Question

The ratio of the measures of the consecutive angles of a quadrilateral is \[1:2:3:4\]. What type of quadrilateral is it?
A. Trapezium
B. Kite
C. Parallelogram
D. Rectangle

Answer 1

Hint: We use a constant variable to write the values of consecutive angles of a quadrilateral using a given ratio. Use the property of the sum of interior angles of the quadrilateral to find the value of the variable and calculate each angle. Check if any pair of lines is parallel in the quadrilateral. Classify the quadrilateral on the basis of given definitions.
* Ratio is comparison of two or more values in the form of a relation. Ratio of \[a:b = \dfrac{a}{b}\] where \[\dfrac{a}{b} = \dfrac{{ka}}{{kb}}\], where ‘k’ is a constant value.
* Sum of alternate interior angles of a quadrilateral is \[{360^ \circ }\]

Complete step-by-step solution:
We draw a quadrilateral ABCD

Then consecutive angles of quadrilateral ABCD are \[\angle A,\angle B,\angle C,\angle D\]
We are given the ratio of consecutive angles of a quadrilateral as \[1:2:3:4\]
Let us assume constant value as ‘k’
Then we can write the values of angles of quadrilateral as
\[ \Rightarrow \angle A = k,\angle B = 2k,\angle C = 3k,\angle D = 4k\]................… (1)
Since we know sum of all interior angles of a quadrilateral is \[{360^ \circ }\]
Calculate sum of angles using equation (1)
\[ \Rightarrow k + 2k + 3k + 4k = {360^ \circ }\]
Calculate the sum in LHS of the equation
\[ \Rightarrow 10k = {360^ \circ }\]
Divide both sides of the equation by 10
\[ \Rightarrow \dfrac{{10k}}{{10}} = \dfrac{{{{360}^ \circ }}}{{10}}\]
Cancel same factors from numerator and denominator on both sides of the equation
\[ \Rightarrow k = {36^ \circ }\]
Substitute the value of ‘k’ in equation (1) to calculate each angle of the quadrilateral ABCD
\[ \Rightarrow \angle A = 1 \times {36^ \circ },\angle B = 2 \times {36^ \circ },\angle C = 3 \times {36^ \circ },\angle D = 4 \times {36^ \circ }\]
Calculate each product
\[ \Rightarrow \angle A = {36^ \circ },\angle B = {72^ \circ },\angle C = {108^ \circ },\angle D = {144^ \circ }\]................… (2)
We can see that \[\angle B + \angle C = {72^ \circ } + {108^ \circ } = {180^ \circ }\]and \[\angle A + \angle D = {36^ \circ } + {144^ \circ } = {180^ \circ }\]
\[ \Rightarrow \]Alternate interior angles are supplementary
Since we know that when two parallel lines are cut by a transversal then alternate interior angles are supplementary, so in quadrilateral ABCD \[AB\parallel CD\] which are cut by transversals AD and BC.
\[ \Rightarrow \]Quadrilateral has a pair of opposite sides parallel to each other
Since we know a quadrilateral having one pair of opposite sides parallel is called a trapezium, Then ABCD is a trapezium.

\[\therefore \]Option A is correct

Note: * Trapezium: A quadrilateral having one pair of opposite sides parallel is called a trapezium
* Kite: A quadrilateral with two distinct pairs of equal adjacent sides is called a Kite.
* Parallelogram: A quadrilateral having opposite sides equal and parallel to each other.
* Rectangle: A quadrilateral having opposite sides parallel to each other.