Final answer:
To find the degree measure of angle XWZ in the inscribed convex quadrilateral WXYZ with given angle XYZ equal to 54 degrees, we utilize the properties of a circle. The opposite angles of an inscribed quadrilateral are supplementary. After calculating, we determine angle XWZ is 126 degrees.
Step-by-step explanation:
To determine the degree measure of angle XWZ in a convex quadrilateral WXYZ inscribed in a circle, where angle XYZ is equal to 54 degrees, we can use the properties of inscribed angles and the fact that the sum of opposite angles in such a quadrilateral equals 180 degrees since they are supplementary angles spanned by the same arc. This is due to the Inscribed Angle Theorem, which states that an angle inscribed in a circle is half the measure of its intercepted arc.
Since angle XYZ is 54 degrees, the arc XWZ (which is the intercepted arc for angle XYZ) would measure 108 degrees (double the inscribed angle XYZ). Therefore, the remaining arc WXY (the entire circle is 360 degrees minus arc XWZ which is 108 degrees) would be 360 - 108 = 252 degrees. Now, using the Inscribed Angle Theorem again, we can find the angle XWZ by halving the measure of arc WXY.
Angle XWZ would then be 252 / 2 = 126 degrees. This is because angle XWZ corresponds to the angle opposite of XYZ, and as such, is inscribed by arc WXY. Thus, the degree measure of angle XWZ in the inscribed convex quadrilateral WXYZ is 126 degrees.