Question

an angle is equal to four times it's complement determine its measure. Step by step explanation

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Answer 1

$\bf{\underline{Given :}}$ An angle is equal to four times it's complement.

$\bf{\underline{To \: Find :}}$ What is the angle?

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❍ Let us assume, that its supplement and the angle is 'x' and '4x' respectively.

$\mathbf{ \dag \: As \: we \: know,}$

If the sum of two angles is 90°, then one angle is complement of another.

Therefore,

$\tt{ \dashrightarrow \: x + 4x° = 90°}$

$\sf{ \dashrightarrow \: 5x = 90°}$

$\sf{ \dashrightarrow \: x = \dfrac{ \cancel{90°}^{18} }{ \cancel{5}_{1} } }$

$\purple{\bf{ \dashrightarrow \: x = 18°}}$

∴ Its measure is (4 × 18)° = 72°.

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$\bf{\underline{Verification :}}$

❐ Substituting x = 18, we get

$\tt{ \Rightarrow \: (x + 4x)° = 90°}$

$\sf{ \Rightarrow \: [18 + 4(18)]° = 90°}$

$\sf{ \Rightarrow \; (18 + 72)° = 90°}$

$\red{\bf{ \Rightarrow \: 90° = 90°}}$

Hence Verified. $\checkmark$

Answer 2

Complete step-by-step answer:

We need to find the angle which is four times its complement. Thus, we will first assume the angle to be x. Let, the complement of this angle be y. Now, by definition of the complement of angle, that is the sum of angles that add up to 90 degrees. Thus, we have,

x + y = 90

y = 90 – x -- (1)

Further, we know that the angle is four times the complement, thus, we have,

x = 4y

From eqn (1), we have,

⇒ x = 4(90-x)

⇒ x = 360 – 4x

⇒ 5x = 360

⇒ x = 72

Hence, the angle which is four times its complement is 72 degrees.

Note: In general, when we are asked to find the angle which is n times its complement (here, n can be any value like 1,2,3,4 and so on). We divide 90 into n+1 parts equally (in the above problem, since n was 4, we divide 90 into 4+1=5 parts). Thus, each part is equivalent to 905 = 18. Now, we multiply any one part by n. Thus, in the above problem (for n=4), we have 18×4=72. Thus, we get the same answer as the above solution.