Properties of Isosceles Triangles

An isosceles triangle is a triangle that has two sides of equal length. The properties of isosceles triangles are fundamental in geometry. The most important property is that the angles opposite the equal sides are also equal. This relationship provides a powerful tool for solving geometric problems involving triangles. This property also extends to its converse, meaning if two angles in a triangle are equal, the sides opposite those angles are also equal.

Formulae

Let’s denote an isosceles triangle as $\triangle ABC$, where $AB = AC$. The properties can be summarized using the following concepts:

• If $AB = AC$, then $\angle B = \angle C$.
• If $\angle B = \angle C$, then $AB = AC$.

Examples

Here are a couple of examples to illustrate the concept:

Example-1: Consider an isosceles triangle $\triangle XYZ$ where $XY = XZ$. If $\angle Y = 50^\circ$, find $\angle Z$.

Since $XY = XZ$, we know that $\angle Y = \angle Z$ (angles opposite equal sides are equal). Therefore, $\angle Z = 50^\circ$.

Example-2: In triangle $\triangle PQR$, $\angle P = 70^\circ$ and $\angle Q = 70^\circ$. Determine the relationship between the sides of the triangle.

Since $\angle P = \angle Q$, the sides opposite these angles are equal. Therefore, $QR = PR$. This means the triangle is isosceles.

Theorem with Proof

Theorem: The angles opposite equal sides of an isosceles triangle are equal.

Proof:

Let’s consider an isosceles triangle $\triangle ABC$ where $AB = AC$. We want to prove that $\angle B = \angle C$.

1. Draw the angle bisector of $\angle A$, and let it intersect $BC$ at point $D$.
2. Now, we have two triangles: $\triangle ABD$ and $\triangle ACD$.
3. In $\triangle ABD$ and $\triangle ACD$:
   • $AB = AC$ (Given)
   • $\angle BAD = \angle CAD$ (AD is the angle bisector)
   • $AD = AD$ (Common side)
4. Therefore, $\triangle ABD \cong \triangle ACD$ (By Side-Angle-Side (SAS) congruence).
5. Since the triangles are congruent, their corresponding angles are equal. Hence, $\angle B = \angle C$. This completes the proof.

Common mistakes by students

Students often make these mistakes when working with isosceles triangles:

• Assuming all angles are equal: Incorrectly assuming all angles of an isosceles triangle are equal. Remember, only *two* angles are equal.
• Confusing sides and angles: Not understanding which sides are equal based on the equal angles, or vice versa.
• Incorrect application of converse: Not recognizing when the converse of the property applies (i.e., when equal angles imply equal sides).

Real Life Application

The properties of isosceles triangles are found in real-world applications like:

• Architecture: The design of roofs, bridges, and building facades often incorporate isosceles triangles for stability and aesthetics.
• Engineering: Isosceles triangles are employed in trusses, frameworks, and structural support systems.
• Art and Design: Artists use isosceles triangles for geometric compositions.

Fun Fact

The word “isosceles” comes from the Greek words “isos” (equal) and “skelos” (leg). So, it literally means “equal legs.”

Recommended YouTube Videos for Deeper Understanding

Q.1 If in an isosceles triangle, one of the angles is $100^\circ$, then the other two angles are:

Check Solution

Ans: A

Since it’s an isosceles triangle, either the $100^\circ$ angle is the vertex angle or one of the base angles. If $100^\circ$ is a base angle, then the sum would exceed $180^\circ$. Therefore, $100^\circ$ is the vertex angle, and the other two angles must be equal, hence $(180 – 100)/2 = 40^\circ$.

Q.2 In $\triangle ABC$, $AB = AC$ and $\angle B = 70^\circ$. The measure of $\angle A$ is:

Check Solution

Ans: B

Since $AB = AC$, $\angle B = \angle C$. Therefore, $\angle C = 70^\circ$. $\angle A = 180^\circ – \angle B – \angle C = 180^\circ – 70^\circ – 70^\circ = 40^\circ$.

Q.3 If the angles of a triangle are in the ratio $2:5:5$, then the triangle is:

Check Solution

Ans: C

Let the angles be $2x, 5x, 5x$. Then $2x + 5x + 5x = 180^\circ \implies 12x = 180^\circ \implies x = 15^\circ$. The angles are $30^\circ, 75^\circ, 75^\circ$. Since two angles are equal, the triangle is isosceles.

Q.4 In an isosceles triangle, the angle opposite the unequal side is $50^\circ$. What are the other two angles?

Check Solution

Ans: A

The angle opposite the unequal side is the vertex angle. Since the sum of the angles is $180^\circ$, the sum of the two base angles is $180^\circ – 50^\circ = 130^\circ$. Since they are equal, each is $130^\circ/2 = 65^\circ$.

Q.5 $\triangle PQR$ is an isosceles triangle with $PQ=PR$. If $\angle Q = 65^\circ$, which of the following is true?

Check Solution

Ans: B

Since $PQ = PR$, the angles opposite to them must be equal. Therefore, $\angle Q = \angle R = 65^\circ$. Hence, $\angle P = 180^\circ – \angle Q – \angle R = 180^\circ – 65^\circ – 65^\circ = 50^\circ$.

Next Topic: Triangle Inequalities