Question

what is the measure of angle abc? 42.5° 67.5° 85° 135°

Explanation:

Step1: Find the measure of the arc \( CDE \)

The measure of an inscribed angle is half the measure of its intercepted arc. The angle adjacent to \( 25^{\circ} \) (i.e., \( \angle BDE\)) is \( 180^{\circ}- 25^{\circ}=155^{\circ}\).
The measure of the arc \( CDE \) is \( 2\times155^{\circ} = 310^{\circ}\) (by the central - angle/inscribed - angle relationship). But we know that the total measure of a circle is \( 360^{\circ}\). Let the measure of arc \( CE\) be \(x\). Then \(x + 110^{\circ}=360^{\circ}\), so \(x = 250^{\circ}\).

Step2: Use the formula for the angle formed by two secants

The formula for the angle formed by two secants \( \angle ABC=\frac{1}{2}(\text{measure of arc } CE-\text{measure of arc } AD)\).
We know that the measure of arc \( CE = 250^{\circ}\) and the measure of arc \( AD\): Let the measure of arc \( AD=y\). Since \(x + 110^{\circ}=360^{\circ}\) (total circle), and using the angle - arc relationship for the angle outside the circle.
Another formula: The angle formed outside the circle \( \angle ABC=\frac{1}{2}(\text{major arc}-\text{minor arc})\). The major arc \( CDE = 250^{\circ}\) and the minor arc \( AD\):
The measure of the angle formed by a secant and a tangent - like situation (here two secants) is \( \angle ABC=\frac{1}{2}( \text{arc } CE-\text{arc } AD)\).
We can also use the property: \( \angle ABC=\frac{1}{2}(110^{\circ}-25^{\circ})\times1.7\) (incorrect approach).
The correct formula: \( \angle ABC=\frac{1}{2}(110^{\circ}-25^{\circ})\) (using the theorem that the measure of an angle formed by two secants outside the circle \( \angle=\frac{1}{2}(\text{larger arc}-\text{smaller arc})\).

$$ \angle ABC=\frac{1}{2}(110^{\circ}-25^{\circ})=\frac{1}{2}\times85^{\circ} = 42.5^{\circ} $$

Answer:

\(42.5^{\circ}\)