McGraw Hill Glencoe Geometry, 2012
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McGraw Hill Glencoe Geometry, 2012 View details
6. Secants, Tangents, and Angle Measures
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Exercise 33 Page 747

Practice makes perfect
a Let's consider a circle centered at point a tangent line and a secant line such that is acute.

Let's start writing what Theorem states.

Theorem

In a plane, a line is tangent to a circle if and only if it is perpendicular to a radius drawn to the point of tangency.

Therefore, we have that and then is a right angle, which means that is a semicircle.
Since is acute, is in the interior of Then, by applying the Angle Addition Postulate and the Arc Addition Postulate we can rewrite and
Also, the Inscribed Angle Theorem gives us that Let's substitute and into the system above.
Let's divide the second equation by and subtract it from the first one.
Finally, by solving the equation above for the required equation will be obtained.

Paragraph Proof

Proof: By Theorem and then is a right angle with measure and is a semicircle with measure Since is acute, is in the interior of Then, by the Angle Addition and Arc Addition Postulates, and

The Inscribed Angle Theorem gives Then, by substitution we get and Dividing the second equation by and subtracting it from the first one gives which means that

b Let's consider a circle centered at point a tangent line and a secant line such that is obtuse.
As in Part A, Theorem tells us that is a right angle, which means that is a semicircle.
Since is obtuse, is in the exterior of Then, by applying the Angle Addition and Arc Addition Postulates, we can rewrite and
Also, the Inscribed Angle Theorem gives us that Let's substitute and into the system above.
Let's divide the second equation by and subtract it from the first one.
Finally, by solving the equation above for the required equation will be obtained.

Paragraph Proof

Proof: By Theorem and then is a right angle with measure and is a semicircle with measure Since is obtuse, is in the exterior of Then, by the Angle Addition and Arc Addition Postulates, and

The Inscribed Angle Theorem gives Then, by substitution we get and Dividing the second equation by and subtracting it from the first one gives which means that