Pairs of Angles
In geometry, certain pairs of angles can have special relationships. Using our knowledge of acute, right, and obtuse angles, along with properties of parallel lines, we will begin to study the relations between pairs of angles.
Two angles are complementary angles if their degree measurements add up to 90°. That is, if we attach both angles and fit them side by side (by putting the vertices and one side on top of each other), they will form a right angle. We can also say that one of the angles is the complement of the other.
Complementary angles are angles whose sum is 90°
Another special pair of angles is called supplementary angles. One angle is said to be the supplement of the other if the sum of their degree measurements is 180°. In other words, if we put the angles side by side, the result would be a straight line.
Supplementary angles are angles whose sum is 180°
Vertical angles are the angles opposite of each other at the intersection of two lines. They are called vertical angles because they share a common vertex. Vertical angles always have equal measures.
?JKL and ?MKN are vertical angles. Another pair of vertical angles in the picture is ?JKM and ?LKN.
Alternate Interior Angles
Alternate interior angles are formed when there exists a transversal. They are the angles on opposite sides of the transversal, but inside the two lines the transversal intersects. Alternate interior angles are congruent to each other if (and only if) the two lines intersected by the transversal are parallel.
An easy way of identifying alternate interior angles is by drawing the letter "Z" (forwards and backwards) on the lines as shown below.
In the figure on the left, ?ADH and ?GHD are alternate interior angles. Note that ?CDH and ?EHD are also alternate interior angles. The figure on the right has alternate interior angles that are congruent because there is a set of parallel lines.
Alternate Exterior Angles
Similar to alternate interior angles, alternate exterior angles are also congruent to each other if (and only if) the two lines intersected by the transversal are parallel. These angles are on opposite sides of the transversal, but outside the two lines the transversal intersects.
In the figure on the left, ?ADB and ?GHF are alternate exterior angles. So are ?CDB and ?EHF. The figure on the left does not have alternate enterior angles that are congruent, but the figure on the right does.
Corresponding angles are the pairs of angles on the same side of the transversal and on corresponding sides of the two other lines. These angles are equal in degree measure when the two lines intersected by the transversal are parallel.
It may help to draw the letter "F" (forwards and backwards) in order to help identify corresponding angles. This method is illustrated below.
Drawing the letter "F" backwards helps us see that ?ADH and ?EHF are corresponding angles. We have three other pairs of corresponding angles in this figure.
Now that we have familiarized ourselves with pairs of angles, let's practice applying some of their properties in the following exercises.
(1) Find the value of x in the figure below.
Notice that the pair of highlighted angles are vertical angles. Because they have this relationship, their angle measures are equal. Thus, we have
We have found that the value of x is 37. We can go one step further to make sure that the angles are equal by plugging 37 in for x. Indeed, the vertical angles highlighted above are equal.
(2) Find the measures of ?QRT and ?TRS shown below.
In order to solve this problem, it will be important to use our knowledge of supplementary angles. The figure shows two angles that, when combined, form straight angle ?QRS, which is 180°. So, we have
However, we are still not done. The question asks for the measures of ?QRT and ?TRS. We still have to plug in 15 for x. We get
(3) Find the values of x and y using the figure below. Lines MG and NJ run parallel to each other.
There are several ways to work this problem out. Regardless of which path we decide to take it will be necessary to use supplementary angles. We know that the sum of the measure of ?HIJ and ?JIK must be 180°. Thus, we write
Next, we must find a relationship between ?GHI, ?HIJ, and ?JIK. Notice that ?GHI and ?JIK are corresponding angles. Since we were given that MG and NJ are parallel, we know that these angles are equal. Through the transitive property, we can reason that ?GHI and ?HIJ are supplements of each other:
We can now add the measures of ?GHI and ?HIJ to get
Solving a system of equations will ultimately allow us to solve for x and y. We have
In order to eliminate a variable, which in this case will be y, we multiply the bottom equation by -1/5. Then we add the two equations and solve for x as shown below.
(Note: Rather than multiplying the bottom equation by -1/5 in the previous step, we could have multiplied the top equation by -5 to cancel out y. We get x = 16 in either case.)
We can solve for y by plugging our value for x into either of the equations we were given. In this case, we use the first equation.