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.
Types of Angle Pairs
Adjacent angles: two angles with a common vertex, sharing a common side and no overlap.
Angles ∠1 and ∠2 are adjacent.
Complementary angles: two angles, the sum of whose measures is 90°.
Angles ∠1 and ∠2 are complementary.
Complementary are these angles too(their sum is 90°):
Supplementary angles: two angles, the sum of whose measures is 180°.
Angles ∠1 and ∠2 are supplementary.
Angle pairs formed by parallel lines cut by a transversal
When two parallel lines are given in a figure, there are two main areas: the interior and the exterior.
When two parallel lines are cut by a third line, the third line is called the transversal. In the example below, eight angles are formed when parallel lines m and n are cut by a transversal line, t.
There are several special pairs of angles formed from this figure. Some pairs have already been reviewed:
∠1 and ∠4
∠2 and ∠3
∠5 and ∠8
∠6 and ∠7
Recall that all pairs of vertical angles are congruent.
∠1 and ∠2
∠2 and ∠4
∠3 and ∠4
∠1 and ∠3
∠5 and ∠6
∠6 and ∠8
∠7 and ∠8
∠5 and ∠7
Recall that supplementary angles are angles whose angle measure adds up to 180°. All of these supplementary pairs are linear pairs. There are other supplementary pairs described in the shortcut later in this section. There are three other special pairs of angles. These pairs are congruent pairs.
Alternate interior angles two angles in the interior of the parallel lines, and on opposite (alternate) sides of the transversal. Alternate interior angles are non-adjacent and congruent.
Alternate exterior angles two angles in the exterior of the parallel lines, and on opposite (alternate) sides of the transversal. Alternate exterior angles are non-adjacent and congruent.
Corresponding angles two angles, one in the interior and one in the exterior, that are on the same side of the transversal. Corresponding angles are non-adjacent and congruent.
Use the following diagram of parallel lines cut by a transversal to answer the example problems.
What is the measure of ∠8?
The angle marked with measure 53° and ∠8 are alternate exterior angles. They are in the exterior, on opposite sides of the transversal. Because they are congruent, the measure of ∠8 = 53°.
What is the measure of ∠7?
∠8 and ∠7 are a linear pair; they are supplementary. Their measures add up to 180°. Therefore, ∠7 = 180° – 53° = 127°.
1. When a transversal cuts parallel lines, all of the acute angles formed are congruent, and all of the obtuse angles formed are congruent.
In the figure above ∠1, ∠4, ∠5, and ∠7 are all acute angles. They are all congruent to each other. ∠1 ≅ ∠4 are vertical angles. ∠4 ≅ ∠5 are alternate interior angles, and ∠5 ≅ ∠7 are vertical angles. The same reasoning applies to the obtuse angles in the figure: ∠2, ∠3, ∠6, and ∠8 are all congruent to each other.
2. When parallel lines are cut by a transversal line, any one acute angle formed and any one obtuse angle formed are supplementary.
From the figure, you can see that ∠3 and ∠4 are supplementary because they are a linear pair.
Notice also that ∠3 ≅ ∠7, since they are corresponding angles. Therefore, you can substitute ∠7 for ∠3 and know that ∠7 and ∠4 are supplementary.
In the following figure, there are two parallel lines cut by a transversal. Which marked angle is supplementary to ∠1?
The angle supplementary to ∠1 is ∠6. ∠1 is an obtuse angle, and any one acute angle, paired with any obtuse angle are supplementary angles. This is the only angle marked that is acute.
Angles - Problems with Solutions
Types of angles
Parallel lines cut by a transversal Test