Question

PQRS is a Rhombus. mL P = (3x - 5)
and mL S = (2x - 15), then mL QRT = ?
(1) 115
(3) 70
(2) 55
(4) 65​

Answer 1

Required Answer :

${\underline {\boxed {\Large {\rm { Option \: 4 \: ( 65^\circ) } }}}}$

As per the provided question, we have :

• PQRS is a rhombus.
• Measure of angle P = (3x - 5)°
• Measure of angle S = (2x - 15)°

We have to find the measure of angle QRT.

Rhombus is a parallelogram, so opposite angles of a rhombus is also equal. Hence,

$\dashrightarrow \sf { \angle R = \angle P}$

$\dashrightarrow \sf { \angle R = (3x - 5)^\circ}$

Similarly,

$\dashrightarrow \sf { \angle Q = \angle S}$

$\dashrightarrow \sf { \angle Q = (2x - 15)^\circ}$

Now, as we know that the sum of all the angles lie on a straight line 180° as the measure of straight angle is 180°, so

$\longrightarrow \sf { \angle QRS + \angle QRT = 180^\circ}$

• Reason : Linear Pair

Measure of $\sf \angle QRS$ is (3x - 5)°.

$\longrightarrow \sf { (3x -5)^\circ + \angle QRT = 180^\circ}$

Let it be the equation (i).

Firstly, we'll find the value of x.

We know that sum of the measure of all the interior angles of a quadrilateral is 360°. Since, rhombus is also a quadrilateral, so

$\longrightarrow \sf { \angle P + \angle Q + \angle R + \angle S = 360^\circ}$

$\longrightarrow \sf { (3x - 5) + (2x - 15) + (3x - 5) + (2x -15) = 360}$

$\longrightarrow \sf { 3x - 5 + 2x - 15 + 3x - 5 + 2x -15 = 360}$

$\longrightarrow \sf { 10x - 40= 360}$

$\longrightarrow \sf { 10x = 360 + 40}$

$\longrightarrow \sf { 10x = 400}$

$\longrightarrow \sf { x = \cancel{\dfrac{400}{10}} }$

$\longrightarrow \boxed{\sf { x = 40}}$

Now, substitute the value of x in the equation (i).

$\longrightarrow \sf { (3x -5)^\circ + \angle QRT = 180^\circ}$

$\longrightarrow \sf { (3 \times 40 -5)^\circ + \angle QRT = 180^\circ}$

$\longrightarrow \sf { (120 -5)^\circ + \angle QRT = 180^\circ}$

$\longrightarrow \sf { 115^\circ + \angle QRT = 180^\circ}$

$\longrightarrow \sf { \angle QRT = 180^\circ - 115^\circ}$

$\longrightarrow \boxed{\sf { \angle QRT = 65^\circ }}$

Therefore, measure of angle QRT is 65°.