### Inscribed angles

Before we begin, let’s state a few important theorems.

THEOREM:  If two angles inscribed in a circle intercept the same arc, then they are equal to each other.

THEOREM:  If an angle inside a circle intercepts a diameter, then the angle has a measure of $$90^\circ$$.

Now let’s use these theorems to find the values of some angles!

EXAMPLE:  Find the measure of the angle indicated.

SOLUTION:  First we see that $$\measuredangle DNC$$ and $$\measuredangle DEC$$ both intercept the same arc.  So, by our first theorem, we know that $$\measuredangle DNC = \measuredangle DEC$$. And since $$\measuredangle DNC = 37^\circ$$, we conclude that $$\measuredangle DEC = 37^\circ$$.

Next, since $$\measuredangle EDC$$ intercepts a diameter of the circle, we can conclude, by our second theorem, that $$\measuredangle EDC = 90^\circ$$.

So two of the angles in triangle $$EDC$$ are $$90^\circ$$ and $$37^\circ$$. Then, by using the fact that the tree angles of a triangle must sum to $$180^\circ$$, we know that the measure of the desired angle is $$53^\circ$$.

EXAMPLE:  Find the measure of the indicated angle.

SOLUTION:  Since $$\measuredangle NBM$$ and $$\measuredangle NLM$$ intercept the same arc, they are equal, so that $$\measuredangle NLM = 52^\circ$$. Next, since $$\measuredangle NML$$ intersects a diameter of the circle, we know that $$\measuredangle NML = 90^\circ$$. Then since two of the angles of triangle $$LMN$$ are $$90^\circ$$ and $$52^\circ$$, we know that the desired angle must be $$38^\circ$$, by the fact that the three angles of a triangle must sum to $$180^\circ$$. Solution:  The desired angle is $$38^\circ$$.

8790 x

State if each angle is an inscribed angle. If it is, name the angle and the intercepted arc.

This free worksheet contains 10 assignments each with 24 questions with answers.

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7590 x

Find the measure of the arc or angle indicated.

This free worksheet contains 10 assignments each with 24 questions with answers.

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6877 x

Solve for x and y.

This free worksheet contains 10 assignments each with 24 questions with answers.

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### Geometry

Circles
Congruent Triangles
Constructions
Parallel Lines and the Coordinate Plane
Properties of Triangles

### Algebra and Pre-Algebra

Beginning Algebra
Beginning Trigonometry
Equations
Exponents
Factoring
Linear Equations and Inequalities
Percents
Polynomials