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Isosceles Triangles

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# Isosceles Triangles

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Picture yourself in Paris, France in front of the Eiffel Tower and observe its structure.

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:

1. AC = BC

2. ∠ACD = ∠BCD

3. 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:

1. ∠ACD = ∠BCD

2. CD = CD

3. ∠ADC = ∠BDC = 90o

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

1. Isosceles acute;

2. Isosceles right;

3. 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 Consists of two equal sides and one unequal side.The two angles opposite the sides are equal.If each of the equal angles is more than 45o and less than 90o, the apex angle will be an acute angle. Isosceles Right Right isosceles triangle, StudySmarter Originals Consist of two equal sides: one of which act as the perpendicular and the other as the base of the triangle. The third unequal side acts as the hypotenuse of the triangle. If each of the equal angles is exactly 45o, then the apex angle is a right angle. By Pythagoras Theoremh2 = a2 + a2 = 2a2 Isosceles Obtuse Obtuse isosceles triangle, StudySmarter Originals Consists of two equal sides and one obtuse angle.If each of the equal angles is less than 45o, then the vertex angle is an obtuse angle.

## Formulas of Isosceles Triangles

In this section, we shall look at three important formulas involving isosceles triangles, namely

1. The height of an isosceles triangle;

2. The perimeter of an isosceles triangle;

3. 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 right-angled triangles where a is the hypotenuse. Thus, to find the height, we can simply adopt Pythagoras Theorem as

id="2851376" role="math" ${a}^{2}={h}^{2}+{\left(\frac{b}{2}\right)}^{2}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}⇒{h}^{2}={a}^{2}-{\left(\frac{b}{2}\right)}^{2}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}⇒h=\sqrt{{a}^{2}-{\left(\frac{b}{2}\right)}^{2}}$

### 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

$P=2\left(9\right)+7\phantom{\rule{0ex}{0ex}}⇒P=25units$

### 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

$Area=\frac{1}{2}×b×h$,

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 right-angled triangles. Since the base is 6 units, then AD = BD = 3 units. The height is found by applying Pythagoras Theorem as

${13}^{2}={h}^{2}+{3}^{2}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}⇒{h}^{2}={13}^{2}-{3}^{2}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}⇒h=\sqrt{{13}^{2}-{3}^{2}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}⇒h=4\sqrt{10}units$

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

$A=\frac{1}{2}×4\sqrt{10}×6\phantom{\rule{0ex}{0ex}}⇒A=12\sqrt{10}unit{s}^{2}$

## 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 right-angled 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 right-angled 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.

$AC=BC\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}⇒4x-22=2x+14\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}⇒2x=36\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}⇒x=18\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}AC=4\left(18\right)-22\phantom{\rule{0ex}{0ex}}⇒AC=50units\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}AC=BC=50units$

Given the triangles ABD and BDC below, determine the value of ∠X if AB = BD = CD and ∠C is 23o.

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 = 23o.

As the sum of the interior angles of a triangle is 180o, the ∠BDC is 130o, 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 180o. Thus ∠ADB is 50o.

As AB = BD, ∠A = ∠ADB = 50o. As before, since the sum of the interior angles of a triangle, is 180o, the ∠X is 80o, 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 = 31o.

Example 5, StudySmarter Originals

Solution

As ∠Y and ∠ACB are vertical angles then ∠Y = ∠ACB = 31o.

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 180o, we obtain

$\angle X+\angle B+{31}^{o}={180}^{o}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}⇒2\angle X={180}^{o}-{31}^{o}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}⇒2\angle X={149}^{o}\phantom{\rule{0ex}{0ex}}⇒\angle X=\frac{{149}^{o}}{2}\phantom{\rule{0ex}{0ex}}⇒AngleX=74.{5}^{o}$

Thus, ∠X = ∠Z = 74.5o.

## Comparing Triangles

There are three types of triangles we shall often see throughout this syllabus, namely

1. Isosceles triangle

2. Equilateral triangle

3. 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 $A=\frac{1}{2}×h×borA=\frac{1}{2}×b×\sqrt{{a}^{2}-{\left(\frac{b}{2}\right)}^{2}}$
• The perimeter is given by $P=2a+b$

An isosceles triangle is a triangle with two equal sides

We can find the height of an isosceles triangle by using Pythagoras Theorem

The area of an isosceles triangle is the product of the base and height multiplied by half

The base angles of an isosceles triangle are equal

Acute, right and obtuse isosceles triangles

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## Join over 22 million students in learning with our StudySmarter App

The first learning app that truly has everything you need to ace your exams in one place

• Flashcards & Quizzes
• AI Study Assistant
• Study Planner
• Mock-Exams
• Smart Note-Taking