1. **Stating the problem:** We have a geometric figure with angles 45°, 80°, and unknown angles $a^\circ$, $b^\circ$, $c^\circ$, and $d^\circ$. We need to find the values of $a$, $b$, $c$, and $d$ using properties of angles. 2. **Important angle rules:** - Angles on a straight line add up to 180°. - Vertically opposite angles are equal. - Angles in parallel lines cut by a transversal have special relationships (alternate interior angles are equal, corresponding angles are equal). 3. **Find $a$:** Since $a$ and 80° are on a straight line, they add to 180°: $$a + 80 = 180$$ Subtract 80 from both sides: $$a = 180 - 80 = 100$$ 4. **Find $b$:** Angles 45° and $b$ are vertically opposite angles (formed by intersecting lines), so they are equal: $$b = 45$$ 5. **Find $c$:** Since the two lines are parallel, angle $c$ is alternate interior to angle 80°, so: $$c = 80$$ 6. **Find $d$:** Angles $c$ and $d$ are on a straight line, so: $$c + d = 180$$ Substitute $c = 80$: $$80 + d = 180$$ Subtract 80: $$d = 100$$ **Final answers:** $$a = 100, b = 45, c = 80, d = 100$$