The measures of three angles of a triangle form an AP. If the greatest angle is twice the smallest angle, find all the angles of the triangle.
Given:
The measures of three angles of a triangle form an AP.
The greatest angle is twice the smallest angle.
To do:
We have to find the angles of the triangle.
Solution:
Let the angles be $a−d,\ a,\ a+d$
According to the question,
$a+d=2(a−d)$
$\Rightarrow a+d=2a-2d$
$\Rightarrow 2a-a=2d+d$
$\Rightarrow a=3d\ .....( i)$
$\Rightarrow a−d+a+a+d=180^o$
$\Rightarrow 3a=180^o$
$\Rightarrow a=\frac{180^{o}}{3}=60^o$
This implies,
$\Rightarrow d=\frac{a}{3}=\frac{60^{o}}{3}=20^o$ [From $( i)\ d=\frac{a}{3}$]
$\Rightarrow a−d=60^o-20^o=40^o$
$\Rightarrow a=60^o$
$\Rightarrow a+d=60^o+20^o=80^o$
Hence, the angles of the triangle are $40^o$, $60^o$ and $80^o$.
Related Articles
- The angles of a triangle are in \( \mathrm{AP} \). The greatest angle is twice the least. Find all the angles of the triangle.
- The angles of a triangle are in A.P. The greatest angle is twice the least. Find all the angles of the triangle.
- The angles of a triangle are in A.P. The greatest angle is twice the least. Find all the angles.
- In an isosceles triangle, if the vertex angle is twice the sum of the base angles, calculate the angles of the triangle.
- The second angle of a triangle is 15° less than the first angle. If the third angle is 20° more than the second angle, find all three angles of the triangle.
- Find the measure of all the angles of a parallelogram, if one angle is $24^o$ less than twice the smallest angle.
- Two angles of a triangle are in ration 4:5. If the sum of these angle is equal to the angle, find the angles of the triangle
- One of the exterior angles of a triangle is 100°. The interior opposite angle is 75°. Find the measure of all the angles of the triangle?
- In a triangle $\triangle ABC, \angle A = 50^o, \angle B = 60^o$ and $\angle C = 70^o$. Find the measures of the angles of the triangle formed by joining the mid-points of the sides of this triangle.
- If the bisectors of the base angles of a triangle enclose an angle of $135^o$. Prove that the triangle is a right triangle.
- Two angles of a triangle are equal and the third angle is greater than each of those angles by $30^o$. Determine all the angles of the triangle.
- In an isosceles triangle, the vertex angle is twice either base angle. Find the angles, and represent the situation as an equation.
- The Vertical Angle Of An Isosceles Triangle Is 100°. Find It's Base Angles.
- In a $\triangle ABC, \angle C = 3 \angle B = 2(\angle A + \angle B)$. Find the three angles.
- The angles of a triangle are arranged in ascending order of magnitude. If the difference between two consecutive angle is $10^o$, find the three angles.
Kickstart Your Career
Get certified by completing the course
Get Started