Question

The consecutive angles of a trapezium form an arithmetic sequence. If the smallest angle is ${75^ \circ }$ , then the largest angle is
A) ${100^ \circ }$
B) ${105^ \circ }$
C) ${110^ \circ }$
D) ${115^ \circ }$

Answer 1

Hint: The sum of interior angles of a trapezium is 360. Given one angle is 75 and the consecutive angles are in A.P. which means there is a common difference between each consecutive angle. The first term is given which is 75 so there will be 4 terms in A.P. whose sum is 360. Using this, find the common difference and the largest angle.

Complete step-by-step answer:

We are given the consecutive angles of a trapezium are in Arithmetic Sequence and the smallest angle is ${75^ \circ }$.
This means the remaining angles are greater than ${75^ \circ }$
In an arithmetic sequence, every term has an addition of d in it from its previous term, and the difference between consecutive terms is called common difference d.
So the first angle here is $75$
The second angle will be $75 + d$
The third angle will be $75 + 2d$
The fourth angle will be $75 + 3d$
The sum of all angles of a trapezium is 360
$
\to 75 + 75 + d + 75 + 2d + 75 + 3d = 360 \\
75 + 75 + 75 + 75 + d + 2d + 3d = 360 \\
300 + 6d = 360 \\
 $
Send 300 to the right hand side
$
  6d = 360 - 300 \\
  6d = 60 \\
  d = \dfrac{{60}}{6} \\
  d = 10 \\
 $
Therefore, the difference between two successive angles is 10, which means every angle starting from 75 will increase by 10 degrees.
The largest angle is $75 + 3d$
Substitute the value of d in $75 + 3d$
$
  75 + 3\left( {10} \right) \\
   = 75 + 30 \\
   = 105 \\
 $
The measure of the largest angle of the given trapezium is ${105^ \circ }$
Therefore, from among the options given in the question Option B is correct, which is ${105^ \circ }$.
So, the correct answer is “Option B”.

Note: A trapezium is a quadrilateral with one pair of parallel sides and other two non-parallel sides. A trapezium has a perimeter and area just as another quadrilateral. Arithmetic sequence is a sequence of numbers such that the difference of any two successive members is a constant which is called common difference.