Question

Exterior angle, ACD of a triangle is 100° and
angle A and B are in the ratio 3 : 2. Find all the
angles of the triangle.​

Answer 1

$\huge\bf\underline\mathfrak\red{Answer :}$

• $\text{Angles of the ∆ABC = 80°, 40° and 60°.}$

$\huge\bf\underline\mathfrak\red{Step \: by \: step \: explanation :}$

$\huge\bf\underline\mathfrak\red{Given :}$

• $\text{ABC is a triangle.}$
• $\text{Exterior Angle, ∠ACD = 100°.}$
• $\text{∠A : ∠B = 3 : 2}$

$\huge\bf\underline\mathfrak\red{To \: find :}$

• $\text{Measure of all the angles of ∆ABC.}$

$\huge\bf\underline\mathfrak\red{Solution :}$

Given, ratio of ∠CAB : ∠ABC = 3 : 2

⠀⠀⠀⠀⠀⠀$\sf\text{Let ∠CAB be 3x.}$

⠀⠀⠀⠀⠀⠀$\sf\text{Let ∠ABC be 2x.}$

Now, $\text{100° = ∠CAB + ∠ABC (Exterior angle property)}$

⇒ $\sf\text{100° = 3x + 2x}$

⇒ $\sf\text{100° = 5x}$

⇒ $\sf\text{x = 20°}$

⠀⠀⠀⠀⠀⠀⠀⠀⠀∴ $\sf\purple{∠CAB = 3x = 60°}$

⠀⠀⠀⠀⠀ ⠀ And, $\sf\purple{∠ABC = 2x = 40°}$

Now,

$\sf\text{∠CAB + ∠ABC + ∠ACB = 180° ( Angle sum property )}$

$\sf\text{40° + 60° + ∠ACB = 180°}$

⇒ $\sf\text{100° + ∠ACB = 180°}$

⇒ $\sf\purple{∠ACB = 180°-100° = 80°}$

Hence,

• $\sf\red{∠CAB = 60°}$
• $\sf\red{∠ABC = 40°}$, and
• $\sf\red{∠ACB = 80°}$

$\huge\bf\underline\mathfrak{Learn \: More :}$

1. A triangle is considered as a close polygon that has three sides and three angles whose sum makes up 180°.
2. Triangle can be divided into following types :-

• Equilateral Triangle - It is a triangle having all the three sides and the corresponding three angles as equal.
• Isosceles Triangle - It is a triangle in which a pair of sides is considered equal and the angles that lies opposite to the equal sides are said to be equal in measure.
• Scelene Triangle - It a triangle having no pair of equal sides as well as equal angles.
• Right Angles Triangle - It is triangle in which one angle has a measure of 90° and it also obeys the Pythagoras theorem.⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀

Answer 2

★ How to do :-

Here, we are given with one of the exterior angle of a triangle. We are also given with ratio of other two angles of the same triangle namely A and B. Here, we are going to use some other concepts such as a straight line angle always measures 180°. All the interior angles of any triangle always measures 180° altogether. So, first we'll use the first concept given to find the third angle of the triangle. Then, we use the second concept and the ratio given and then we'll find other two angles. So, let's solve!!

$\:$

➤ Solution :-

Value of ∠x :-

${\sf \: \: \: \: \: \: \underline{\boxed{\sf Straight \: line \: angle = {180}^{\circ}}}}$

Apply the values here.

${\tt \leadsto {100}^{\circ} + x = {180}^{\circ}}$

Shift the number 100 from LHS to RHS, changing it's sign.

${\tt \leadsto x = 180 - 100}$

Subtract the values to get the value of x.

${\tt \leadsto x = {80}^{\circ}}$

$\:$

Now, we can find the value of other two angles.

${\sf \: \: \: \underline{\boxed{\sf {Sum \: of \: all \: angles}_{(Triangle)} = {180}^{\circ}}}}$

Substitute the values.

${\tt \leadsto 3 : 2 + 80 = 180}$

Shift the number 80 from LHS to RHS, changing it's sign.

${\tt \leadsto 3 : 2 = 180 - 80}$

Subtract the values on RHS.

${\tt \leadsto 3 : 2 = 100}$

Take a variable y on both part of ratio.

${\tt \leadsto 3y + 2y = 100}$

Add both values having same variable on LHS.

${\tt \leadsto 5y = 100}$

Shift the number 5 from LHS to RHS, changing it's sign.

${\tt \leadsto y = \dfrac{100}{5}}$

Simplify the fraction to get the value of y.

${\tt \leadsto y = 20}$

$\:$

Value of ∠A :-

${\tt \leadsto 3y = 3 \times 20}$

${\tt \leadsto \pink{\underline{\boxed{\tt \angle{A} = {60}^{\circ}}}}}$

Value of ∠B :-

${\tt \leadsto 2y = 2 \times 20}$

${\tt \leadsto \pink{\underline{\boxed{\tt \angle{B} = {40}^{\circ}}}}}$

$\:$

Hence solved !!