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

Using The Figure To The Right If RSTU Is A Rhombus Find Measure RST class=

Sagot :

Given:

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

To find:

The [tex]m\angle RST[/tex].

Solution:

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

[tex]m\angle RUV=m\angle TUV[/tex]

[tex](10x-23)^\circ=(3x+19)^\circ[/tex]

[tex]10x-23=3x+19[/tex]

Isolating variable terms, we get

[tex]10x-3x=23+19[/tex]

[tex]7x=42[/tex]

Divide both sides by 7.

[tex]x=\dfrac{42}{7}[/tex]

[tex]x=6[/tex]

Now,

[tex]m\angle RUV=(10x-23)^\circ[/tex]

[tex]m\angle RUV=(10(6)-23)^\circ[/tex]

[tex]m\angle RUV=(60-23)^\circ[/tex]

[tex]m\angle RUV=37^\circ[/tex]

And,

[tex]m\angle TUV=(3x+19)^\circ[/tex].

[tex]m\angle TUV=(3(6)+19)^\circ[/tex]

[tex]m\angle TUV=(18+19)^\circ[/tex]

[tex]m\angle TUV=37^\circ[/tex]

Now,

[tex]m\angle RUT=m\angle RUV+m\angle TUV[/tex]

[tex]m\angle RUT=37^\circ+37^\circ[/tex]

[tex]m\angle RUT=74^\circ[/tex]

We know that opposite angles of a rhombus are equal.

[tex]m\angle RST=m\angle RUT[/tex]

[tex]m\angle RST=74^\circ[/tex]

Therefore, the measure of angle RST is [tex]74^\circ[/tex].

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