Question

if acute angle of right angle triangle are in ratio of 4:5 then sum of smallest and biggest angle is​

Answer 1

Answer:

130°

Step-by-step explanation:

As per the provided information in the given question, we have :

• Acute angles of right angle triangle are in ratio of 4:5.

We are asked to find the sum of smallest and biggest angle.

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Let us assume those acute angles as 4x and 5x respectively as they are in ratio, x is the constant natural number here.

Since, it is a right angled triangle, so measure one of its angles will be 90°.

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Finding the value of x :

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As we know that,

• Angles sum property of a ∆ states that the sum of all the interior angles of triangle is 180°. So,

$\longmapsto$ First angle + Second angle + Third angle = 180°

Here,

• First angles is 4x°
• Second angle is 5x°
• Third angle is 90°.

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Substituting values,

$\longrightarrow \sf {4x^\circ + 5x^\circ + 90^\circ = 180^\circ }$

$\longrightarrow \sf {9x^\circ + 90^\circ = 180^\circ }$

$\longrightarrow \sf {9x^\circ = 180^\circ - 90^\circ}$

$\longrightarrow \sf {9x^\circ = 90^\circ}$

$\longrightarrow \sf {x^\circ =\cancel{ \dfrac{ 90^\circ}{9}} }$

$\longrightarrow\boxed{ \sf {x^\circ = 10^\circ} }$

Therefore,

$\dashrightarrow \sf { First \: angle = 4x^\circ}$

$\dashrightarrow \sf { First \: angle = 4(10)^\circ}$

$\dashrightarrow \bf { First \: angle = 40^\circ}$

And,

$\dashrightarrow \sf { Second \: angle = 5x^\circ}$

$\dashrightarrow \sf { Second \: angle = 5(10)^\circ}$

$\dashrightarrow \bf { Second \: angle = 50^\circ}$

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And the third angle is 90°.

Clearly, the first angle is the smallest angle and the third angle is the biggest angle.

Therefore, sum of the biggest angle and the smallest angle :

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$\longrightarrow \sf{ Sum = First \: angle + Third \: angle}$

$\longrightarrow \sf{ Sum = 40^\circ + 90^\circ}$

$\longrightarrow \underline{\boxed{\sf{ Sum = 130^\circ}}} \: \: \bigstar$

Therefore, sum of smallest and biggest angle is 130°.