Can you construct a triangle all of whose angles are equal to ? If two of the angles are what would the third angle be? If all the angles in a triangle have to be equal, then what must its measure be? Explore and find out.

The sum of angles in any triangle is always . Step 1 — Checking for angles Let us assume a triangle has three angles. Let us call these angles , , and . The problem asks if all angles can be . So, we set each angle to . We need to find the sum of these angles. This sum is not equal to . So, a triangle with three angles cannot exist. > Step 2 — Finding the third angle Let us consider a triangle with two angles given. Let us call these angles and . The problem states that two angles are . So, we set and . Let the third angle be . We know the sum of angles in a triangle is . So, we can write the equation: Now, we substitute the known values into the equation. To find , we subtract from . The third angle would be . > Step 3 — Finding equal angles Let us assume all three angles in a triangle are equal. Let us call the measure of each equal angle . So, the three angles are , , and . We know the sum of angles in a triangle is . So, we can write the equation: We can simplify the left side of the equation. To find , we divide by . Each angle must measure . This type of triangle is called an equilateral triangle. > Answer > (i) No, a triangle all of whose angles are equal to cannot be constructed. > (ii) If two of the angles are , the third angle would be . > (iii) If all the angles in a triangle have to be equal, then its measure must be .

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Question 14

Can you construct a triangle all of whose angles are equal to $70^\circ$ ? If two of the angles are $70^\circ$ what would the third angle be? If all the angles in a triangle have to be equal, then what must its measure be? Explore and find out.

Solution

The sum of angles in any triangle is always $180^\circ$ .

Step 1 — Checking for $70^\circ$ angles

Let us assume a triangle has three angles. Let us call these angles $A$ , $B$ , and $C$ . The problem asks if all angles can be $70^\circ$ . So, we set each angle to $70^\circ$ . We need to find the sum of these angles.

$A + B + C$

$= 70^\circ + 70^\circ + 70^\circ$

$= 210^\circ$

This sum is not equal to $180^\circ$ . So, a triangle with three $70^\circ$ angles cannot exist.

$\boxed{\text{No, a triangle with three } 70^\circ \text{ angles cannot be constructed.}}$

Diagram 1

Step 2 — Finding the third angle

Let us consider a triangle with two angles given. Let us call these angles $A$ and $B$ . The problem states that two angles are $70^\circ$ . So, we set $A = 70^\circ$ and $B = 70^\circ$ . Let the third angle be $C$ . We know the sum of angles in a triangle is $180^\circ$ . So, we can write the equation:

$A + B + C = 180^\circ$

Now, we substitute the known values into the equation.

$70^\circ + 70^\circ + C = 180^\circ$

$140^\circ + C = 180^\circ$

To find $C$ , we subtract $140^\circ$ from $180^\circ$ .

$C = 180^\circ - 140^\circ$

$C = 40^\circ$

The third angle would be $40^\circ$ .

$\boxed{\text{The third angle would be } \mathbf{40^\circ}.}$

Diagram 2

Step 3 — Finding equal angles

Let us assume all three angles in a triangle are equal. Let us call the measure of each equal angle $x$ . So, the three angles are $x$ , $x$ , and $x$ . We know the sum of angles in a triangle is $180^\circ$ . So, we can write the equation:

$x + x + x = 180^\circ$

We can simplify the left side of the equation.

$3x = 180^\circ$

To find $x$ , we divide $180^\circ$ by $3$ .

$x = \frac{180^\circ}{3}$

$x = 60^\circ$

Each angle must measure $60^\circ$ . This type of triangle is called an equilateral triangle.

$\boxed{\text{Each angle must measure } \mathbf{60^\circ}.}$

Diagram 3

Answer

(i) No, a triangle all of whose angles are equal to $70^\circ$ cannot be constructed. (ii) If two of the angles are $70^\circ$ , the third angle would be $\mathbf{40^\circ}$ . (iii) If all the angles in a triangle have to be equal, then its measure must be $\mathbf{60^\circ}$ .

Question 14

Answer

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