Properties of Triangles

In a triangle $A B C, a=(1+\sqrt{3}) \mathrm{cm}, b=2 \mathrm{~cm}$ and angle $C=60^{\circ}$. Then the other two angles are

Paragraph: $A B C$ is a triangular plot with $A B=16 \mathrm{~m}, B C=10 \mathrm{~m}$ and $C A=10 \mathrm{~m}$. A lamp post is situated at the middle point of the side $A B$. The lamp post subtends an angle $45^{\circ}$…

Consider the following statements : If in a triangle $A B C, A=2 B$ and $b=c$, then it must be an obtuse-angled triangle. There exists no triangle $A B C$ with $A=40^{\circ}, B=65^{\circ}$ and $\frac{a}{c}=\sin…

In a triangle $\mathrm{ABC}, a=8, b=10$ and $c=12$. What is angle C equal to?

What is the diameter of a circle inscribed in a regular polygon of 12 sides, each of length 1 cm ?

The sides of a triangle are $m, n$ and $\sqrt{m^2+n^2+m n}$. What is the sum of the acute angles of the triangle?

In triangle $\mathrm{ABC}, \mathrm{BC}=\sqrt{39}, \mathrm{AC}=5$ and $\mathrm{AB}=7$. What is the measure of the angle $A$ ?

If the angles of a triangle are $30^{\circ}$ and $45^{\circ}$ and the included side is $(\sqrt{3}+1)$, then what is the area of the triangle?

Paragraph: Read the following information and answer: $A B C D$ is a trapezium such that $A B$ and $C D$ are parallel and $B C$ is perpendicular to them. Let $\angle A D B=\theta, \angle A B D=\alpha, B C=p$ and $C D=q$…

If the sides of a triangle are in the ratio $2: \sqrt{6}: 1+\sqrt{3}$, then what is the smallest angle of the triangle?

ABC is a triangle inscribed in a circle with centre O . Let $\alpha=\angle \mathrm{BAC}$, where $45^{\circ} \lt \alpha \lt 90^{\circ}$. Let $\beta=\angle \mathrm{BOC}$. Which one of the following is correct?

Consider the following statements : If $A B C$ is a right-angled triangle, right-angled at $A$ and if $\sin B=\frac{1}{3}$, then $\operatorname{cosec} C=3$. If $b \cos B=c \cos C$ and if the triangle $A B C$ is not…