In the first section of our study of triangles, we learned how to classify triangles by the measures of their angles and by the lengths of their sides. However, aside from the names we give triangles, it will be important to understand and recognize the characteristics that make certain triangles special. In this section, we will take a closer look at the properties of some unique figures: isosceles and equilateral triangles.
Let's begin our study of isosceles triangles by learning new terminology that will help us identify various characteristics of these kinds of triangles. Recall that an isosceles triangle is a triangle with at least two congruent sides. These congruent sides are called legs. The point at which these legs meet is called the vertex point of the isosceles triangle, and the angle formed by the legs is called the vertex angle. The other two angles of the triangle are called base angles. A labeled illustration of an isosceles triangle is shown below.
In the past, we determined isosceles triangles by the lengths of their sides. In other words, if we saw that a triangle had two sides with equal lengths, then we classified the triangle as isosceles. There are other characteristics that mark isosceles triangles, however. Let's look at an important theorem that gives us even more information about these kinds of triangles.
If two sides of a triangle are congruent, then the angles opposite those sides are congruent.
Converse also true: If two angles of a triangle are congruent, then the sides opposite those angles are congruent.
We will practice using these theorems to help us solve the following exercises.
Determine the values of x and y.
In the diagram, we are given that ?A is 52°. Because the side opposite of ?A is congruent to the side opposite of ?C, we know that the angles are also congruent by the Isosceles Triangle Theorem. Thus, the value of y is 52.
Now, let's try to determine the value of x. In order to figure this out, we must use the Triangle Angle Sum Theorem to figure out what the total degree measure is at ?B.
Since we've determined that ?B must have a measure of 76°, we can write an algebraic equation to help us solve for x. This method is shown below.
We subtract 6 from both sides of the equation.
Now, we divide by 14 to find x.
Our answers are x = 14 and y = 52.
Let's look at the information we've been given to see which direction we'd like to take this problem in. We are given that ?TUS and ?QSR are congruent.
Now, let's try to find a special relationship that either ?TUS or ?QSR may have with another angle in the diagram. Notice that ?QSR and ?TSU are vertical angles, so by the Vertical Angles Theorem, we can say that they are congruent to each other.
We can now apply the Transitive Property to show that ?TUS and ?TSU are congruent.
Finally, by the Isosceles Triangle Theorem, we know that the sides opposite of two congruent angles are also congruent. Thus, segments TS and TU are congruent to each other. Our new diagram and the two-column geometric proof for this exercise are shown below.
Determine the values of x and y in the figure below.
We first want to notice that ?BCA and ?BCD are supplementary. Recall, that this means that their sum of their degree measures is 180. Thus we will try to determine the measure of ?BCA:
By the Triangle Angle Sum Theorem, we know that the sum of ?A, ?B, and ?BCA is 180°, so we will try to determine the values of x and y by figuring out what the sum of ?A and ?B should be.
Together, ?A and ?B should have a measure of 124.
Let's look at the diagram again. Notice that segment AC is congruent to segment BC. So, by the Isosceles Triangle Theorem, we know that ?A is congruent to ?B (since they are the angles opposite of the congruent sides). Therefore, we can divide the remainder of the angle measures of the triangle, 124, by the two congruent angles to determine what the measure of each angle should be. When we do this, we see that ?A and ?B should come out to 62° each.
To solve for x, we have
To solve for y, we have
So, we have x = 31 and y = 4.
Equilateral triangles are another type of triangle with unique characteristics. Knowledge of these kinds of triangles will assist us in some of the proofs and exercises we will encounter in the future, so let's take a closer look at the traits that make equilateral triangles special.
While the following characteristics of equilateral triangles are not theorems or postulates, they are statements we can use in our proofs. The following statements are called corollaries. Corollaries are proven results that rely heavily on one theorem. The following corollaries of equilateral triangles are a result of the Isosceles Triangle Theorem:
(1) A triangle is equilateral if and only if it is equiangular.
(2) Each angle of an equilateral triangle has a degree measure of 60.
The congruent sides of the triangle imply that all the angles are congruent. We can also use the converse of this, which is that three congruent angles imply three congruent sides in a triangle. Each of the angles above is 60°.
Let's practice using these corollaries in the following exercises.
Determine the values of x and y in the diagram below.
In order to solve this problem, we must recognize the fact that the triangle shown is an equilateral triangle. We notice this by the tick marks on all three sides of the triangle. This indicates to us that all three sides of the triangle are congruent.
Moreover, we must be able to understand the relationship between the angles of the triangle. In order to solve for x, we will need to keep in consideration that every angle of an equilateral triangle is 60°.
We will solve for x first. In order to do this, we need to use the information given to us about the sides of the triangles to solve for x. We will set 2(2x + 1) equal to 14 since equilateral triangles have congruent sides. Thus, we have
Now that we have solved for x, let's determine the value for y. This part of the exercise requires our knowledge of the angles of equilateral triangles. As mentioned before, every angle has a measure of 60, so we have
We have already determined the value of x, so we can plug this value right into our equation to solve for y.
Thus, we get x = 3 and y = 6.
First, we will consider the information we've been given to see if we can derive any more useful information from it. We are given that ?RQS and ?TQS are congruent, as shown in the diagram. Also, we are told that ?RQT is an equilateral triangle. This fact will be of use to us as we continue the exercise.
Since ?RQT is an equilateral triangle, we know that all three sides and angles of the triangle are congruent. Thus, we can say that segments RQ and TQ are congruent to each other.
Now, we have one pair of sides and one pair of angles that are congruent to each other. If we can prove that one more pair of corresponding sides of ?RQS and ?TQS are congruent, then we can use the SAS Postulate to prove that the triangles are congruent. Indeed, if we use the Reflexive Property to show that QS is congruent to itself, we see that the two triangles are congruent to each other. Now, our figure looks like this:
Finally, we can say that segment RS is congruent to segment TS because they are corresponding sides of congruent triangles, so they are congruent. Our two-column proof is shown below.