MCQ
In a $\triangle\text{ABC},\ \angle\text{C}=3\angle\text{B}=2(\angle\text{A}+\angle\text{B}),$ then $\angle\text{B}=?$
- A$20^\circ$
- ✓$40^\circ$
- C$60^\circ$
- D$80^\circ$
✓
Answer
Correct option: B.
$40^\circ$
Give that in a $\triangle\text{ABC},$
$\angle\text{C}=3\angle\text{B}=2(\angle\text{A}+\angle\text{B})$
$\Rightarrow\angle\text{C}=3\angle\text{B}$ and $\angle\text{C}=2(\angle\text{A}+\angle\text{B})$
Consider, $\angle\text{C}=2(\angle\text{A}+\angle\text{B})$
$\Rightarrow3\angle\text{B}=2(\angle\text{A}+\angle\text{B})$
$\Rightarrow\angle\text{B}=2\angle\text{A}$
By the Angle Sum Property
$\angle\text{A}+\angle\text{B}+\angle\text{C}=180^\circ$
$\Rightarrow\angle\text{A}+2\angle\text{A}+2(\angle\text{A}+2\angle\text{A})=180^\circ$
$\Rightarrow\angle\text{A}+2\angle\text{A}+2\angle\text{A}+4\angle\text{A}=180^\circ$
$\Rightarrow9\angle\text{A}=180^\circ$
$\Rightarrow\angle\text{A}=20^\circ$
So, $\angle\text{B}=2\angle\text{A}$
$\Rightarrow\angle\text{B}=40^\circ$
$\angle\text{C}=3\angle\text{B}=2(\angle\text{A}+\angle\text{B})$
$\Rightarrow\angle\text{C}=3\angle\text{B}$ and $\angle\text{C}=2(\angle\text{A}+\angle\text{B})$
Consider, $\angle\text{C}=2(\angle\text{A}+\angle\text{B})$
$\Rightarrow3\angle\text{B}=2(\angle\text{A}+\angle\text{B})$
$\Rightarrow\angle\text{B}=2\angle\text{A}$
By the Angle Sum Property
$\angle\text{A}+\angle\text{B}+\angle\text{C}=180^\circ$
$\Rightarrow\angle\text{A}+2\angle\text{A}+2(\angle\text{A}+2\angle\text{A})=180^\circ$
$\Rightarrow\angle\text{A}+2\angle\text{A}+2\angle\text{A}+4\angle\text{A}=180^\circ$
$\Rightarrow9\angle\text{A}=180^\circ$
$\Rightarrow\angle\text{A}=20^\circ$
So, $\angle\text{B}=2\angle\text{A}$
$\Rightarrow\angle\text{B}=40^\circ$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
The sum of first $40$ positive integers divisible by $6$ is:
→The distance of the point $(-3, 4)$ from $x-$axis is:
→A cone is cut by a plane parallel to its base and the upper part is removed. The part that is left over is called:
If $A$ and $B$ are complementary angles, then:
→If one zero of $3x^2 + 8x + k$ be the reciprocal of the other, then $k = ?$
→If $ax^2 + bx + c = 0$ has equal roots, then $c =$
→If the circumference of a circle and the perimeter of a square are equal, then:
Find the perimeter of a sector of a circle if its measure is 90° and radius is 7 cm.
If $\sin\theta=\frac{\sqrt3}{2}$ then $(\text{cosec }\theta+\cot\theta)=?$
→Out of given triplets, which is not a Pythagoras triplet ?