Given: Points D, E, and F are on circle C and EF ≅ DF.
So, ∠ CDF = ∠ CEF = x (say)
In quadrilateral GDCE, ∠ DGE = 76°, ∠ GDC = 90° and ∠CEG = 90°
To find: the measure of ∠CDF
Calculation: From the given diagram
In quadrilateral GDCE,
Sum of all angles in a quadrilateral = 360°
so, ∠ DGE + ∠ GDC + ∠CEG + ∠DCE = 360°
or, 76° + 90° + 90° + ∠DCE = 360°
or, ∠DCE = 104°
∠DFE = ∠DCE ÷ 2 = 104° ÷2 = 52° ------( by the theorem, the angle subtended by an arc at the center is twice of the angle subtended by the same arc on the remaining part of the circle)
In quadrilateral GDEF,
Sum of all angles in a quadrilateral = 360°
so, ∠ DGE + ∠ GDF + ∠GEF + ∠DFE = 360°
or, 76° + (90° +x) + (90°+x) + 52° = 360°
or, x = 26°
or, ∠CDF = 26°
Hence, the required angle will be ∠CDF = 26°
