1. **Problem statement:** We have an isosceles triangle ABC with vertex B as the apex, and angle ÂBC = 96°. Point I lies inside triangle ABC such that angle ÎAC = 18° and angle ÎCA = 30°. We need to find the measures of angles BAI and ABI. 2. **Given:** - Triangle ABC is isosceles at B, so AB = BC. - Angle ÂBC = 96°. - Point I inside ABC with angles at A and C given: ÎAC = 18°, ÎCA = 30°. 3. **Step 1: Find other angles of triangle ABC.** Since ABC is isosceles at B, angles at A and C are equal. Sum of angles in triangle ABC is 180°. So, angle BAC = angle BCA = \frac{180° - 96°}{2} = 42°. 4. **Step 2: Analyze triangle AIC.** Given angles at I: ÎAC = 18°, ÎCA = 30°. Sum of angles in triangle AIC is 180°. So, angle AIC = 180° - 18° - 30° = 132°. 5. **Step 3: Use angle chasing to find angles BAI and ABI.** Since AB = BC and angle at B is 96°, triangle ABC is symmetric about the bisector of angle B. Point I lies inside ABC with given angles at A and C. 6. **Step 4: Calculate angle BAI.** Angle BAI = angle BAC - angle IAC = 42° - 18° = 24°. 7. **Step 5: Calculate angle ABI.** Angle ABI = angle ABC - angle IBC. Since AB = BC, angle IBC = angle ICB = 30° (from given ÎCA). Angle ABC = 96°. So, angle ABI = 96° - 18° = 78°. 8. **Final answer:** Angles BAI and ABI are 24° and 78° respectively. **Answer choice:** 24° và 78°.