1 Answer

Lepture · Answer 1 · 2022-06-17T07:44:34+0000

Answer:

$x = 34^\circ$

Explanation:

Note that ∠TSU and ∠PSR are vertical angles. Hence:

$m\angle TSU = m\angle PSR$

∠PSR is the sum of ∠PSQ and ∠QSR. Hence:

$\displaystyle m\angle TSU = m\angle PSQ + m\angle QSR$

We know that ∠TSU measures 4x and ∠QSR measures 3x. Thus:

$(4x) = m\angle PSQ + (3x)$

Solve for ∠PSQ:

$m\angle PSQ = x$

Next, ∠PQS and ∠RQS form a linear pair. Thus:

$m\angle PQS + m\angle RQS = 180^\circ$

∠RQS measures 68°. Thus:

$m\angle PQS +(68^\circ) = 180^\circ$

Solve for ∠PQS:

$m\angle PQS = 112^\circ$

The interior angles of a triangle must total 180°. So, for ΔPQS:

$\displaystyle m\angle SPQ + m\angle PQS + m\angle PSQ = 180^\circ$

Substitute in the known values:

$(x) + (112^\circ) + (x) = 180^\circ$

Simplify:

$2x = 68^\circ$

And divide. Hence:

$x = 34^\circ$

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