Question
Circle A is shown. Secant W Y intersects tangent Z Y at point Y outside of the circle. Secant W Y intersects circle A at point X. Arc X Z is 105 degrees and arc W Z is 175 degrees.
In the diagram of circle A, what is the measure of ∠XYZ?
35°
70°
75°
140°
In the diagram of circle A, what is the measure of ∠XYZ?
35°
70°
75°
140°
Answers
Answered by
GPT-4o mini
To find the measure of angle ∠XYZ, we can use the properties of tangent and secant.
The measure of angle formed by a tangent and a secant is equal to half the difference of the measures of the intercepted arcs. The intercepted arcs in this case are arc XZ and arc WZ.
1. The measure of arc XZ is given as 105 degrees.
2. The measure of arc WZ is given as 175 degrees.
Using the formula:
\[
\angle XYZ = \frac{1}{2} ( \text{measure of arc WZ} - \text{measure of arc XZ} )
\]
Substituting the values:
\[
\angle XYZ = \frac{1}{2} (175^\circ - 105^\circ) = \frac{1}{2} (70^\circ) = 35^\circ
\]
Thus, the measure of ∠XYZ is **35°**.
The measure of angle formed by a tangent and a secant is equal to half the difference of the measures of the intercepted arcs. The intercepted arcs in this case are arc XZ and arc WZ.
1. The measure of arc XZ is given as 105 degrees.
2. The measure of arc WZ is given as 175 degrees.
Using the formula:
\[
\angle XYZ = \frac{1}{2} ( \text{measure of arc WZ} - \text{measure of arc XZ} )
\]
Substituting the values:
\[
\angle XYZ = \frac{1}{2} (175^\circ - 105^\circ) = \frac{1}{2} (70^\circ) = 35^\circ
\]
Thus, the measure of ∠XYZ is **35°**.