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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.

Respuesta :

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  [tex]{}[/tex]                                                            Reason

Quadrilateral CDEF is inscribed is circle A  [tex]{}[/tex]       Given

[tex]m\widehat{CDE}[/tex] + [tex]m\widehat{CFE}[/tex] = 360°  [tex]{}[/tex]                                 Measure of angle round a circle

∠CFE and ∠CDE are inscribed angles   [tex]{}[/tex]             Given

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

2 × (∠CFE  + ∠CDE) = ([tex]m\widehat{CDE}[/tex] + [tex]m\widehat{CFE}[/tex])

So, (ii) [tex]m\widehat{CDE}[/tex] = 2 × m∠CFE and [tex]m\widehat{CFE}[/tex] = 2 × m∠CDE From the inscribed angle theorem above (See attached drawing)

2·m∠CFE + 2·m∠CDE = 360°  [tex]{}[/tex]    Using substitution property of equality

(2·m∠CFE + 2·m∠CDE)/2 = 360°/2 → m∠CFE + m∠CDE) = 180° [tex]{}[/tex]                Dividing both sides by 2

m∠CFE and m∠CDE) are supplementary  [tex]{}[/tex]  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

Ver imagen oeerivona

Answer:

1/2 the measure of their intercepted arcs; m arc CDE= 2 ⋅ m∠CFE and arc CFE= 2 ⋅ m∠CDE (or C).

Step-by-step explanation:

I just took the test and got this right :)

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