Question

two adjacent angles of a parallelogram are in the ratio 1:2 find the measures of all the angles of the parallelogram​

Answer 1

★ The measure of all the interior angles of the parallelogram ABCD are 120°, 60°, 120° and 60°, respectively.

Step-by-step explanation:

Analysis -

‎ ‎ ‎ ‎ ‎ ‎In the question, it has been given that two adjacent or consecutive angles of a parallelogram are in the ratio 1:2. We've been asked to find the measures of all the interior angles of the parallelogram. $\:$

Solution -

‎ ‎ ‎ ‎ ‎ ‎First of all, let's know what a parallelogram is! In a quadrilateral, if both the pairs of opposite sides are parallel, then it is called a parallelogram. In a parallelogram, opposite sides are equal and diagonals need not be equal, but they bisect each other.

Now let us head into the given question.

Consider the parallelogram as ABCD. As per the analysis, we can say that the four angles of the parallelogram be 1x, 2x, 1x and 2x, respectively. We know about the angle sum property of a quadrilateral, i.e., all the four sides of any quadrilateral is 360°.

Now, on interpreting this data as a equation, we get:

‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎⇢ ∠A + ∠B + ∠C + ∠D = 360°

‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎⇢ 2x + 1x + 2x + 1x = 360°

‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎⇢ 3x + 3x = 360

‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎⇢ 6x = 360°

‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎⇢ x = 360/6

‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎⇢ x = 60

We've obtained the value of x as 60. So, let's substitute the value of x in the expressions formed for each angle of the parallelogram.

∠A ➛ 2x = 2(60) = 120°

∠B ➛ 1x = 1(60) = 60°

∠C ➛ 2x = 2(60) = 120°

∠D ➛ 1x = 1(60) = 60°

Verification -

‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎Let's verify our answer using angle sum property of a quadrilateral.

⇢ ∠A + ∠B + ∠C + ∠D = 360°

⇢ 120° + 60° + 120° + 60° = 360°

⇢ 180° + 180° = 360°

⇢ 360° = 360°

Since the L.H.S. and the R.H.S. are equal, our answer is correct.

Therefore, the four angles of the parallelogram ABCD are 120°, 60°, 120° and 60° respectively.

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Answer 2

Given : The two adjacent angles of a parallelogram are in the ratio 1 : 2 .

Exigency To Find : The measure of all angles of Parallelogram .

⠀⠀⠀⠀⠀━━━━━━━━━━━━━━━━━━━⠀

❍ Let's Consider two adjacent angles of Parallelogram be x & 2x .

$\qquad \therefore \sf \angle A \:\:\:=\:\:\:x^\circ \: \:\: \bf \& \:\:\sf \angle B \:\:\:=\:\:\:2x^\circ$

As We know that ,

⠀⠀⠀⠀⠀▪︎⠀⠀SUM of ADJACENT ANGLES of Parallelogram are SUPPLEMENTARY .

⠀⠀⠀OR ,

$\qquad \qquad \star \pmb{\bf \:\:\:\:\angle A \:\:\:+\:\:\:\angle B \;\:\: =\:\:180^\circ \:}\:$

$\qquad \dashrightarrow \sf \angle A \:\:\:+\:\:\:\angle B \;\:\: =\:\:180^\circ \:\: \\\\\underline {\boldsymbol{\star\:Now \: By \: Substituting \: the \: known \: Values \::}}\\\\ \dashrightarrow \sf \angle A \:\:\:+\:\:\:\angle B \;\:\: =\:\:180^\circ \:\: \\\\\dashrightarrow \sf x \:\:\:+\:\:\:2x \;\:\: =\:\:180^\circ \:\: \\\\\dashrightarrow \sf 3x\;\:\: =\:\:180^\circ \:\: \\\\ \dashrightarrow \sf x\;\:\: =\:\:\dfrac{180}{3}\:\: \\\\\dashrightarrow \sf x\;\:\: =\:\:\cancel {\dfrac{180}{3}}\:\: \\\\\dashrightarrow \sf x\;\:\: =\:\:60^\circ\:\: \\\\\qquad \therefore \pmb{\underline{\purple{\:x\;\:\: =\:\:60^\circ\:\: }} }\bigstar$

⠀⠀⠀⠀⠀Therefore,

$\qquad \leadsto \sf \angle A \:\:\:=\:x\:\:=\:\bf \:60^\circ \:\\\\ \qquad \bf AND , \:\:\\\\ \qquad \leadsto \sf \angle B \:\:\:=\:2x\:\:=\:\:2\:\: \times \:\:60^\circ \:\:=\:\bf \:120^\circ$

⠀⠀⠀⠀⠀Now ,

⠀⠀As , We know that ,

⠀⠀⠀⠀⠀▪︎⠀⠀OPPOSITE ANGLES of Parallelogram are EQUAL :

$\qquad \therefore \sf \:\:\: \angle A \:\:=\:\angle C \:\:=\:\:\:60^\circ \qquad \bf \& \:\: \qquad \sf \angle B \:=\:\angle D \:\:\:=\:\:\:120^\circ$

$\qquad \therefore \underline {\sf Hence, \: The \:measure \:of \: all \: angles \:of \:Parallelogram \:are \:\bf 60^\circ \:,\: 120^\circ \:, \: 60^\circ \:\& \:\: 120 ^\circ} \:.$