Question

Using the figure to the right, if RSTU is a rhombus, find measure RST.

Answer 1

Given:

In rhombus RSTU, $m\angle RUV=(10x-23)^\circ$ and $m\angle TUV=(3x+19)^\circ$.

To find:

The $m\angle RST$.

Solution:

We know that, diagonals of a rhombus are angle bisector. So,

$m\angle RUV=m\angle TUV$

$(10x-23)^\circ=(3x+19)^\circ$

$10x-23=3x+19$

Isolating variable terms, we get

$10x-3x=23+19$

$7x=42$

Divide both sides by 7.

$x=\dfrac{42}{7}$

$x=6$

Now,

$m\angle RUV=(10x-23)^\circ$

$m\angle RUV=(10(6)-23)^\circ$

$m\angle RUV=(60-23)^\circ$

$m\angle RUV=37^\circ$

And,

$m\angle TUV=(3x+19)^\circ$.

$m\angle TUV=(3(6)+19)^\circ$

$m\angle TUV=(18+19)^\circ$

$m\angle TUV=37^\circ$

Now,

$m\angle RUT=m\angle RUV+m\angle TUV$

$m\angle RUT=37^\circ+37^\circ$

$m\angle RUT=74^\circ$

We know that opposite angles of a rhombus are equal.

$m\angle RST=m\angle RUT$

$m\angle RST=74^\circ$

Therefore, the measure of angle RST is $74^\circ$.