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Eiffel Tower and isosceles triangle, StudySmarter Originals
From above, we see that the structure of the Eiffel Tower represents a triangle. Now further inspect the dimensions of this triangle. Notice how the two opposite sides are equal while the base is different. This means we can rule out the Eiffel Tower being the shape of an equilateral triangle, which you recall is a triangle of three equal sides. So what kind of triangle might this be? To answer your question, this is called an isosceles triangle.
An isosceles triangle is a triangle with two equal sides.
Components of an Isosceles Triangle
Consider the isosceles triangle ABC below.
Isosceles triangle, StudySmarter Originals
Here are the most important components of an isosceles triangle:
Sides and vertices
The legs of the isosceles triangle are represented by the variable a.
The base is defined by the variable b.
The vertex $C$ is the topmost section of the isosceles triangle. This is also called the apex.
The altitude$CD$is a perpendicular line segment drawn from the vertex to the base of an isosceles triangle.
The length of the altitude is called height and is described by the variable h.
Angles
The angle ${}_{\angle}C$ between the legs of the isosceles triangle is called the vertex angle (or apex angle).
Each of the two angles ${}_{\angle}B$ and ${}_{\angle}A$ between a leg and the base of the isosceles triangle is called base angle.
Properties of an Isosceles Triangle
There are several significant properties of isosceles triangles you should familiarise yourself with in order to fully understand the composition of an isosceles triangle. The table below describes this in detail.
Property  Description 
There are two equal sides  AC = BC 
The base angles are equal  ∠A = ∠B 
The altitude from the vertex angle bisects both the vertex angle and the base  AD = BD∠ACD = ∠BCD 
The altitude drawn from the apex angle divides the isosceles triangle into two congruent triangles  Triangle ACD is congruent to Triangle BCD 
Identifying the Leg and Base of an Isosceles Triangle
Say we are given a triangle with three sides. We are told that the triangle is indeed an isosceles triangle. However, we need to determine which sides are the legs of the isosceles triangle and which side is the base.
To determine the legs of the isosceles triangle, take note of the characteristics below:
Exactly two equal sides are present, which are the legs
Both legs branch out from the vertex of the triangle
The altitude is adjacent to the two legs
In contrast, the base should fulfil the following properties.
The two angles at the ends of the base are equal
An altitude drawn from the vertex is perpendicular to the base
A perpendicular line segment through the vertex bisects the base, i.e. cuts the base into two equal halves
Isosceles Triangle Theorems
With that in mind, let us now discuss two notable theorems involving isosceles triangles that take a closer look into the two main properties as described above.
Theorem 1
The angles opposite the equal sides of an isosceles triangle are equal.
Proof of Theorem 1
Consider the isosceles triangle ABC below where AC = BC. Draw a bisector passing through ∠C. We shall name this line segment CD.
Isosceles triangle theorem 1, StudySmarter Originals
We aim to prove that the angles opposite the sides AC and BC are equal.
Essentially, we want to show that ∠A = ∠B.
Notice that in triangles ACD and BCD:
AC = BC
∠ACD = ∠BCD
CD = CD
SAS Congruence
If two sides and an included angle of one triangle is equal to the two sides and included angle of the second triangle, then the two triangles are said to be congruent.
By the SAS Congruence rule above, triangles ACD and BCD must be congruent. As the two triangles are congruent, the corresponding angles must also be congruent. Thus, ∠A must be equal to ∠B.
Theorem 2
The sides opposite the equal angles of an isosceles triangle are equal.
Proof of Theorem 2
Consider the isosceles triangle ABC below where ∠A = ∠B. We shall construct a bisector CD that meets the side AB at right angles.
Isosceles triangle theorem 2, StudySmarter Originals
We aim to prove that AC = BC to show that triangle ABC is indeed an isosceles triangle.
Notice that in triangles ACD and BCD:
∠ACD = ∠BCD
CD = CD
∠ADC = ∠BDC = 90^{o}
ASA Congruence
If two angles and an included side between the angles of one triangle are equal to the corresponding two angles and included side between the angles of the second triangle, then the two triangles are said to be congruent.
By the ASA Congruence rule above, triangles ACD and BCD must be congruent. As the two triangles are congruent, the corresponding sides must also be congruent. Thus, AC must be equal to BC and so triangle ABC is an isosceles triangle.
Types of Isosceles Triangles
There are three types of isosceles triangles to consider, namely
Isosceles acute;
Isosceles right;
Isosceles obtuse.
The table below compares each of these types of isosceles triangles.
Type of Isosceles Triangle  Diagram  Description 
Isosceles Acute  Acute isosceles triangle, StudySmarter Originals 

Isosceles Right  Right isosceles triangle, StudySmarter Originals 

Isosceles Obtuse  Obtuse isosceles triangle, StudySmarter Originals 

Formulas of Isosceles Triangles
In this section, we shall look at three important formulas involving isosceles triangles, namely
The height of an isosceles triangle;
The perimeter of an isosceles triangle;
The area of an isosceles triangle.
The Height of an Isosceles Triangle
The height of an isosceles triangle can be found by applying Pythagoras Theorem. Say we have the isosceles triangle ABC below where the measures of a leg a and the base b are given.
Height of an isosceles triangle, StudySmarter Originals
We know that the altitude (line segment CD) from the vertex angle bisects the base of the isosceles triangle. This means that
$AD=BD=\frac{1}{2}b$.
Furthermore, ADC and BDC are rightangled triangles where a is the hypotenuse. Thus, to find the height, we can simply adopt Pythagoras Theorem as
id="2851376" role="math" ${a}^{2}={h}^{2}+{\left(\frac{b}{2}\right)}^{2}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\Rightarrow {h}^{2}={a}^{2}{\left(\frac{b}{2}\right)}^{2}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\Rightarrow h=\sqrt{{a}^{2}{\left(\frac{b}{2}\right)}^{2}}$
The Perimeter of an Isosceles Triangle
The perimeter of an isosceles triangle is given by the following formula.
$P=2a+b$
where a is the length of the two equal sides and b is the base of the isosceles triangle. Let us demonstrate this with a worked example.
Given the triangle below, calculate its perimeter.
Example 1, StudySmarter Originals
Solution
By the perimeter formula, we find that the perimeter of this isosceles triangle is
$P=2\left(9\right)+7\phantom{\rule{0ex}{0ex}}\Rightarrow P=25units$
The Area of an Isosceles Triangle
Once you know the height of an isosceles triangle, calculating the area is a breeze. The formula for this is
$Area=\frac{1}{2}\times b\times h$,
where b is the base and h is the height of the isosceles triangle. Below is a worked example applying this method.
Find the area of an isosceles triangle whose base is 6 units and side is 13 units.
Solution
Let us begin by making a sketch of this isosceles triangle. Construct an altitude from the vertex angle of this isosceles triangle to the base.
Example 2, StudySmarter Originals
We know that the altitude bisects the base of the isosceles triangle and creates two congruent rightangled triangles. Since the base is 6 units, then AD = BD = 3 units. The height is found by applying Pythagoras Theorem as
${13}^{2}={h}^{2}+{3}^{2}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\Rightarrow {h}^{2}={13}^{2}{3}^{2}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\Rightarrow h=\sqrt{{13}^{2}{3}^{2}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\Rightarrow h=4\sqrt{10}units$
Now that we have the height of the isosceles triangle, we can use the area formula. We find that the area of this isosceles triangle is
$A=\frac{1}{2}\times 4\sqrt{10}\times 6\phantom{\rule{0ex}{0ex}}\Rightarrow A=12\sqrt{10}unit{s}^{2}$
Altitudes in Isosceles Triangles
Let us now define an altitude of a triangle.
An altitude is a line that passes through the vertex of a triangle that is perpendicular to the opposite side.
Do not confuse this term with perpendicular bisectors! A perpendicular bisector divides a segment into two equal parts and is perpendicular to that segment.
Now that we have established the definition of an altitude, we shall now link this idea with our subject at hand. The following are two theorems that relate the altitude to isosceles triangles.
Theorem 1
The altitude to the base of an isosceles triangle bisects the vertex angle.
Theorem 2
The altitude to the base of an isosceles triangle bisects the base.
Proof of Theorem 1 and 2
Consider the isosceles triangle shown below.
The altitude of an isosceles triangle, StudySmarter Originals
Say we draw an altitude to the base of the isosceles triangle. We find that two congruent triangles are formed. The altitude creates two rightangled triangles ADC and BDC and becomes the shared side between the two triangles. The congruent sides of the triangle become the hypotenuse for triangles ADC and BDC and are of equal length.
Since constructing an altitude to the base of the isosceles triangle forms two congruent rightangled triangles, we conclude that the altitude bisects both the base and vertex of the isosceles triangle.
Worked Examples Involving Isosceles Triangles
Given the triangle, ABC below, determine lengths AC and BC if ∠A = ∠B.
Example 3, StudySmarter Originals
Solution
Since the two angles of the triangle above are congruent, the sides opposite them are also congruent. In other words, as ∠A = ∠B, then AC = BC.
$AC=BC\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\Rightarrow 4x22=2x+14\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\Rightarrow 2x=36\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\Rightarrow x=18\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}AC=4\left(18\right)22\phantom{\rule{0ex}{0ex}}\Rightarrow AC=50units\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}AC=BC=50units$
Given the triangles ABD and BDC below, determine the value of ∠X if AB = BD = CD and ∠C is 23^{o}.
Example 4, StudySmarter Originals
Solution
We know that if two sides of a triangle are equal then the angles opposite them are also equal. This means that since BD = CD then ∠C = ∠CBD = 23^{o}.
As the sum of the interior angles of a triangle is 180^{o}, the ∠BDC is 130^{o}, for triangle BDC.
The ∠ADB is the exterior angle of triangle BDC. The sum of the exterior angle and its adjacent interior angle of a triangle is 180^{o}. Thus ∠ADB is 50^{o}.
As AB = BD, ∠A = ∠ADB = 50^{o}. As before, since the sum of the interior angles of a triangle, is 180^{o}, the ∠X is 80^{o}, for triangle ABD.
Given the triangles ACB and DCE below, determine the value of angles X, Y and Z if AC = BC, DC = EC and ∠ACB = 31^{o}.
Example 5, StudySmarter Originals
Solution
As ∠Y and ∠ACB are vertical angles then ∠Y = ∠ACB = 31^{o}.
We know that if two sides of a triangle are congruent the angles opposite them are also congruent. ∠X = ∠B = ∠D = ∠Z since the vertex angle for triangles ACB and DCE are equal. Noting that the sum of the interior angles of a triangle is 180^{o}, we obtain
$\angle X+\angle B+{31}^{o}={180}^{o}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\Rightarrow 2\angle X={180}^{o}{31}^{o}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\Rightarrow 2\angle X={149}^{o}\phantom{\rule{0ex}{0ex}}\Rightarrow \angle X=\frac{{149}^{o}}{2}\phantom{\rule{0ex}{0ex}}\Rightarrow AngleX=74.{5}^{o}$
Thus, ∠X = ∠Z = 74.5^{o}.
Comparing Triangles
There are three types of triangles we shall often see throughout this syllabus, namely
Isosceles triangle
Equilateral triangle
Scalene triangle
In this final section, we shall look at the differences between these three triangles. By familiarising ourselves with these contrasts, we can properly distinguish each type we are dealing with and perform the correct calculations. The table below compares these three triangles with respect to sides, angles and altitudes.
Property  Isosceles Triangle  Equilateral Triangle  Scalene Triangle 
Diagram  Isosceles triangle, StudySmarter Originals  Equilateral triangle, StudySmarter Originals  Scalene triangle, StudySmarter Originals 
Sides  Two sides of equal length  Three sides of equal length  Three sides of different length 
Angles  Two angles of equal value  Three angles of equal value  Three angles of different value 
Altitude  An altitude drawn from the vertex angle bisects that angle and the unequal side of the triangle.  An altitude drawn from any angle bisects that angle and the opposite side of the triangle.  No special criteria 
Isosceles Triangles  Key takeaways
 An isosceles triangle
 Consists of two equal sides and two equal angles
 Have equal base angles
 The altitude from the vertex angle bisects the base and the vertex angle
 The altitude drawn from the apex angle divides the isosceles triangle into two congruent triangles
 There are three types of isosceles triangles: Acute, Right and Obtuse
 The area is given by $A=\frac{1}{2}\times h\times borA=\frac{1}{2}\times b\times \sqrt{{a}^{2}{\left(\frac{b}{2}\right)}^{2}}$
 The perimeter is given by $P=2a+b$
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Frequently Asked Questions about Isosceles Triangles
What is an isosceles triangle?
An isosceles triangle is a triangle with two equal sides
How do you find the height of an isosceles triangle?
We can find the height of an isosceles triangle by using Pythagoras Theorem
How do you find the area of an isosceles triangle?
The area of an isosceles triangle is the product of the base and height multiplied by half
What is the rule for isosceles triangles?
The base angles of an isosceles triangle are equal
What is an example of isosceles triangles?
Acute, right and obtuse isosceles triangles
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