that describes two numbers, and congruent is a word that describes two geometric objects. Equality of measure is
often one of the conditions for congruence. If the students have been correctly using the naming conventions for
a segment and its measure and an angle and its measure in previous lessons they will be less likely to confuse the
words congruent and equal now.
The Number of Tick Marks or Arcs Does Not Give Relative Length –A common misconception is that a pair of
segments marked with one tick, are longer than a pair of segments marked with two ticks in the same figure. Clarify
that the number of ticks just groups the segments; it does not give any relationship in measure between the groups.
An analogous problem occurs for angles.
Midpoint or Bisector –Midpoint is a location, a noun, and bisect is an action, a verb. One geometric object can
bisect another by passing through its midpoint. This link to English grammar often helps students differentiate
between these similar terms.
Intersects vs. Bisects –Many students replace the word intersects with bisects. Remind the students that if a
segment or angle is bisected it is intersected, and it is know that the intersection takes place at the exact middle.
Orientation Does Not Affect Congruence –The only stipulation for segments or angles to be congruent is that they
have the same measure. How they are twisted or turned on the page does not matter. This becomes more important
when considering congruent polygons later, so it is worth making a point of now.
Labeling a Bisector or Midpoint –Creating a well-labeled picture is an important step in solving many Geometry
problems. How to label a midpoint or a bisector is not obvious to many students. It is often best to explicitly explain
that in these situations, one marks the congruent segments or angles created by the bisector.
Additional Exercises:
- Does it make sense for a line to have a midpoint?
Answer: No, a line is infinite in one dimension, so there is not a distinct middle.
Angle Pairs
Complementary or Supplementary –The quantity of vocabulary in Geometry is frequently challenging for stu-
dents. It is common for students to interchange the words complementary and supplementary. A good mnemonic
device for these words is that they, like many math words, go in alphabetical order; the smaller one, complementary,
comes first.
Linear Pair and Supplementary –All linear pairs have supplementary angles, but not all supplementary angles
form linear pairs. Understanding how Geometry terms are related helps students remember them.
Angles formed by Two Intersection Lines –Students frequently have to determine the measures of the four angles
formed by intersecting lines. They can check their results quickly when they realize that there will always be two
sets of congruent angles, and that angles that are not congruent must be supplementary. They can also check that all
four angles measures have a sum of 360 degrees.
Write on the Picture –In a complex picture that contains many angle measures which need to be found, students
should write angle measures on the figure as they find them. Once they know an angle they can use it to find other
angles. This may require them to draw or trace the picture on their paper. It is worth taking the time to do this. The
act of drawing the picture will help them gain a deeper understanding of the angle relationships.
Proofs –The word proof strikes fear into the heart of many Geometry students. It is important to define what a
mathematical proof is, and let the students know what is expected of them regarding each proof.
Definition: Amathematicalproof is a mathematical argument that begins with a truth and proceeds by logical steps
to a conclusion which then must be true.
2.1. Basics of Geometry