Question

Find the possible pair(s) of interior angles of an obtuse-angled triangle whose third angle is ${75}^o.$

• A. ${65}^o$ and ${40}^o$
• B. ${91}^o$ and ${14}^o$
• C. ${99.5}^o$ and ${5.5}^o$
• D. ${105}^o$ and $0^o$

Answer 1

Answer: (B), (C)

${99.5}^o$ and ${5.5}^o$

Let the three angles of the obtuse-angled triangle be $\angle a,\angle b$ and $\angle c.$

We know that the sum of the angles of a triangle is ${180}^o.$
$\therefore \angle a+\angle b+\angle c={180}^o$
$\Rightarrow \angle a+\angle b={180}^o+\angle c$

It is given that $\angle c={75}^o$
$\therefore \angle a+\angle b={180}^o-{75}^o={105}^o$

Given that the triangle is obtuse angled triangle, hence either $\angle a$ or $\angle b$ must be greater than ${90}^o$

Now check each given pair of angles.
Option(a)
${65}^o+{40}^o={105}^o$
Hence, this is not a possible pair because all three angles are less than ${90}^o$

Option(b)
${91}^o+{14}^o={105}^o$
Hence, this is a possible pair because one angle is greater than ${90}^o$

Option(c)
${99.5}^o+{5.5}^o={105}^o$
Hence, this is a possible pair because one angle is greater than ${90}^o$

Option(d)
But the pair ${105}^o$ and $0^o$ do not correspond to a triangle as one of the angles is $0^o.$
$\therefore$ This is not a valid option.