Final answer:
The measure of ∠ABC in triangle ABC is 81°, and the measure of ∠FEC is 121°30' because triangle ABC is isosceles with AB congruent to BC and BE as a median and altitude bisecting ∠ABC.
Step-by-step explanation:
To find the measure of ∠ABC and ∠FEC in triangle ABC where AB is congruent to BC and BE is a median, we start by analyzing the information given. Since triangle ABC is isosceles with AB ≅ BC, and BE is a median, it is also an altitude and bisects ∠ABC. Therefore, ∠ABE and ∠CBE are congruent, where m∠ABE = 40°30' indicates that m∠CBE is also 40°30'. The entire angle ∠ABC is twice the size of ∠ABE, so m∠ABC = 2(40°30') = 81°.
Since ∠FEC is an exterior angle to △BEC, and BEC is a straight line, m∠FEC is equal to the sum of the measures of the remote interior angles of △BEC, which are ∠ABE and ∠ABC. As we know, m∠ABE is 40°30' and m∠ABC is 81°, thus m∠FEC = m∠ABE + m∠ABC = 40°30' + 81° = 121°30'. Therefore, the measure of ∠ABC is 81° and the measure of ∠FEC is 121°30'.