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.
We will utilize the following properties to help us reason through several geometric proofs.
A quantity is equal to itself.
If A = B, then B = A.
If A = B and B = C, then A = C.
If A = B, then A + C = B + C.
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.
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.
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.
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.
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.
If two angles are complements of the same angle (or of congruent angles), then the two angles are congruent.
If two angles are supplements of the same angle (or of congruent angles), then the two angles are congruent.
All right angles are congruent.
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°.
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.
(1) Given: m?DGH = 131
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)
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.