square root. In other words, remind them that:
Triangle Congruence Using ASA, AAS 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.
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 not 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.
Congruent Angle/Segment Pairs - It is important to review what pairs of angles will be congruent in diagrams. Students may forget that they can mark vertical angles and shared (overlapping) angles congruent. They may also need a reminder that if segments are parallel, then alternate interior angles will be congruent. Remind them as well that shared sides can be marked congruent.
Marking the Diagram - Once again, students should be encouraged to mark all given information and all deduced congruencies in their diagrams. Seeing the markings will help them determine which triangle congruence theorem is being utilized.
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? Sometimes either can be used to prove triangles are congruent, 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. This is a good time to practice identifying the “included” side between two angles. This will help students see when a side is included and when it is not.
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.
Patterns and Structure - All of the shortcuts to triangle congruence require three pieces of information. Reinforce this concept with students are you complete proofs. Students often wonder why we need to include statements such as BC¯¯¯¯¯¯¯¯≅BC¯¯¯¯¯¯¯¯ when it is so obvious to them that it is the same segment and that it has to be congruent to itself. It is important to the structure of the proof that we include exactly which segment and/or angle pairs we are using in order to conclude the triangles are congruent by a particular triangle congruence theorem.
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.
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.
The Process of Writing a Proof - 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 cannot 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.