Question

If ∠dbc = 40°, then match the following to their values. hint: first find the measure of arc cd. with that, use the fact that bd is a diameter to find the measure of arc bc. 1. m∠ecb = 40 2. m = 100 3. m = 80 4. m∠dcf = 50

Answer 1

To solve the student's question, we calculate the measure of arc CD as 40° (same as ∠DBC), and then find the measure of arc BC as 140° (subtracting arc CD from 180° given that BD is a diameter). This allows us to match the measure of angle ∠ECB as 40°.

The correct answer is A.

Step-by-step explanation:

To find the values that are being asked for in the question, we need to apply the concepts of circle geometry and the measures of arcs and angles.

Here are the steps to find the relevant measures:

1. First, determine the measure of arc CD based on angle ∠DBC. Since ∠DBC is a central angle of a circle, the measure of arc CD will also be 40°.
2. Knowing that BD is a diameter implies that it splits the circle into two semicircles. Since a circle has 360°, a semicircle will have 180°. Thus, to find the measure of arc BC, subtract the measure of arc CD (40°) from 180°, which gives us an arc measure of 140° for BC.
3. We can now match the values to their corresponding angles and arcs:

• The measure of angle ∠ECB is equal to the measure of arc CD which is 40° (Option 1).
• If we represent the measure of arc BC by m, then m = 140°.
• Given that CD is 40°, and BD is a diameter, the angle ∠DCF subtended by arc CD is half the measure of arc BC. Therefore, ∠DCF = 70°, which is not listed in the given options.

The provided passage discusses the approximation of an arc length and its corresponding angle measure in context with small segments of circles. Yet, the details given are not directly related to the question asked.