Can two sides of a triangle be equal to the third

Can two sides of a triangle be equal to the third
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Determining if three side lengths can make a triangle is easier than it looks. All you have to do is use the Triangle Inequality Theorem, which states that the sum of two side lengths of a triangle is always greater than the third side. If this is true for all three combinations of added side lengths, then you will have a triangle.[1] X Research source Go to source

  1. 1

    Learn the Triangle Inequality Theorem. This theorem simply states that the sum of two sides of a triangle must be greater than the third side. If this is true for all three combinations, then you will have a valid triangle. You'll have to go through these combinations one by one to make sure that the triangle is possible. You can also think of the triangle as having the side lengths a, b, and c and the theorem being an inequality, which states: a+b > c, a+c > b, and b+c > a.[2] X Research source Go to source

    • For this example, a = 7, b = 10, and c = 5.

  2. 2

    Check to see if the sum of the first two sides is greater than the third. In this case, you can add the sides a and b, or 7 + 10, to get 17, which is greater than 5. You can also think of it as 17 > 5.[3] X Research source Go to source

  3. 3

    Check to see if the sum of the next combination of two sides is greater than the remaining side. Now, just see if the sum of sides a and c are greater than the side b.[4] X Research source Go to source This means you should see if 7 + 5, or 12, is greater than 10. 12 > 10, so it is.

  4. 4

    Check to see if the sum of the last combination of two sides is greater than the remaining side. You need to see if the sum of side b and side c is greater than side a. To do this, you'll need to see if 10 + 5 is greater than 7. 10 + 5 = 15, and 15 > 7, so the triangle passes on all sides.[5] X Research source Go to source

  5. 5

    Check your work. Now that you've checked the side combinations one by one, you can double check that the rule is true for all three combinations. If the sum of any two side lengths is greater than the third in every combination, as it is for this triangle, then you've determined that the triangle is valid. If the rule is invalid for even just one combination, then the triangle is invalid. Since the following statements are true, you've found a valid triangle:[6] X Research source Go to source

    • a + b > c = 17 > 5
    • a + c > b = 12 > 10
    • b + c > a = 15 > 7

  6. 6

    Know how to spot an invalid triangle. Just for practice, you should make sure you can spot a triangle that doesn't work as well.[7] X Research source Go to source Let's say you're working with these three side lengths: 5, 8, and 3. Let's see if it passes the test:

    • 5 + 8 > 3 = 13 > 3, so one side passes.
    • 5 + 3 > 8 = 8 > 8. Since this is invalid, you can stop right here. This triangle is not valid.

  • Question

    What if the sides are equal?

    Then you have an equilateral triangle.

  • Question

    Can three equal side lengths form a triangle?

    Yes. It's called an equilateral triangle, and it can work because two side lengths added together are bigger than the third side.

  • Question

    What if the addition of the two sides is the same as the other side?

    The resulting figure is not a triangle, because the two smaller sides must be on top of the larger side in order to connect to the larger segment's endpoints. This figure has no area, and is a line segment, not a triangle.

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Consider two triangles $\triangle abc$ and $\triangle def$ such that $ab=de$ and $ac=df$.Also area of $\triangle abc$ is equal to area of $\triangle def$.Now draw $cm$ perpendicular to $ab$ and $fn$ perpendicular to $de$.$ab$ and $de$ are equal and area of triangles is also equal so $cm$ should be equal to $fn$.Now $\triangle amc$ and$\triangle dnf$ are congruent by right angle triangle congruence(since hypoteneous $ac$ and $df$ are equal).therefore $\angle bac$ is equal to $\angle edf$.Now in $\triangle abc$ and $\triangle def$ by SAS both $\triangle abc$ and $\triangle def$ are congruent so $bc=ef$.I don't know where i am wrong.

edit:Please read my proof and point out what's wrong



Triangle is a closed figure which is formed by three line segments. It consists of three angles and three vertices. The angles of triangles can be the same or different depending on the type of triangle. There are different types of triangles based on line and angles properties.

Properties of a Triangle:

1. Each triangle has 3 sides and 3 angles.

2. Sum of all the angles of triangles is 180°

3. Perimeter of a triangle is the sum of all three sides of the triangle.

4. A triangle has 3 vertices.

Types of Triangles based on line Properties

Scalene Triangle: Scalene Triangle is a type of triangle in which all the sides are of different lengths. All the angles of a scalene triangle are different from one another.

Isosceles Triangle: Isosceles Triangle is another type of triangle in which two sides are equal and the third side is unequal. In this triangle, the two angles are also equal and the third angle is different.

Right-angled Triangle: A right-angled triangle is one that follows the Pythagoras Theorem and one angle of such triangles is 90 degrees which is formed by the base and perpendicular. The hypotenuse is the longest side in such triangles.

Equilateral Triangle: An equilateral triangle is a triangle in which all the three sides are of equal size and all the angles of such triangles are also equal.

Finding Third Side of a Triangle given Two Sides

Lets assume that the triangle is Right Angled Triangle because to find a third side provided two sides are given is only possible in a right angled triangle.

We know that the right-angled triangle follows Pythagoras Theorem

According to Pythagoras Theorem, the sum of squares of two sides is equal to the square of the third side. 

(Perpendicular)2 + (Base)2 = (Hypotenuse)2 

Using the above equation third side can be calculated if two sides are known.

Example: Suppose two sides are given one of 3 cm and the other of 4 cm then find the third side.

Lets take perpendicular P = 3 cm and Base B = 4 cm.

using Pythagoras theorem 

P2 + B2 = H2

(3)2 + (4)2 = H2

9 + 16 = H2

25 = H2

H = 5

Sample Questions

Question 1: Find the measure of base if perpendicular and hypotenuse is given, perpendicular = 12 cm and hypotenuse = 13 cm.


Perpendicular = 12 cm

Hypotenuse = 13 cm

Using Pythagoras Theorem 

P2 + B2 = H2

B2 = H2 – P2

B2 = 132 – 122

B2 = 169 – 144

B2 = 25

B = 5

Question 2: Perimeter of the equilateral triangle is 63 cm find the side of the triangle.


Perimeter of an equilateral triangle =  3×side

3×side = 64

side = 63/3

side = 21 cm

Question 3: Find the measure of the third side of a right-angled triangle if the two sides are 6 cm and 8 cm.


Perpendicular = 6 cm

Base = 8 cm

Using Pythagoras Theorem

H2 = P2 + B2 

H2 = P2 + B2

H2 = 62 + 82 

H2 = 36 + 64

H2 = 100

H = 10 cm

Question 4: Find whether the given triangle is a right-angled triangle or not, sides are 48, 55, 73?


A right-angled triangle follows the Pythagorean theorem so we need to check it .

Sum of squares of two small sides should be equal to the square of the longest side

so 482 + 552 must be equal to 732

2304 + 3025 = 5329 which is equal to 732 = 5329

Hence the given triangle is a right-angled triangle because it is satisfying the Pythagorean theorem.

Question 5: Find the hypotenuse of a right angled triangle whose base is 8 cm and whose height is 15 cm?


Using Pythagorean theorem, a2 + b2 = c2

So 82 + 152 = c2  

hence c = √(64 + 225)

          c = √289

          c = 17 cm

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