Question

explore the properties of angles and arcs formed by two intersecting secants.

1. multiply the angle measure by 2.

2(m∠abc) = 56 °

1. move points a and c to change the angle and arcs and make a conjecture.

the difference of arc measures always equals dropdown times the angle measure.
dropdown options: 1/2, 2
m∠abc = 28°
mac = 80°
mde = 24°
(diagram of a circle with points a, c, e, d on it, and point b outside the circle with lines ba (intersecting the circle at d) and bc (intersecting the circle at e))

Explanation:

Step1: Recall the theorem for angles formed by two secants

The measure of an angle formed by two secants intersecting outside a circle is half the difference of the measures of the intercepted arcs. The formula is \( m\angle ABC=\frac{1}{2}(m\widehat{AC} - m\widehat{DE}) \). Rearranging this formula to find the difference of the arc measures, we get \( m\widehat{AC}-m\widehat{DE} = 2(m\angle ABC) \). So the difference of arc measures equals 2 times the angle measure.

Step2: Verify with given values

Given \( m\angle ABC = 28^{\circ} \), \( m\widehat{AC}=80^{\circ} \), \( m\widehat{DE} = 24^{\circ} \). Calculate the difference of arcs: \( 80 - 24=56^{\circ} \). Calculate 2 times the angle measure: \( 2\times28 = 56^{\circ} \). This confirms that the difference of arc measures equals 2 times the angle measure.

Answer:

2