1. **State the problem:** We have a right triangle with two equal sides of length 8 and a right angle at the bottom center. We need to find the measure of angle $R$, which is the angle between the vertical segment from the top vertex to the base and the slanted segment down to the lower-right interior point. An angle of $18^\circ$ is given between the slanted interior segment and the outer right side. 2. **Analyze the triangle:** Since the two sides from the top vertex to the base endpoints are equal (both 8), the large triangle is isosceles with the right angle at the base center. 3. **Use the right triangle properties:** The right triangle has legs of length 8 and 8, so the hypotenuse is $$\sqrt{8^2 + 8^2} = \sqrt{64 + 64} = \sqrt{128} = 8\sqrt{2}.$$ The vertical segment from the top vertex to the base is one leg, and the slanted segment to the lower-right interior point forms an angle $R$ with it. 4. **Use angle relationships:** The $18^\circ$ angle is between the slanted interior segment and the outer right side. Since the outer right side is one leg of the triangle, and the vertical segment is perpendicular to the base, angle $R$ is complementary to $18^\circ$. 5. **Calculate angle $R$:** $$ R = 90^\circ - 18^\circ = 72^\circ $$ **Final answer:** $$ \boxed{72^\circ} $$