Answer:
The statements that complete the proofs are;
First statement
<u>One half the measure of their intercepted arcs</u>
Second statement
<u>mARC CDE = 2·m∠CFE and mARC CFE = 2·m∠CDE</u>
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) <em><u>one half the measure of their intercepted arcs</u></em>
2 × (∠CFE + ∠CDE) = ( + )
So, (ii) <u> = 2 × m∠CFE and </u><u> = 2 × m∠CDE</u> 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