In order to study geometry
in a logical way, it will be important to understand key mathematical properties
and to know how to apply useful postulates and theorems. A postulate is a
proposition that has not been proven true, but is considered to be true on the basis
for mathematical reasoning. Theorems, on the other hand, are statements that
have been proven to be true with the use of other theorems or statements. While
some postulates and theorems have been introduced in the previous sections, others
are new to our study of geometry. We will apply these properties, postulates, and
theorems to help drive our mathematical proofs in a very logical, reason-based way.
Before we begin, we must introduce the concept of congruency. Angles are congruent
if their measures, in degrees, are equal. Note: “congruent” does not
mean “equal.” While they seem quite similar, congruent angles do not have to point
in the same direction. The only way to get equal angles is by piling two angles
of equal measure on top of each other.
Properties
We will utilize the following properties to help us reason through several geometric
proofs.
Reflexive Property
A quantity is equal to itself.
Symmetric Property
If A = B, then B = A.
Transitive Property
If A = B and B = C, then A = C.
Addition Property of Equality
If A = B, then A + C = B + C.
Angle Postulates
Angle Addition Postulate
If a point lies on the interior of an angle, that angle is the sum of two smaller
angles with legs that go through the given point.
Consider the figure below in which point T lies on the interior of
?QRS. By this postulate, we have that ?QRS = ?QRT + ?TRS.
We have actually applied this postulate when we practiced finding the complements
and supplements of angles in the previous section.
Corresponding Angles Postulate
If a transversal intersects two parallel lines, the pairs of corresponding
angles are congruent.
Converse also true: If a transversal intersects two lines and the corresponding
angles are congruent, then the lines are parallel.
The figure above yields four pairs of corresponding angles.
Parallel Postulate
Given a line and a point not on that line, there exists a unique line through the
point parallel to the given line.
The parallel postulate is what sets Euclidean geometry apart from non-Euclidean geometry.
There are an infinite number of lines that pass through point E, but only
the red line runs parallel to line CD. Any other line through E will
eventually intersect line CD.
Angle Theorems
Alternate Exterior Angles Theorem
If a transversal intersects two parallel lines, then the alternate exterior
angles are congruent.
Converse also true: If a transversal intersects two lines and the alternate
exterior angles are congruent, then the lines are parallel.
The alternate exterior angles have the same degree measures because the lines are
parallel to each other.
Alternate Interior Angles Theorem
If a transversal intersects two parallel lines, then the alternate interior
angles are congruent.
Converse also true: If a transversal intersects two lines and the alternate
interior angles are congruent, then the lines are parallel.
The alternate interior angles have the same degree measures because the lines are
parallel to each other.
Congruent Complements Theorem
If two angles are complements of the same angle (or of congruent angles), then the
two angles are congruent.
Congruent Supplements Theorem
If two angles are supplements of the same angle (or of congruent angles), then the
two angles are congruent.
Right Angles Theorem
All right angles are congruent.
Same-Side Interior Angles Theorem
If a transversal intersects two parallel lines, then the interior angles
on the same side of the transversal are supplementary.
Converse also true: If a transversal intersects two lines and the interior
angles on the same side of the transversal are supplementary, then the lines are
parallel.
The sum of the degree measures of the same-side interior angles is 180°.
Vertical Angles Theorem
If two angles are vertical angles, then they have equal measures.
The vertical angles have equal degree measures. There are two pairs of vertical angles.
Exercises
(1) Given: m?DGH = 131
Find: m?GHK
First, we must rely on the information we are given to begin our proof. In this
exercise, we note that the measure of ?DGH is 131°.
From the illustration provided, we also see that lines DJ and EK
are parallel to each other. Therefore, we can utilize some of the angle theorems
above in order to find the measure of ?GHK.
We realize that there exists a relationship between ?DGH and ?EHI:
they are corresponding angles. Thus, we can utilize the Corresponding Angles Postulate
to determine that ?DGH??EHI.
Directly opposite from ?EHI is ?GHK. Since they are
vertical angles, we can use the Vertical Angles Theorem, to see that ?EHI??GHK.
Now, by transitivity, we have that ?DGH??GHK.
Congruent angles have equal degree measures, so the measure of ?DGH
is equal to the measure of ?GHK.
Finally, we use substitution to conclude that the measure of ?GHK
is 131°. This argument is organized in two-column proof form below.
(2) Given: m?1 = m?3
Prove: m?PTR = m?STQ
We begin our proof with the fact that the measures of ?1 and ?3
are equal.
In our second step, we use the Reflexive Property to show that ?2
is equal to itself.
Though trivial, the previous step was necessary because it set us up to use the
Addition Property of Equality by showing that adding the measure of ?2
to two equal angles preserves equality.
Then, by the Angle Addition Postulate we see that ?PTR is the
sum of ?1 and ?2, whereas ?STQ is the
sum of ?3 and ?2.
Ultimately, through substitution, it is clear that the measures of ?PTR
and ?STQ are equal. The two-column proof for this exercise is shown
below.
(3) Given: m?DCJ = 71, m?GFJ = 46
Prove: m?AJH = 117
We are given the measure of ?DCJ and ?GFJ to begin the
exercise. Also, notice that the three lines that run horizontally in the illustration
are parallel to each other. The diagram also shows us that the final steps of our
proof may require us to add up the two angles that compose ?AJH.
We find that there exists a relationship between ?DCJ and ?AJI:
they are alternate interior angles. Thus, we can use the Alternate Interior Angles
Theorem to claim that they are congruent to each other.
By the definition of congruence, their angles have the same measures, so
they are equal.
Now, we substitute the measure of ?DCJ with 71
since we were given that quantity. This tells us that ?AJI is also
71°.
Since ?GFJ and ?HJI are also alternate interior angles,
we claim congruence between them by the Alternate Interior Angles Theorem.
The definition of congruent angles once again proves that the angles have equal
measures. Since we knew the measure of ?GFJ, we just substitute
to show that 46 is the degree measure of ?HJI.
As predicted above, we can use the Angle Addition Postulate to get the sum
of ?AJI and ?HJI since they compose ?AJH.
Ultimately, we see that the sum of these two angles gives us 117°.
The two-column proof for this exercise is shown below.
(4) Given: m?1 = 4x + 9, m?2 = 7(x + 4)
Find: m?3
In this exercise, we are not given specific degree measures for the angles shown.
Rather, we must use some algebra
to help us determine the measure of ?3. As always, we begin with the
information given in the problem. In this case, we are given equations for the measures
of ?1 and ?2. Also, we note that there exists two pairs
of parallel lines in the diagram.
By the Same-Side Interior Angles Theorem, we know that that sum of ?1
and ?2 is 180 because they are supplementary.
After substituting these angles by the measures given to us and simplifying,
we have 11x + 37 = 180. In order to solve for x, we
first subtract both sides of the equation by 37, and then divide both sides by 11.
Once we have determined that the value of x is 13, we plug it back in to the equation for the measure
of ?2 with the intention of eventually using the Corresponding Angles
Postulate. Plugging 13 in for x gives us a measure of
119 for ?2.
Finally, we conclude that ?3
must have this degree measure as well since ?2 and ?3
are congruent. The two-column proof that shows this argument is shown below.
