Given: Triangle [tex]\( PQR \)[/tex] with [tex]\( m \angle P = x^\circ \)[/tex], [tex]\( m \angle Q = 3x^\circ \)[/tex], and [tex]\( m \angle R = 5x^\circ \)[/tex].

Prove: [tex]\( x = 20 \)[/tex]

By the triangle sum theorem, the sum of the angles in a triangle is equal to [tex]\( 180^\circ \)[/tex]. Therefore, using the given and the triangle sum theorem:

[tex]\[ m \angle P + m \angle Q + m \angle R = 180^\circ \][/tex]

Using the substitution property:

[tex]\[ x^\circ + 3x^\circ + 5x^\circ = 180^\circ \][/tex]

Simplifying the equation:

[tex]\[ 9x = 180 \][/tex]

Finally, using the division property of equality:

[tex]\[ x = 20 \][/tex]