Question

In a trapezium \[abcd\], \[ab||cd\] and \[ab\] is a diameter. Angle \[cab\] is \[{28^ \circ }\]. Find the difference between angle \[adc\] and angle \[abc\].

Answer 1

Hint: As we know that angle subtended by a diameter on any point of the circle is \[{90^ \circ }\]. Using this we will find \[\angle acb\]. Then using the angle sum property of a triangle in \[\vartriangle acb\], we will find \[\angle abc\]. The sum of opposite angles of a trapezium is equal to \[{180^ \circ }\]. So, using this we will get \[\angle adc\]. Now we will subtract \[\angle abc\] from \[\angle adc\] to find the difference between angle \[adc\] and angle \[abc\].

Complete step by step answer:

Given a trapezium \[abcd\],\[ab||cd\] and \[ab\] is a diameter.
As we know that angle subtended by a diameter on any point of the circle is \[{90^ \circ }\]. Therefore, \[\angle acb = {90^ \circ }\].
The sum of all the interior angles of a triangle is \[{180^ \circ }\]. So, we can write
\[ \Rightarrow \angle cab + \angle abc + \angle acb = {180^ \circ }\]
Putting the values, we get
\[ \Rightarrow {28^ \circ } + \angle abc + {90^ \circ } = {180^ \circ }\]
\[ \Rightarrow {118^ \circ } + \angle abc = {180^ \circ }\]
Subtracting \[{118^ \circ }\] from both the sides, we get
\[ \Rightarrow \angle abc = {180^ \circ } - {118^ \circ }\]
On simplification, we get
\[ \Rightarrow \angle abc = {62^ \circ }\]
The sum of opposite angles of a trapezium is equal to \[{180^ \circ }\]. So, we can write
\[ \Rightarrow \angle adc + \angle abc = {180^ \circ }\]
Putting the values, we get
\[ \Rightarrow \angle adc + {62^ \circ } = {180^ \circ }\]
Subtracting \[{62^ \circ }\] from both the sides, we get
\[ \Rightarrow \angle adc = {180^ \circ } - {62^ \circ }\]
\[ \Rightarrow \angle adc = {118^ \circ }\]
So, we get \[\angle adc = {118^ \circ }\] and \[\angle abc = {62^ \circ }\].
Therefore, the difference between \[\angle adc\] and \[\angle abc\] is \[\left( {{{118}^ \circ } - {{62}^ \circ }} \right)\] i.e., \[{56^ \circ }\].

Note:Except for isosceles trapezium, trapezium has non-parallel sides unequal and the sum of interior angles is \[{360^ \circ }\]. Exactly one pair of opposite sides are parallel and diagonal intersect each other. Two angles of a trapezium are supplementary to each other i.e., their sum is equal to \[{180^ \circ }\].