Quadrilateral CDEF is inscribed in circle A. Which statements complete the proof to show that ∠CFE and ∠CDE are supplementary? Quadrilateral CDEF is inscribed in circle A, so mARC CDE + mARC CFE= 360°. ∠CFE and ∠CDE are inscribed angles, which means that their measures are . So, . Using the substitution property of equality, 2 ⋅ m∠CFE + 2 ⋅ m∠CDE = 360°. Using the division property of equality, divide both sides of the equation by 2, resulting in m∠CFE + m∠CDE = 180°. Therefore, ∠CFE and ∠CDE are supplementary.

The statements that complete the proofs are;

First statement

One half the measure of their intercepted arcs

Second statement

mARC CDE = 2·m∠CFE and mARC CFE = 2·m∠CDE

Step-by-step explanation:

The two column proof that ∠CFE and ∠CDE are supplementary is presented as follows;

Statement                                                              Reason

Quadrilateral CDEF is inscribed is circle A         Given

+ = 360°                                   Measure of angle round a circle

∠CFE and ∠CDE are inscribed angles               Given

∠CFE  + ∠CDE = 1/2 × ( + )     Inscribed angles are (i) one half the measure of their intercepted arcs

2 × (∠CFE  + ∠CDE) = ( + )

So, (ii) = 2 × m∠CFE and = 2 × m∠CDE From the inscribed angle theorem above (See attached drawing)

2·m∠CFE + 2·m∠CDE = 360°      Using substitution property of equality

(2·m∠CFE + 2·m∠CDE)/2 = 360°/2 → m∠CFE + m∠CDE) = 180°                Dividing both sides by 2

m∠CFE and m∠CDE) are supplementary    Angles that sum up to 180°

The statements that complete the proofs are;

(i) One half the measure of their intercepted arcs

(ii) mARC CDE = 2·m∠CFE and mARC CFE = 2·m∠CDE