7.4. Some Properties of a Triangle

Construct a triangle in which two sides are equal, say each equal to 3.5 cm and the third side equal to 5 cm. Do you remember what is such a triangle called?

A triangle in which two sides are equal is called an isosceles triangle. So, ∆ ABC of Fig. 7.24 is an isosceles triangle with AB = AC.

An isosceles triangle is defined as a triangle with at least two sides that are equal in length. In this activity, you construct a triangle with two equal sides measuring 3.5 cm. The triangle resulting from this construction is referred to as an isosceles triangle, denoted as ∆ ABC, where AB is equal to AC.

Think of a pair of scissors; the two blades (or sides of the triangle) of a pair of scissors can be compared to the equal sides of an isosceles triangle. Both sides are the same length, allowing the scissors to function properly, just as the equal sides of an isosceles triangle ensure that the triangle holds its shape.

Now, measure ∠ B and ∠ C. What do you observe? Repeat this activity with other isosceles triangles with different sides. You may observe that in each such triangle, the angles opposite to the equal sides are equal. This is a very important result and is indeed true for any isosceles triangle.

In the triangle constructed, when you measure angles B and C (the angles opposite the equal sides AB and AC), you will find that they are equal. This observation holds for all isosceles triangles, demonstrating a key property: the angles opposite to the equal sides are always equal.

Imagine a swing that is perfectly balanced on a pivot. If you were to look at the lengths of the chains on either side (akin to the equal sides of our triangle), you'd see that they are the same length. Consequently, the angle that each chain makes with the horizontal is also the same, akin to the angles in an isosceles triangle being equal.

Theorem 7.2: Angles opposite to equal sides of an isosceles triangle are equal. This result can be proved in many ways. One of the proofs is given here. Proof: We are given an isosceles triangle ABC in which AB = AC. We need to prove that ∠ B = ∠ C.

The theorem states that in an isosceles triangle, the angles opposite to the equal sides are equal. To prove this, we could draw a bisector (AD) from the vertex angle to the base (BC) and show that the two smaller triangles formed, ∆ BAD and ∆ CAD, are congruent. Since they are congruent, their corresponding angles are equal, thus proving that ∠ B = ∠ C.

Think of a balance scale: if both sides (or equal sides of the triangle) weigh the same, then both items must be of equal weight and will balance out. Similarly, in our geometric triangle, if the sides are equal, the angles opposite those sides are also equal, just like the balance.

Is the converse also true? That is: If two angles of any triangle are equal, can we conclude that the sides opposite to them are also equal? Perform the following activity. Construct a triangle ABC with BC of any length and ∠ B = ∠ C = 50°. ...

The converse theorem states that if two angles of a triangle are equal, then the sides opposite those angles must also be equal. To establish this, you can construct a triangle where angles B and C are both 50°. When you fold the triangle along the angle bisector, you will see that the two sides (AC and AB) align perfectly. This shows that AC = AB, confirming the converse of the isosceles triangle theorem.

Consider a pair of identical tuning forks. If you strike one fork and it produces a certain pitch (representing the equal angles), the other tuning fork will produce the same pitch (representing the equal sides) if it's the same. The equality of angles ensures the equality of opposite sides.

Example 4: In ∆ ABC, the bisector AD of ∠ A is perpendicular to side BC. Show that AB = AC and ∆ ABC is isosceles. In ∆ ABD and ∆ ACD, ∠ BAD = ∠ CAD (Given)...

In this example, when the bisector AD of angle A intersects side BC perpendicularly, it splits ∆ ABC into two congruent triangles, ∆ ABD and ∆ ACD. Since we establish that the angles and common sides align, the triangles are congruent, which confirms that sides AB and AC must be equal. Hence, ∆ ABC is proven to be isosceles.

Think about a perfectly symmetrical bridge supported by two equal arches from a central point (like the angle A). If you know that the cables defining the angle are equal in strength (just as the angles are equally split), the arches will hold equal weight and remain balanced (the sides equal).

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