1. **Problem statement:** We have two parallel lines $a$ and $b$ with a 45° right triangle placed between them. Given that angle $\angle 1 = 15^\circ$, we need to find the size of angle $\angle 2$. 2. **Key facts and formulas:** - The triangle is a right triangle with one angle $45^\circ$, so the angles inside the triangle are $45^\circ$, $45^\circ$, and $90^\circ$. - Lines $a$ and $b$ are parallel, so alternate interior angles formed by a transversal are equal. - The sum of angles on a straight line is $180^\circ$. 3. **Step-by-step solution:** - Since the triangle is a 45° right triangle, the angle adjacent to line $a$ inside the triangle is $45^\circ$. - Given $\angle 1 = 15^\circ$ is outside the triangle but adjacent to line $a$, the angle between the transversal (hypotenuse) and line $a$ is $15^\circ$. - The angle between the hypotenuse and line $a$ inside the triangle is $45^\circ$, so the angle between the hypotenuse and line $a$ outside the triangle is $180^\circ - 45^\circ = 135^\circ$. - But $\angle 1 = 15^\circ$ is given, so the transversal forms an angle of $15^\circ$ with line $a$ on the outside. - The difference between $135^\circ$ and $15^\circ$ is $120^\circ$, which is the angle between the transversal and line $a$ on the other side. - Since lines $a$ and $b$ are parallel, the angle between the transversal and line $b$ is also $15^\circ$ (alternate interior angles). - Inside the triangle, the angle adjacent to line $b$ is $\angle 2$. The sum of angles around the point on line $b$ is $180^\circ$. - Therefore, $\angle 2 = 45^\circ - 15^\circ = 30^\circ$. 4. **Final answer:** $$\boxed{\angle 2 = 30^\circ}$$