Chapter Test
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m∠1=29^(∘), m∠2=66^(∘), m∠3=37^(∘)
Consider the given diagram.
We will find the measures of ∠1, ∠2, and ∠3 one at a time.
Looking at the diagram, we can see that ∠1 is formed by the intersection of two secants outside the circle. In this case, we can consider the Angles Outside the Circle Theorem.
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Angles Outside the Circle Theorem |
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If a tangent and a secant, two tangents, or two secants intersect outside a circle, then the measure of the angle formed is one-half the difference of the measures of the intercepted arcs. |
We can see that the measures of the intercepted arcs are 96^(∘) and 38^(∘).
Looking at the diagram, we can see that ∠2 is formed by the intersection of two chords inside the circle. In this case, we can consider the Angles Inside the Circle Theorem.
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Angles Inside the Circle Theorem |
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If two chords intersect inside a circle, then the measure of the angle formed is one-half the sum of the measures of the arcs intercepted by the angle and its vertical angle. |
We can see that the measures of the intercepted arcs are 96^(∘) and 36^(∘).
Note that the angle ∠3 is an inscribed angle and the measure of its intercepted arc is given. In this case, we can consider using the Inscribed Angle Theorem.
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Inscribed Angle Theorem |
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The measure of an inscribed angle is one-half the measure of its intercepted arc. |
With this in mind, we can find the measure of ∠3 in the given diagram.
