Question

explore: measurements of inscribed and central angles enter the measures for angle i and angle c into the table below. change the measure of angle ∠acb, and record the new measurements. use the rows in the table below to collect four sets of measurements. measure of ∠i measure of ∠c \\(\frac{(∠i)}{∠c}\\) 39.8

Explanation:

Step1: Identify the relationship

In a circle, the measure of a central angle (∠C) is twice the measure of the inscribed angle (∠I) that subtends the same arc. So \( m\angle C = 2\times m\angle I \).

Step2: Calculate measure of ∠C

Given \( m\angle I = 39.8^\circ \), then \( m\angle C = 2\times39.8^\circ = 79.6^\circ \).

Step3: Calculate the ratio \( \frac{\angle I}{\angle C} \)

Substitute the values: \( \frac{39.8}{79.6} = 0.5 \).

Answer:

For the first row:

• Measure of \( \angle I \): \( 39.8^\circ \)
• Measure of \( \angle C \): \( 79.6^\circ \)
• \( \frac{\angle I}{\angle C} \): \( 0.5 \)

(For the other rows, you can change the measure of \( \angle ACB \) (which is related to \( \angle I \)), then use the same relationship \( m\angle C = 2\times m\angle I \) to find \( \angle C \) and the ratio will always be \( 0.5 \) as long as the arcs subtended are the same.)