Chapter Test
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By the Angles Outside the Circle Theorem, 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.
m∠1=14.5^(∘), m∠2=83^(∘)
Consider the given diagram.
We will find the values of ∠1, and ∠2 one at a time.
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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 77^(∘), and 48^(∘).
Notice that the angle ∠2 is formed by the intersection of two tangents outside the circle. In this case, we can consider the Angles Outside the Circle Theorem.
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Angles Outside the Circle Theorem |
|
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. |
Before applying the theorem, we should calculate the undefined arc. Let's call it x.
Considering that a full circle is 360^(∘), we can calculate x as follows.
x+215^(∘)+77^(∘)+48^(∘)=360^(∘) ⇔ x= 20^(∘)
Now, we will find the measure of ∠2 by applying the theorem.
Notice that the measure of one of the inscribed angle is the sum of 215^(∘) and 48^(∘), and the measure of the other is the sum of ∠77 and ∠20. Thus, we can compute ∠2 as follows.