Aptitude Discussion

Q. |
The difference between two angles of a triangle is $24^circ$. The average of the same two angles is $54^circ$ .Which one of the following is the value of the greatest angle of the triangle? |

✖ A. |
$45^\circ$ |

✖ B. |
$60^\circ$ |

✖ C. |
$66^\circ$ |

✔ D. |
$72^\circ$ |

**Solution:**

Option(**D**) is correct

Let $a$ and $b$ be the two angles in the question, with $a > b$. We are given that the difference between the angles is $24^\circ$.

$\Rightarrow a – b = 24$.

Since the average of the two angles is $54^\circ$, we have \(\dfrac{(a+b)}{2}=54\)

Solving for $b$ in the first equation yields $b=a–24$, and substituting this into the second equation yields

\(\dfrac{a+(a+24)}{2}=54\)

\(\dfrac{2a-24}{2}=54\)

$2a−24=54×2$

$2a−24=108$

$2a=108+24$

$2a=132$

$a=66$

Also,

$b=a−24=66−24=42$.

Now, let $c$ be the third angle of the triangle. Since the sum of the angles in the triangle is $180^\circ$,

$a+b+c=180$.

Plugging the previous results into the equation yields $66+42+c=180$.

Solving for $c$ yields $c=72$

Hence, the greatest of the three angles $a, b$ and $c$ is $c$, which equals $72^\circ$.

**Edit:** For a quick alternative solution, check comment by **Sravan Reddy.**

**Venki**

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**Sravan**

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Sorry for typo. It's $180-108=72$ as Divya Pointed it out :)

**Divya**

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180-54 = 126 not 72 small correction its 180-108=72 (sum of the angles in the triangle-sum of two angles= third side)

**Sunil Singh**

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multiply 54 by 2 and 180-108 = 72 ans.

**Sravan Reddy**

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Killing it in 5 sec:

Average of two angles is 54 => Sum of two angles is 108 and hence 3rd angle is $180-54 = 72$.

In option's that's the largest value given so no need to check whether any other angle is bigger than this :)

for these probelm only these shortcut is ok ,but if answer >72 we must follow process