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The word ‘isosceles’ is derived from the Greek words ‘iso’ meaning ‘same’ or ‘equal’ and ‘skeles’ meaning ‘leg’. Thus in geometry, the word ‘isosceles’ is used for figures(or shapes) having equal sides. An isosceles triangle is a triangle having two equal sides.
Let’s understand what is an isosceles triangle and the properties of an isosceles triangle with examples.
What is an Isosceles Triangle?
An isosceles triangle is a triangle that has two congruent or equal sides. Also, in an isosceles triangle, the two angles opposite the two equal sides are equal.
In the above figure, $\triangle \text{ABC}$ is an isosceles triangle, where $\text{AB} = \text{AC}$. Also, the angles opposite to equal sides are equal, i.e., $\angle \text{B} = \angle \text{C}$.
Properties of Isosceles Triangle
The following are the properties of isosceles triangle, that help in distinguishing it from other types of triangles.
- In an isosceles triangle, any of the two sides is equal
- The two equal sides of an isosceles triangle are called its legs
- The angle between the legs of an isosceles triangle is called the vertex angle or apex angle
- Angles opposite to equal sides are equal in an isosceles triangle
- The equal angles are called the base angles
- The perpendicular drawn from the vertex angle bisects the base and the vertex angle
- The perpendicular drawn from the vertex angle divides the isosceles triangle into two congruent triangles
Isosceles Triangle Theorem
In an isosceles triangle, angles opposite to the two sides are also congruent.
Let $\triangle \text{ABC}$ be an isosceles triangle, such that $\text{AB} = \text{AC}$, then $\angle \text{B} = \angle \text{B}$.
Let’s prove the above statement. For this, a small construction is required. Draw a perpendicular from the vertex angle $\text{A}$ on the opposite side $\text{BC}$, meeting at a point $\text{D}$.
In $\triangle \text{ABD}$ and $\triangle \text{ADC}$
$\text{AB} = \text{AC}$ (Given)
$\text{AD} = \text{AD}$ (Common)
$\angle \text{ADB} = \angle \text{ADC} = 90^{\circ}$ (By construction, $\text{AD} \perp \text{BC}$)
Therefore, $\triangle \text{ABD} \cong \triangle \text{ADC}$
Thus, $\angle \text{B} = \angle \text{C}$
Perpendicular Drawn From the Vertex Angle Divides the Isosceles Triangle Into Two Congruent Triangles
In an isosceles triangle, the perpendicular drawn from the vertex angle divides the isosceles triangle into two congruent triangles.
Let $\triangle \text{ABC}$ be an isosceles triangle, such that $\text{AB} = \text{AC}$, and $\text{AD} \perp \text{BC}$, then $\triangle \text{ABD} \cong \triangle \text{ADC}$.
Let’s prove the above statement.
In $\triangle \text{ABD}$ and $\triangle \text{ADC}$
$\text{AB} = \text{AC}$ (Given)
$\text{AD} = \text{AD}$ (Common)
$\angle \text{ADB} = \angle \text{ADC} = 90^{\circ}$ (By construction, $\text{AD} \perp \text{BC}$)
Therefore, $\triangle \text{ABD} \cong \triangle \text{ADC}$.
Perpendicular Drawn From the Vertex Angle Bisects the Base and the Vertex Angle
In an isosceles triangle, the perpendicular drawn from the vertex angle bisects the base and the vertex angle.
Let $\triangle \text{ABC}$ be an isosceles triangle, such that $\text{AB} = \text{AC}$, and $\text{AD} \perp \text{BC}$, then $\text{BD} = \text{DC}$ and $\angle \text{BAD} = \angle \text{CAD}$
Let’s prove the above statement.
In $\triangle \text{ABD}$ and $\triangle \text{ADC}$
$\text{AB} = \text{AC}$ (Given)
$\text{AD} = \text{AD}$ (Common)
$\angle \text{ADB} = \angle \text{ADC} = 90^{\circ}$ (By construction, $\text{AD} \perp \text{BC}$)
Therefore, $\triangle \text{ABD} \cong \triangle \text{ADC}$.
Thus, $\text{BD} = \text{BD}$ and $\angle \text{BAD} = \angle \text{CAD}$ (Corresponding Parts of Congruent Triangles)
Practice Problems
- What is meant by an isosceles triangle?
- Prove that angles opposite to equal sides of an isosceles triangle are equal.
- Prove that in an isosceles triangle, the perpendicular drawn from the vertex angle bisects the base and the vertex angle.
- Prove that in an isosceles triangle, the perpendicular drawn from the vertex angle divides the isosceles triangle into two congruent triangles.
FAQs
What is an isosceles triangle?
An isosceles triangle is a triangle that has two congruent or equal sides. Also, in an isosceles triangle, the two angles opposite the two equal sides are equal. The perpendicular drawn from the vertex angle divides the isosceles triangle into two congruent triangles.
What are the 3 properties of an isosceles triangle?
The three important properties of an isosceles triangle are
a) Any of the two sides is equal
b) Angles opposite to equal sides are equal
c) The perpendicular drawn from the vertex angle bisects the base and the vertex angle
Can an isosceles triangle be a right triangle?
Yes, a triangle can be an isosceles right triangle. In such a triangle, the two equal angles will be of $45^{\circ}$ each.
Conclusion
An isosceles triangle is a triangle that has two congruent or equal sides. Also, in an isosceles triangle, the two angles opposite the two equal sides are equal.
Recommended Reading
- How to Construct a Rectangle Using Compass and Ruler – Methods, Steps & Examples
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