In this unit, we have focused on studying the relationship between different types of triangles. This section we will focus on special triangles and their properties: Isosceles triangles and equilateral triangles.
ISOSCELES TRIANGLES
As you may recall, an isosceles triangle is a triangle with two sides congruent. These two congruent sides are call its legs, and the third side is called its base. The angle opposite of the base is called the vertex angle, whereas the angles opposite each of the legs is called its base angles. Here’s a picture to show you:
ISOSCELES TRIANGLES
As you may recall, an isosceles triangle is a triangle with two sides congruent. These two congruent sides are call its legs, and the third side is called its base. The angle opposite of the base is called the vertex angle, whereas the angles opposite each of the legs is called its base angles. Here’s a picture to show you:
Please note: The base of an isosceles doesn’t always need to be on the bottom of the triangle. First determine which sides are the legs based on which two sides are congruent, then its base will be the non-congruent side. Here’s another example of an isosceles triangle and its parts. Click on each of the points to move them around and notice how each of the parts of the isosceles triangle rotates with it.
Now that we know what an isosceles triangle is, we can learn about its properties. The following will help you determine what is true about the base angles of an isosceles triangle.
Theorem
If two sides of a triangle are congruent (which are the legs of an isosceles triangle), then the angles opposite them (base angles) are congruent. (If \(\cong\) sides, then \(\cong\) angles.)
If two sides of a triangle are congruent (which are the legs of an isosceles triangle), then the angles opposite them (base angles) are congruent. (If \(\cong\) sides, then \(\cong\) angles.)
Its converse is also true:
Theorem
If two angles of a triangle are congruent (base angles), then the sides opposite them (legs) are congruent. (If \(\cong\) angles, then \(\cong\) sides.)
If two angles of a triangle are congruent (base angles), then the sides opposite them (legs) are congruent. (If \(\cong\) angles, then \(\cong\) sides.)
Example 1:
Given \(\triangle ARM\), where \(m\angle R = (5x + 6)^{\circ}\), and \(m\angle M = (10x - 44)^{\circ}\). Find the measure of each angle.
Given \(\triangle ARM\), where \(m\angle R = (5x + 6)^{\circ}\), and \(m\angle M = (10x - 44)^{\circ}\). Find the measure of each angle.
Solution:
Notice that \(\overline{AR}\) and \(\overline{AM}\) are congruent, therefore they are the legs of the isosceles triangle. According to our theorem, if two sides of a triangle are congruent, then the angles opposite them are also congruent. Therefore, we know that the base angles, which are opposite of \(\overline{AR}\) and \(\overline{AM}\), must also be congruent:
Notice that \(\overline{AR}\) and \(\overline{AM}\) are congruent, therefore they are the legs of the isosceles triangle. According to our theorem, if two sides of a triangle are congruent, then the angles opposite them are also congruent. Therefore, we know that the base angles, which are opposite of \(\overline{AR}\) and \(\overline{AM}\), must also be congruent:
Since \(\angle R\cong\angle M\), we know that the measures are equal. So,
\(\begin{align*}\\
m\angle R &= m\angle M\\
5x + 6 &= 10x - 44\\
5x + 50 &= 10x\\
50 &= 5x\\
10 &= x
\end{align*}\)
Substituting \(x = 10\) into each angle measure:
\(m\angle R = 5(10) + 6 = 50 + 6 = 56^{\circ}\)
\(m\angle M = 10(10) - 44 = 100 - 44 = 56^{\circ}\)
\(m\angle A = 180^{\circ} - m\angle R + m\angle M = 180^{\circ} - 56^{\circ} - 56^{\circ} = 68^{\circ}\)
Example 2:
Given \(\triangle JEM\), where \(JE = x + 8\), \(EM = -2x + 16\), and \(JM = 4x +2\). Find the length of each side.
\(\begin{align*}\\
m\angle R &= m\angle M\\
5x + 6 &= 10x - 44\\
5x + 50 &= 10x\\
50 &= 5x\\
10 &= x
\end{align*}\)
Substituting \(x = 10\) into each angle measure:
\(m\angle R = 5(10) + 6 = 50 + 6 = 56^{\circ}\)
\(m\angle M = 10(10) - 44 = 100 - 44 = 56^{\circ}\)
\(m\angle A = 180^{\circ} - m\angle R + m\angle M = 180^{\circ} - 56^{\circ} - 56^{\circ} = 68^{\circ}\)
Example 2:
Given \(\triangle JEM\), where \(JE = x + 8\), \(EM = -2x + 16\), and \(JM = 4x +2\). Find the length of each side.
Solution: Notice the angles marked congruent. According to our theorem, if two angles are congruent, then the sides opposite them are also congruent:
So \(\overline{JM}\cong\overline{JE}\). Therefore their lengths are equal, so we can set up the following equation:
\(\begin{align*}\\
JM &= JE\\
x+ 8 &= 4x + 2\\
8 &= 3x + 2\\
6 &= 3x\\
2 &= x
\end{align*}\)
Substituting \(x = 2\) into each side length:
\(JM = 2 + 8 = 10\) units
\(JE = 4(2) + 2 = 8 + 2 = 10\) units
\(EM = -2(2) + 16 = -4 + 10 = 6\) units
Example 3:
Given \(\triangle ABG\) where \(AB = 2x - 3\), \(BG = 5x - 12\), and \(AG = x + 1\). Find \(AG\).
\(\begin{align*}\\
JM &= JE\\
x+ 8 &= 4x + 2\\
8 &= 3x + 2\\
6 &= 3x\\
2 &= x
\end{align*}\)
Substituting \(x = 2\) into each side length:
\(JM = 2 + 8 = 10\) units
\(JE = 4(2) + 2 = 8 + 2 = 10\) units
\(EM = -2(2) + 16 = -4 + 10 = 6\) units
Example 3:
Given \(\triangle ABG\) where \(AB = 2x - 3\), \(BG = 5x - 12\), and \(AG = x + 1\). Find \(AG\).
Solution:
Let’s watch the video for the solution.
Let’s watch the video for the solution.
Example 4:
Find the measures of the missing angles.
Solution
Notice that \(\angle 1\cong\angle 2\) because base angles of an isosceles triangle are congruent.
\(180^{\circ} - 110^{\circ} = 70^{\circ}\)
\(m\angle 1 = m\angle 2 = \Large \frac{70^{\circ}}{2}\)\(= 35^{\circ}\)
\(m\angle 3 = 180^{\circ} - (35^{\circ} + 32^{\circ}) = 180^{\circ} - 67^{\circ} = 113^{\circ}\)
Notice that \(\angle 1\cong\angle 2\) because base angles of an isosceles triangle are congruent.
\(180^{\circ} - 110^{\circ} = 70^{\circ}\)
\(m\angle 1 = m\angle 2 = \Large \frac{70^{\circ}}{2}\)\(= 35^{\circ}\)
\(m\angle 3 = 180^{\circ} - (35^{\circ} + 32^{\circ}) = 180^{\circ} - 67^{\circ} = 113^{\circ}\)
EQUILATERAL TRIANGLES
An equilateral triangle is a triangle that has three congruent sides. Equilateral triangles are even more special than isosceles triangles! Let’s take a look here:
Move around points A and B to see if these patterns change.
Notice that each of the angles in the triangle have the same measure. We call this an equiangular triangle. Also notice, that since the sum of the measures of a triangle must be \(180^{\circ}\), that means that each angle must have the measure: \(60^{\circ}\). That brings us to our new theorem:
Notice that each of the angles in the triangle have the same measure. We call this an equiangular triangle. Also notice, that since the sum of the measures of a triangle must be \(180^{\circ}\), that means that each angle must have the measure: \(60^{\circ}\). That brings us to our new theorem:
THEOREM:
A triangle is equilateral if and only if it is also equiangular.
A triangle is equilateral if and only if it is also equiangular.
Let’s try a few problems utilizing this theorem:
Example 5:
Given the following diagram, find the value of \(x\).
Example 5:
Given the following diagram, find the value of \(x\).
Solution:
Notice the the tick marks in the diagram show that the triangle is equilateral. According to our theorem, then we know that \(\triangle GER\) is also equiangular and thus each angle must measure \(60^{\circ}\).
\(\begin{align*}\\
m\angle G &= 60^{\circ}\\
(3x - 5)^{\circ} &= 60^{\circ}\\
3x &= 65\\
x &= \frac{65}{3}\approx 21.7
\end{align*}\)
Example 6:
Given the triangle below, find the values of each variable.
Notice the the tick marks in the diagram show that the triangle is equilateral. According to our theorem, then we know that \(\triangle GER\) is also equiangular and thus each angle must measure \(60^{\circ}\).
\(\begin{align*}\\
m\angle G &= 60^{\circ}\\
(3x - 5)^{\circ} &= 60^{\circ}\\
3x &= 65\\
x &= \frac{65}{3}\approx 21.7
\end{align*}\)
Example 6:
Given the triangle below, find the values of each variable.
Solution:
Notice that the triangle is an equiangular triangle. According to our theorem, the triangle will also be equilateral. Therefore, each of the sides are congruent! There are several different ways to solve for the missing variables. Here’s one way:
Since we know \(AR = 12\) units, that means the length of each of the other sides must also be \(12\) units:
\(AD = 12\)
\(3y = 12\)
\(y = 4\)
Similarly,
\(DR = 12\)
\(0.5x = 12\)
\(x = 24\)
Therefore, \(x = 24\) and \(y = 4\). Let’s substitute these values back into the expression to make sure we didn’t make a mistake:
Notice that the triangle is an equiangular triangle. According to our theorem, the triangle will also be equilateral. Therefore, each of the sides are congruent! There are several different ways to solve for the missing variables. Here’s one way:
Since we know \(AR = 12\) units, that means the length of each of the other sides must also be \(12\) units:
\(AD = 12\)
\(3y = 12\)
\(y = 4\)
Similarly,
\(DR = 12\)
\(0.5x = 12\)
\(x = 24\)
Therefore, \(x = 24\) and \(y = 4\). Let’s substitute these values back into the expression to make sure we didn’t make a mistake:
Since each of the side lengths is \(12\) units, the values for our variables checks out!
Quick Check
1) If \(m\angle K = (3x - 10)^{\circ}\) and \(m\angle W = (x)^{\circ}\) , find the measure of each angle in the triangle.
1) If \(m\angle K = (3x - 10)^{\circ}\) and \(m\angle W = (x)^{\circ}\) , find the measure of each angle in the triangle.
2) Given \(TS = -x+10\), \( TG = x+4\), and SG = 5x-20. Find the length of each side of the triangle.
3) Given the following, find the values of \(x\) and \(y\).
4) Given the following, find the values of \(x\) and \( y\).