Identification of an Isosceles Triangle

🏆Practice types of triangles
5 Questions
Triangle
Types of triangles
Practice
Add a new subject

When we have atriangle, we can identify that it is an isosceles if at least one of the following conditions is met:

1) If the triangle has two equal angles - The triangle is isosceles.
2) If in the triangle the height also bisects the angle of the vertex - The triangle is isosceles.
3) If in the triangle the height is also the median - The triangle is isosceles.
4) If in the triangle the median is also the bisector - The triangle is isosceles.

Start practice

Test yourself on types of triangles!

einstein

Which triangle is given in the drawing?

90°90°90°AAABBBCCC

Practice more now

Identification of an Isosceles Triangle

Before we talk about how to identify an isosceles triangle, let's remember that it is a triangle with two sides (or edges) of the same length - This means that the base angles are also equal.
Moreover, in an isosceles triangle, the median of the base, the bisector, and the height are the same, that is, they coincide.

Let's see it illustrated

These magnificent properties of the isosceles triangle cannot prove by themselves that it is an isosceles triangle.
So, how can we prove that our triangle is isosceles?

If at least one of the following conditions is met:
1) If our triangle has two equal angles - The triangle is isosceles.
This derives from the fact that the sides opposite to equal angles are also equal, therefore, if the angles are equal, the sides are too.

2) If in the triangle the altitude also bisects the vertex angle - The triangle is isosceles.
3) If in the triangle the altitude is also the median - The triangle is isosceles.
4) If in the triangle the median is also the angle bisector - The triangle is isosceles.
In fact, we can summarize guidelines 2 2 and 4 4 and write a single condition:
If two of these coincide - the median, the altitude, and the bisector - The triangle is isosceles.

Great, now you know how to identify isosceles triangles easily and quickly.

If you are interested in learning more about other angle topics, you can enter one of the following articles:

  • Sum of the interior angles of a polygon
  • Angles in regular hexagons and octagons
  • Measure of an angle of a regular polygon
  • Sum of the exterior angles of a polygon
  • Exterior angle of a triangle
  • Relationships between angles and sides of the triangle
  • The relationship between the lengths of the sides of a triangle

In the blog of Tutorela you will find a variety of articles about mathematics.

Examples and exercises with solutions for identifying an isosceles triangle

examples.example_title

Given an equilateral triangle:

XXX

The perimeter of the triangle is 33 cm, what is the value of X?

examples.explanation_title

We know that in an equilateral triangle all sides are equal,

Therefore, if we know that one side is equal to X, all sides are equal to X.

We know that the perimeter of the triangle is 33.

The perimeter of the triangle is equal to the sum of the sides together.

We replace the data:

x+x+x=33 x+x+x=33

3x=33 3x=33

We divide the two sections by 3:

3x3=333 \frac{3x}{3}=\frac{33}{3}

x=11 x=11

examples.solution_title

11

examples.example_title

AAABBBCCCDDD

ABCD is a square with AC as its diagonal.

What kind of triangles are ABC and ACD?

(There may be more than one correct answer!)

examples.explanation_title

Since ABCD is a square, all its angles measure 90 degrees.

Therefore, angles D and B are equal to 90°, that is, they are right angles,

Therefore, the two triangles ABC and ADC are right triangles.

In a square all sides are equal, therefore:

AB=BC=CD=DA AB=BC=CD=DA

But the diagonal AC is not equal to them.

Therefore, the two previous triangles are isosceles:

AD=DC AD=DC

AB=BC AB=BC

examples.solution_title

Right triangles

examples.example_title

ABC is an isosceles triangle.

AD is the height of triangle ABC.

555333171717888AAABBBCCCDDDEEEFFFGGG

AF = 5

AB = 17
AG = 3

AD = 8

What is the perimeter of the trapezoid EFBC?

examples.explanation_title

To find the perimeter of the trapezoid, all its sides must be added:

We will focus on finding the bases.

To find GF we use the Pythagorean theorem: A2+B2=C2 A^2+B^2=C^2 in the triangle AFG

We replace

32+GF2=52 3^2+GF^2=5^2

We isolate GF and solve:

9+GF2=25 9+GF^2=25

GF2=25−9=16 GF^2=25-9=16

GF=4 GF=4

We perform the same process with the side DB of the triangle ABD:

82+DB2=172 8^2+DB^2=17^2

64+DB2=289 64+DB^2=289

DB2=289−64=225 DB^2=289-64=225

DB=15 DB=15

We start by finding FB:

FB=AB−AF=17−5=12 FB=AB-AF=17-5=12

Now we reveal EF and CB:

GF=GE=4 GF=GE=4

DB=DC=15 DB=DC=15

This is because in an isosceles triangle, the height divides the base into two equal parts so:

EF=GF×2=4×2=8 EF=GF\times2=4\times2=8

CB=DB×2=15×2=30 CB=DB\times2=15\times2=30

All that's left is to calculate:

30+8+12×2=30+8+24=62 30+8+12\times2=30+8+24=62

examples.solution_title

62

Join Over 30,000 Students Excelling in Math!
Endless Practice, Expert Guidance - Elevate Your Math Skills Today
Test your knowledge
Question 1

Which triangle is given in the drawing?

404040707070707070AAABBBCCC

Question 2

Which triangle is the following

393939107107107343434AAABBBCCC

Question 3

What is the triangle given in the drawing?

999555999AAABBBCCC

Start practice
Related Subjects
👉  Click here  to practice