1. **State the problem:** We need to find the size of angle $m$ in a quadrilateral where three angles are given: $67^\circ$, $86^\circ$, and a right angle $90^\circ$.\n\n2. **Recall the formula:** The sum of interior angles in any quadrilateral is always $360^\circ$.\n\n3. **Set up the equation:** Let the four angles be $67^\circ$, $86^\circ$, $90^\circ$, and $m$. Then:\n$$67 + 86 + 90 + m = 360$$\n\n4. **Calculate the sum of known angles:**\n$$67 + 86 = 153$$\n$$153 + 90 = 243$$\n\n5. **Solve for $m$:**\n$$243 + m = 360$$\n$$m = 360 - 243$$\n$$m = 117$$\n\n6. **Answer:** The size of angle $m$ is $117^\circ$.