Question

Theorem:– the measure of an exterior angle of a triangle is equal to the sum of its remote interior angles​

Answer 1

$\huge\bf\mathcal\blue{Question }$

the measure of an exterior angle of a triangle is equal to the sum of its remote interior angles

$\huge\bf\mathcal\blue{Answer:--}$

$\pink{▬▬▬}$$\red{▬▬▬}$$\green{▬▬▬}$$\blue{▬▬▬}$$\orange{▬▬▬}$

What is Exterior Angle of a Triangle?

The exterior angle of a triangle is the angle formed between one side of a triangle and the extension of its adjacent side.

$\pink{▬▬▬}$$\red{▬▬▬}$$\green{▬▬▬}$$\blue{▬▬▬}$$\orange{▬▬▬}$

${\huge{\boxed{\tt{\color{red}{Explanation }}}}}$

${\huge{\boxed{\tt{\color{red}{°}}}}}$Triangle Exterior Angle Theorem?

${\huge{\boxed{\tt{\color{red}{> }}}}}$ The exterior angle theorem states that the measure of each exterior angle of a triangle is equal to the sum of the opposite and non-adjacent interior angles.

${\huge{\boxed{\tt{\color{red}{Example }}}}}$

For example, in triangle ABC above;

⇒ d = b + a

⇒ e = a + c

⇒ f = b + c

$\huge\bold\purple{Property\:of\:exterior\:angles }$

Properties of exterior angles

An exterior angle of a triangle is equal to the sum of the two opposite interior angles.

The sum of exterior angle and interior angle is equal to 180 degrees.

⇒ c + d = 180°

. ⇒ a + f = 180°

. ⇒ b + e = 180°

All exterior angles of a triangle add up to 360°.

$\huge\bold\purple{Proof:-}$

⇒ d + e + f = b + a + a + c + b + c

⇒ d +e + f = 2a + 2b + 2c

= 2(a + b + c)

But, according to triangle angle sum theorem,

a + b + c = 180 degrees

Therefore, ⇒ d +e + f = 2(180°)

= 360°

⇒ How to Find the Exterior Angles of a Triangle?

⇒ Rules to find the exterior angles of a triangle are pretty similar to the rules to find the interior angles of a triangle. It is because wherever there is an exterior angle, there exists an interior angle with it, and both of them add up to 180 degrees.

$\pink{▬▬▬}$$\red{▬▬▬}$$\green{▬▬▬}$$\blue{▬▬▬}$$\orange{▬▬▬}$

HOPE ITS HELP YOU

$\pink{▬▬▬}$$\red{▬▬▬}$$\green{▬▬▬}$$\blue{▬▬▬}$$\orange{▬▬▬}$

Answer 2

Answer:

Step 1:

Consider the triangle ABC.

We know that the sum of all the angles of a triangle adds up to 180°.

So, in ΔABC,

∠ABC + ∠BAC + ∠ACB = 180°

i.e., ∠B + ∠A + ∠C = 180° ...(i)

Step 2:

The side BC is extended further to point D. This creates ∠ACD exterior to ∠ACB.

The straight line BD has an angle of 180°. If we draw a traversal to this line, the two angles formed will sum up to 180°. Side AC is the traversal in this case.

Therefore,

∠ACB + ∠ACD = 180°

i.e., ∠C + ∠ACD = 180°

∠ACD = 180° - ∠C ...(ii)

Step 3:

Substituting value of ∠C from equation (i)

∠ACD = 180° - (180° - ∠A - ∠B)

∠ACD = 180° - 180° + ∠A + ∠B

∠ACD = ∠A + ∠B

Therefore, the measure of the exterior angle of a triangle is equal to the sum of its remote interior angles.

#SPJ2