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Angles $4$ and $7$ are alternating angles. Therefore, their measures are equal; $6x + 20 = 10x - 40$, $4x = 60$, $x = 15$. Since $x = 15$, the measure of angles $4$ and $7$ is $6(15) + 20 = 90 + 20 = 110$. Notice that replacing $x$ with $15$ in the measure of angle $7$ also yields $110$: $10(15) - 40 = 150 - 40 = 110$. Since angles $6$ and $7$ are vertical angles, the measure of angle $6$ is also $110^\circ$.