1. **Problem statement:** In triangle $ABC$, point $D$ lies on $AC$ such that $BD \perp AC$ and $\angle DBC = 77^\circ$. Point $E$ lies on $BC$ such that $\angle CAE = 54^\circ$. We need to find the measure of $\angle AEB$. 2. **Given:** - $BD \perp AC$ means $\angle BDA = 90^\circ$. - $\angle DBC = 77^\circ$. - $\angle CAE = 54^\circ$. 3. **Goal:** Find $\angle AEB$. 4. **Step 1: Analyze triangle $BDC$** Since $BD$ is perpendicular to $AC$, $\angle BDA = 90^\circ$. In triangle $BDC$, angles are $\angle DBC = 77^\circ$, $\angle BDC = 90^\circ$, so $$\angle BCD = 180^\circ - 90^\circ - 77^\circ = 13^\circ.$$ 5. **Step 2: Use $\angle CAE = 54^\circ$** $E$ lies on $BC$, so $\angle CAE$ is an angle at $A$ formed by points $C$ and $E$. 6. **Step 3: Use triangle $AEC$** Since $E$ is on $BC$, and $\angle CAE = 54^\circ$, we consider triangle $AEC$. 7. **Step 4: Find $\angle ACE$** $\angle ACE = \angle ACB - \angle ECB$. From step 4, $\angle BCD = 13^\circ$, so $\angle ACB = 13^\circ$. Since $E$ lies on $BC$, $\angle ECB = 0^\circ$ (point $E$ lies on $BC$). So $\angle ACE = 13^\circ$. 8. **Step 5: Find $\angle AEB$** In quadrilateral $AEBD$, angles around point $E$ satisfy: $$\angle AEB = 180^\circ - \angle CAE - \angle DBC = 180^\circ - 54^\circ - 77^\circ = 49^\circ.$$ But this contradicts the options, so re-examine. 9. **Step 6: Use cyclic quadrilateral property** Since $BD \perp AC$, and $E$ lies on $BC$, $A$, $E$, $B$, and $D$ lie on a circle. Then $\angle AEB = \angle ADB$. 10. **Step 7: Calculate $\angle ADB$** $\angle ADB = 90^\circ$ (since $BD \perp AC$). 11. **Step 8: Calculate $\angle AEB$** $\angle AEB = 180^\circ - \angle DBC = 180^\circ - 77^\circ = 103^\circ$. This is not in options, so check again. 12. **Step 9: Use triangle angle sum in $ABC$** $\angle ABC = 180^\circ - \angle BAC - \angle ACB$. 13. **Step 10: Calculate $\angle BAC$** $\angle CAE = 54^\circ$ and $E$ lies on $BC$, so $\angle BAC = 54^\circ$. 14. **Step 11: Calculate $\angle ABC$** $\angle ABC = 180^\circ - 54^\circ - 13^\circ = 113^\circ$. 15. **Step 12: Calculate $\angle AEB$** Since $E$ lies on $BC$, $\angle AEB = 180^\circ - \angle ABC = 180^\circ - 113^\circ = 67^\circ$. **Final answer:** $\boxed{67^\circ}$. This matches option 1.