2.4 Congruent Triangles
Triangle Sums
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 degrees verses 180 degrees. 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 180 degrees, all of the interior angles should add to 180 degrees,
and the measures of the exterior angles total 360 degrees.
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.
Key Exercises:
- An acute triangle has two congruent angles each measuring 70 degrees. What is the measure of the third angle?
Answer: 180− 2 ∗ 70 =40 degrees
- An obtuse triangle has two congruent angles. One angle of the triangle measures 130 degrees. 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.
( 180 − 130 )÷ 2 =25 degrees
Congruent Figures
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
Chapter 2. Geometry TE - Common Errors