Question 10

We will explore patterns in shapes. We will learn how to be sure.

Step 1 — Angles in a Triangle

We draw many different triangles. We measure their three inside angles. Let us call them Angle A, Angle B, Angle C. We add these angles together.

For example, let us take one triangle. Its angles are $\mathbf{60}$ degrees, $\mathbf{70}$ degrees, $\mathbf{50}$ degrees.

$60^\circ + 70^\circ + 50^\circ$

$= 130^\circ + 50^\circ$

$\boxed{180^\circ}$

We always find the sum is $\mathbf{180}$ degrees. This is a general law we observe.

Step 2 — Sides of a Triangle

We try to draw triangles. We use different side lengths. Let us try sides $\mathbf{3}$ cm, $\mathbf{4}$ cm, $\mathbf{5}$ cm. We can easily draw this triangle. Let us check the side rule.

Is $3 + 4$ greater than $5$? Yes, $\mathbf{7} > \mathbf{5}$. Is $3 + 5$ greater than $4$? Yes, $\mathbf{8} > \mathbf{4}$. Is $4 + 5$ greater than $3$? Yes, $\mathbf{9} > \mathbf{3}$.

Now let us try sides $\mathbf{2}$ cm, $\mathbf{3}$ cm, $\mathbf{6}$ cm. We try to draw this triangle. The two short sides are $\mathbf{2}$ cm and $\mathbf{3}$ cm. Their sum is $\mathbf{2} + \mathbf{3} = \mathbf{5}$ cm. The longest side is $\mathbf{6}$ cm. Is $\mathbf{5}$ cm greater than $\mathbf{6}$ cm? No. We cannot draw this triangle.

Two sides added together must be longer than the third. This is another general law we observe.

Step 3 — Why Angle Sum is Always True

We want to be sure about the angle sum. Let us draw any triangle on paper. We cut out this triangle. We tear off its three corner angles. We place the three corners next to each other. We arrange them along a straight line. Look at the figure below. They will always form a perfect straight line. A straight line has an angle of $\mathbf{180}$ degrees. This shows the angle sum is always $\mathbf{180}$ degrees. This demonstration works for any triangle.

Step 4 — Why Side Rule is Always True

We want to be sure about the side rule. Imagine we have three sticks. Let their lengths be $\mathbf{A}$, $\mathbf{B}$, $\mathbf{C}$. We want to make a triangle with them. We lay the longest stick, say $\mathbf{C}$, flat. We try to connect stick $\mathbf{A}$ and stick $\mathbf{B}$ to its ends. If $\mathbf{A} + \mathbf{B}$ is shorter than $\mathbf{C}$. Then sticks $\mathbf{A}$ and $\mathbf{B}$ will not meet. They cannot form the third corner. So, no triangle can be made. This shows $\mathbf{A} + \mathbf{B}$ must be more than $\mathbf{C}$. This logic applies to any pair of sides. This makes us sure the rule is always true.

Answer

(i) We observed two general laws for triangles:

1. Angle Sum Law: When we add the three angles inside any triangle, we always get $\mathbf{180}$ degrees.
2. Side Length Law (Triangle Inequality): The sum of the lengths of any two sides of a triangle is always greater than the length of the third side.

(ii) We can be sure these laws are always true by:

1. For Angle Sum Law: We can cut out any triangle. We tear off its three corners. We place them next to each other. They always form a straight line. A straight line is $\mathbf{180}$ degrees. This shows the rule is always true.
2. For Side Length Law: Imagine trying to make a triangle with three sticks. If two shorter sticks do not add up to more than the longest stick, they will not meet. They cannot form a triangle. This shows the rule must always be true.