Question

Q50 The measures of angles of a triangle if their measures are proportional to 4, 5 and 6
are
A) 40°, 50°, 60°
B) 30°, 40º, 110°
C) 48, 60°, 72°
D) 500, 600, 700°
E) None of these

Answer 1

Answer:

Given :-

• A triangle if their measures are proportional to 4, 5 and 6.

To Find :-

• What is the measures of angles of a triangle.

Solution :-

Let,

$\mapsto \bf First\: Angle_{(Triangle)} =\: 4x$

$\mapsto \bf Second\: Angle_{(Triangle)} =\: 5x$

$\mapsto \bf Third\: Angle_{(Triangle)} =\: 6x$

As we know that :

$\small \bigstar \: \: \sf\boxed{\bold{Sum\: of\: all\: angles_{(Triangle)} =\: 180^{\circ}}}\: \: \: \bigstar$

According to the question by using the formula we get,

$\implies \bf 4x + 5x + 6x =\: 180^{\circ}$

$\implies \sf 9x + 6x =\: 180^{\circ}$

$\implies \sf 15x =\: 180^{\circ}$

$\implies \sf x =\: \dfrac{180^{\circ}}{15}$

$\implies \sf\bold{x =\: 12^{\circ}}$

Hence, the required measures of angles of a triangle are :

$\dag$ First Angle Of Triangle :

$\small \dashrightarrow \sf First\: Angle_{(Triangle)} =\: 4x =\: (4 \times 12^{\circ}) =\: \bf 48^{\circ}$

$\dag$ Second Angle Of Triangle :

$\small \dashrightarrow \sf Second\: Angle_{(Triangle)} =\: 5x =\: (5 \times 12^{\circ}) =\: \bf 60^{\circ}$

$\dag$ Third Angle Of Triangle :

$\small \dashrightarrow \sf Third\: Angle_{(Triangle)} =\: 6x =\: (6 \times 12^{\circ}) =\: \bf 72^{\circ}$

$\therefore$ The measures of angles of a triangle are 48° , 60° and 72° .

Hence, the correct options is option no (C) 48°, 60°, 72° .

Answer 2

Given:-

• 3 sides of angles which are proportional in ratio 4,5,6.

To find:-

• Measures of angles

Solution:-

Let the number be x for all.

Hence,

Measure of 1st angle = 4x

Measure of 2nd angle = 5x

Measure of 3rd angle = 6x

We know that the sum of interior angles of a triangle are 180° . So we can add all the ratio to get their measure by Cross-multiplication.

So,

$\large \green{ \underline{ \blue{ \boxed{\bf{ \red{ \implies a + b + c = 180 \degree}}}}}}$

Where,

• a = first angle
• b = second angle
• c = Third angle

Substituting values we get:-

$\large\underline{\sf{ \implies 4x + 5x + 6x = 180 \degree}}$

$\large\underline{\sf{ \implies9x + 6x = 180 \degree}}$

$\large\underline{\sf{ \implies 15x = 180 \degree}}$

$\large\underline{\sf{ \implies x = \frac{180 \degree}{15} }}$

$\large\underline{\sf{ \implies x = 12 \degree}}$

Calculating for angles:-

$\large \blue{\underline{ \green{\boxed{\sf{ \maltese \: \: \: \: \red{4x = 4 \times 12 \degree = 48 \degree}}}}}}$

$\large \blue{\underline{ \green{\boxed{\sf{ \maltese \: \: \: \: \: \red{5x = 5 \times 12 \degree = 60 \degree}}}}}}$

$\large \blue{\underline{ \green{\boxed{\sf \sf{{\maltese \red{ \: \: \: \: \: 6x = 6 \times 12 \degree = 72 \degree}}}}}}}$

______________________________________

Henceforth,Option(C) is correct and required answer.