Question

The measure of two angles ofa triangle is in the ratio 4:5 .if the sum of these two measure is equal to the measure of third angle. find the measure of all angles​

Answer 1

Given :

• Ratio of measure of two angles of a triangle = 4 : 5

• Sum of measure of two angles is equal to the measure of third angle.

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To find :

• Measure of all the three angles

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Concept :

In this question we are asked to find the measure of three angles in the triangle. Following the criterias given in the question we would form the equation. Solving those equation we would get the value of each angles.

We need to use angle sum property of the triangle here which is as follows :

$\sf\color{aqua}{\dag\:Sum~of~angles~in~a~triangle~=~180°}$

Hope am clear let's solve :D~

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Assumption :

Let the common term in the ratio be x.

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Solution :

Form Criteria 1 -

Ratio of measure of two angles of a triangle = 4 : 5

Therefore,

• $\tt\:1^{st}~angle~=~4x$

• $\tt\:2^{nd}~angle~=~5x$

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Form Criteria 2 -

Sum of measure of two angles is equal to the measure of third angle.

$\tt\:1^{st}~angle~+~2^{nd}~angle~=~3^{rd}~angle$

$\tt\implies\:4x+~5x=~3^{rd}~angle$

$\tt\implies\:9x=~~3^{rd}~angle$

$\tt\color{olive}{oR\:3^{rd}~angle~=~9x}$

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Now let's use the angle sum property -

$\tt\:1^{st}~angle~+~2^{nd}~angle~+~3^{rd}~angle~=~180°$

Plugging the values

$\tt\implies\:4x+5x+9x=180°$

Adding the like terms

$\tt\implies\:9x+9x=180°$

$\tt\implies\:18x=180°$

Transposing 18 to RHS it goes to the denominator

$\tt\implies\:x=\frac{180°}{18}$

Reducing the fraction

$\tt\implies\:x=\cancel{\frac{180°}{18}}$

$\small{\underline{\boxed{\mathrm{\implies\:x=10°}}}}$ ★

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Calculating the measure of all the angles by substituting x as 10° :

❒ $\sf\:1^{st}~angle~$ = 4x = 4 × 10 = $\large{\mathfrak\red{40~degree}}$

❒ $\sf\:2^{nd}~angle~$ = 5x = 5 × 10 = $\large{\mathfrak\red{50~degree}}$

❒ $\sf\:3^{rd}~angle~$ = 9x = 9 × 10 = $\large{\mathfrak\red{90~degree}}$

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Verification :

As we know that sum of angles in triangle is 180°.

Let's see by adding the values of angles if it is equal to 180° ‎ .

$\tt\:1^{st}~angle~+~2^{nd}~angle~+~3^{rd}~angle~=~180°$

Plugging the values

$\tt\dashrightarrow\:40°~+~50°~+~90°~=~180°$

$\tt\dashrightarrow\:90°~+~90°~=~180°$

$\tt\dashrightarrow\:180°~=~180°$

$\bf\dashrightarrow\:LHS~=~RHS°$

Hence Verified!~

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Henceforth,

The three angles are 40° , 50° , and 90° .

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$\sf\color{purple}{~~~~~~~~~~~~~~~~~~~~~~~~“~Thank~uH~\heartsuit~”~}$

Answer 2

Solution :

The measures of two angles in a triangle are in the ratio of 4:5. Let them be 4x and 5x respectively. Now, its given that the sum of these two angles is equal to the measure of the third angle. Hence, the third angle is 9x. Now, in a triangle, a angles add up to 180 degrees. Hence, 18x= 180 ; x = 10. The angles of the triangles are 40, 50 and 90 respectively. This is the required answer.

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