What they will sometimes do is include both of these angles when using the Exterior Angle Sum theorem. Reinforce
that the number of exterior angles is the same as the number of interior angles and sides, one at each vertex.
Interior or Exterior –Interior and exterior angles come in linear pairs. If one of these angles is known at a particular
vertex, it is simple to find the other. When finding missing angles in a polygon, students need to decide from the
beginning if they are going to use the interior or exterior sum. Most likely, if the majority of the known angle
measures are from interior angles they will use the interior sum. They need to convert the exterior angle measures to
interior angle measures before including them in the sum. If there are more exterior angle measures given, they can
convert the interior angle measures and use the sum of 360 degrees. It is important that they make a clear choice.
They may mix the two types of angles in one summation if they are not careful.
Do a Double Check –Students often do not take the time to think about their answers. Going over the arithmetic
and logic is one way to check work, but it is common to not recognize the error the second time either. A better
strategy is to use other relationships to do the checking. In this lesson if the exterior sum was used, the work can be
checked with the interior sum.
What’s the Interior Sum of a Nonagon Again? –If students do not remember the interior sum for a specific
polygon, and do not remember the formula, they can always convert to the exterior angle measures using the linear
pair relationship. The sum of the exterior angles is always 360 degrees. This strategy will work just as well as using
the interior sum. Remind the students to be creative. When taking a test, they may not know an answer directly, but
many times they can figure out the answer in an alternative way.
Sketchpad Alternative –The activities designed for students to explore interior angles in the pervious section can
be easily adapted for exterior angles, and be used with this section. Using Sketchpad to extend the sides of the
polygon helps students gain an understanding of where the exterior angle is in relation to the polygon.
Classifying Quadrilaterals
Tree Diagram –Most students will need practice working with the classification of quadrilaterals before they
completely understand and remember all of the relationships. The Venn diagram is an important mathematical
tool and should definitely be used to display the relationships among the different types of quadrilaterals. A tree
diagram will also make an informative visual. Using both methods will reinforce the students’ understanding of
quadrilaterals, and their ability to make good diagrams.
Parallel Line Properties –In the second section of Chapter Three: Parallel and Perpendicular Lines, the students
learn about the relationships between the measures of the angles formed by parallel lines and a transversal. Many
of the quadrilaterals studied in this section have parallel sides. The students can apply what they learned in chapter
three to the quadrilaterals in this chapter. They may have trouble seeing the relationships because instead of lines
the quadrilaterals are made of segments. Recommend that the students draw the figures on their papers and extend
the sides of the quadrilaterals so they can see all four angles made by the intersection of the lines. These angles will
be useful when looking for specific information about the quadrilateral.
Show Clear, Organized Work –When using the distance or slope formula to verify information about a quadrilat-
eral on the coordinate plane, students will often do messy scratch work as if they are the only ones that will need to
read it. In this situation, the work is a major part of the answer. They need to communicate their thoughts on the
situation. They should write as if they are trying to convince the reader that they are correct. As students progress
in their study of mathematics, this is more often the case than the need for a single numerical answer. They should
start developing good habits now.
Symmetry –Most students have already studied symmetry at some point in their education. A review here may
be in order. When studying quadrilaterals, symmetry is a good property to consider. Symmetry is also important
when discussing the graphs of key functions that the students will be studying in the next few years. It will serve the
students well to be adept in recognizing different types of symmetry.
2.6. Quadrilaterals