# Equilateral and Isosceles Triangles

Have you seen a book placed on a table? Or a coat hanger? These are nothing but examples of isosceles triangles. Its names come from combining two Greek words Iso and Skelos meaning equal and legs respectively. Let us study in detail about the types of triangles, namely Equilateral triangle, Isosceles Triangle and Scalene Triangle.

## Equilateral Triangles

This is an Equilateral Triangle. In equilateral triangles all the sides are equal. So we can say that they have identical sides and identical angles. No matter what is the length of the sides, angles in equilateral triangles have a measure of 60° each. Since the sum of the angles in the triangle is 180°, therefore, each angle in the equilateral triangle must measure 60°.

As we know that in equilateral Triangle sides are of equal length, we only need the length of one side to calculate its perimeter. The altitude, median, angle bisector, and the perpendicular bisector of a given side are all the same line and is one of the three lines of symmetry of the triangle.

## Isosceles Triangles

The above is an isosceles triangle. Two sides of an isosceles triangle are equal which means two of its angles are also equal. No, matter where the apex or the peak points, it is still going to be an isosceles triangle. The hash mark in the figure denotes the congruency. Any sides or angles with the same number of hashes through them are congruent.  The base of the isosceles triangles is always shorter in length than the sides.

If we know the measure of either of the two angles that we can find the measure of the third angle. Since the angles of the triangle add up to 180°, then the third angle is 180 – 2 times the base angle.  ( 180 – 2b ). So the formula for this is 180 – 2b

If the isosceles triangle has a right angle, then such a triangle is called as a right isosceles triangle. A right isosceles triangle has a right angle and two 45° angles. Right angles are shown by a square at a line intersection rather than a curve.

## Scalene Triangle

When it comes to a scalene triangle, all side lengths have different measures. None of the sides is equal in length to any of the other sides in this case. In a scalene triangle, even the interior angles are all different.

## Solved Examples for You

Question 1. In the figure (not drawn to scale), ABC is an equilateral triangle and ABD is an isosceles triangle with DA = DB, find x

1. 14°
2. 16°
3. 12°
4. 32°

Answer : A. Since ABC is an equilateral triangle
∴ ∠CAB = ∠ ABC = ∠ BCA = 60°
and ∠ DBA = ∠ DAB∠ (60°- x)

In ΔDAB we have,
∠DBA + ∠ DAB + ∠ ADB = 180°
⇒ 2 (60° – x) + 88° = 180°
⇒ 2 (60° – x) = 92°
⇒ 60° – x = 46°
⇒ x = 14°

Question 2: In triangle PQR, QR = PR and ∠P = 36°. Which is the largest side of the traingle?

1. QR
2. PR
3. PQ

Answer : C. Since QR = PR, the angles opposite to them will be equal as well,
⇒  ∠P =  ∠Q =   36°

Since the sum of angles in a triangle =  180°
∠R = 180° –  36° – 36° = 180°

Out of the three angles, ∠R is largest. This means the side opposite to it, which is PQ will be the greatest side of the triangle.

Question 3: What are the angles in an isosceles triangle?

Answer: An isosceles triangle has two base angles and one other angle. The two base angles are said to be equal to each other. Thus, you will have an isosceles triangle when only two sides of that particular triangle will be equal to each other. Finally, you get 36 degrees as one of your base angles.

Question 4: How do you know if a triangle is isosceles?

Answer: An isosceles triangle consists of two equal sides (or three, theoretically) and two equal angles or three, technically. Legs are the equal sides and the base is the third side. The two angles that touch the base are referred to as base angles.

Question 5: Where is the Circumcenter of an isosceles triangle?

Answer: The circumcenter of an obtuse isosceles triangle is found outside the triangle and the perpendicular bisector passing through the obtuse angle of the triangle. Moreover, it is also the centre of the circle that the triangle is constrained inside of.

Question 6: What are equilateral triangles?

Answer: Equilateral triangles are ones having all equal sides. Thus, you can say that they consist of identical sides and identical angles. The length of the sides does not matter as the angles in equilateral triangles measure 60° each. In view of the fact that the sum of the angles in the triangle is 180°, consequently, each angle in the equilateral triangle will have to measure 60°.

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