Question

The circle below has center C. Suppose angle EDF = 84º and DF is tangent to the circle at D.

Find the following. Type your numerical answers (without units) in each blank.
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Answer 1

Applying the inscribed angle theorem, the measure of arc ED = 168°, and the measure of central angle ECD = 168°.

What is the inscribed angle theorem?

The inscribed angle theorem states that, if you have an angle formed by two intersecting chords within a circle, the measure of that angle is equal to half the measure of the central angle that spans the same arc as the chords.

That is:

Inscribed angle = 1/2(Central angle)

Measure of arc subtended = central angle

Applying this, angle EDF is an inscribed angle and measures 84 degrees, therefore, we would solve the missing measures as follows:

Measure of arc ED = 2(m∠EDF)

Measure of arc ED = 2(84)

Measure of arc ED = 168°

Measure of central angle ECD = measure of arc ED

Measure of central angle ECD = 168°

Answer 2

Answer:
`m\overbrace{ED}`= 168° and ∠ECD = 168°

Explanation:

In the given , we have a circle centered at C , ED is a chord and DF is a tangent touching circle at D, ∠EDF = 84°.

Also, CE = CD = radius

⇒ ∠CED = ∠CDE ---(i) [angles opposite to equal sides of triangle are equal]

Since radius makes a right angle with tangent.

So, ∠CDF = 90°

⇒ ∠CDE =∠CDF - ∠EDF

⇒ ∠CDE= 90° - 84°

⇒ ∠CDE = 6°

From (i) ∠CED =∠CDE = 6°

In ΔCDE

⇒∠CED + ∠CDE + ∠ECD = 180°

⇒ 6°+6°+ ∠ECD = 180°

⇒ 12°+ ∠ECD = 180°

⇒∠ECD = 180° - 12°

⇒∠ECD = 168°

Since , Angle measure of the central angle is equal to the measure of the intercepted arc.

Therefore m∠ECD =
`m\overbrace{ED}`

⇒
`m\overbrace{ED}`= 168°

Hence,
`m\overbrace{ED}`= 168° and ∠ECD = 168°