Question

Quadrilateral CDEF is inscribed in circle A. Which statements complete the proof to show that ∠CFE and ∠CDE are supplementary?

Answer 1

Answer:

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

$m\widehat{CDE}$ + $m\widehat{CFE}$ = 360°  ${}$                                 Measure of angle round a circle

∠CFE and ∠CDE are inscribed angles   ${}$             Given

∠CFE  + ∠CDE = 1/2 × ($m\widehat{CDE}$ + $m\widehat{CFE}$)  ${}$   Inscribed angles are (i) one half the measure of their intercepted arcs

2 × (∠CFE  + ∠CDE) = ($m\widehat{CDE}$ + $m\widehat{CFE}$)

So, (ii) $m\widehat{CDE}$ = 2 × m∠CFE and $m\widehat{CFE}$ = 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