Consider two kites where the congruent angle is the one between congruent sides. You'll need to consider a concave kite.
No, see solution.
Practice makes perfect
To determine if this congruence theorem is valid or not, let's begin by considering a kite.
Notice that P is equidistant from the endpoints of QS, and similarly, R is equidistant from the endpoints of QS as well. Then, by the Converse of the Perpendicular Bisector Theorem, both P and S lie on the perpendicular bisector of QS.
Then, any point on PR will generate a pair of consecutive congruent sides. Thus, let's consider a point Z on it such that ZQ = QR.
We now have two kites – namey, PQRS and PQZS. Next, let's list the congruent parts between them.
ccc
RS ≅ ZS & & Side
SP ≅ SP & & Side
∠P ≅ ∠P & & Angle
PQ ≅ PQ & & Side
QR ≅ QZ & & Side
Notice that the two kites meet the conditions of the given theorem but they are not congruent because the corresponding angles are not congruent. Therefore, SSASS is not a valid congruence theorem for kites.
Extra
Extra
When the congruent angles are the ones formed by the noncongruent sides, we have that both kites are the same, so they are congruent in this case. However, since we found an example where the SSASS congruence theorem fails, we can claim that it is not valid.
