We are given that:

• $BE \cong EC$ (so triangle $BEC$ is isosceles),
• $m\angle BEC = 48^\circ$,
• $m\angle A = 40^\circ$,
• and we need to find $m\angle AEB$.

Step-by-step solution:

1. Isosceles Triangle Property: Since $BE \cong EC$, triangle $BEC$ is isosceles, which means that the base angles $\angle EBC$ and $\angle ECB$ are congruent. Let these angles be denoted as $x$.

2. Sum of Angles in Triangle: The sum of the angles in triangle $BEC$ must equal $180^\circ$. So, we can write the equation: $m\angle BEC + m\angle EBC + m\angle ECB = 180^\circ$ Substituting the known values: $48^\circ + x + x = 180^\circ$ Simplifying: $48^\circ + 2x = 180^\circ$ $2x = 132^\circ$ $x = 66^\circ$ Therefore, $m\angle EBC = m\angle ECB = 66^\circ$.

3. Finding $m\angle AEB$: Now consider triangle $ABE$. We know two of its angles:

   • $m\angle A = 40^\circ$,
   • $m\angle ABE = 66^\circ$ (since $\angle ABE = \angle EBC$).

   The sum of the angles in triangle $ABE$ is also $180^\circ$, so: $m\angle A + m\angle ABE + m\angle AEB = 180^\circ$ Substituting the known values: $40^\circ + 66^\circ + m\angle AEB = 180^\circ$ Simplifying: $106^\circ + m\angle AEB = 180^\circ$ $m\angle AEB = 74^\circ$

Thus, the measure of $\angle AEB$ is $74^\circ$.

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Related Questions:

1. What are the properties of an isosceles triangle?
2. How can you use the angle sum property in any triangle?
3. How does knowing the base angles in an isosceles triangle help solve for unknown angles?
4. What is the significance of congruent sides in triangle problems?
5. Can you solve for the missing angle if only two angles are known?

Tip:

In isosceles triangles, the base angles are always equal, which is often the key to solving for unknown angles.