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problem solving inscribed angles

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High school geometry

Unit 8: lesson 7, inscribed angles.

Inscribed Angles

An inscribed angle in a circle is formed by two chords that have a common end point on the circle. This common end point is the vertex of the angle.

Here, the circle with center O has the inscribed angle ∠ A B C . The other end points than the vertex, A and C define the intercepted arc A C ⌢ of the circle. The measure of A C ⌢ is the measure of its central angle . That is, the measure of ∠ A O C .

Inscribed Angle Theorem:

The measure of an inscribed angle is half the measure of the intercepted arc.

That is, m ∠ A B C = 1 2 m ∠ A O C .

This leads to the corollary that in a circle any two inscribed angles with the same intercepted arcs are congruent.

Here, ∠ A D C ≅ ∠ A B C ≅ ∠ A F C .

Find the measure of the inscribed angle ∠ P Q R .

By the inscribed angle theorem, the measure of an inscribed angle is half the measure of the intercepted arc.

The measure of the central angle ∠ P O R of the intercepted arc P R ⌢ is 90 ° .

m ∠ P Q R = 1 2 m ∠ P O R = 1 2 ( 90 ° ) = 45 ° .

Find m ∠ L P N .

In a circle, any two inscribed angles with the same intercepted arcs are congruent.

Here, the inscribed angles ∠ L M N and ∠ L P N have the same intercepted arc L N ⌢ .

So, ∠ L M N ≅ ∠ L P N .

Therefore, m ∠ L M N = m ∠ L P N = 55 ° .

An especially interesting result of the Inscribed Angle Theorem is that an angle inscribed in a semi-circle is a right angle.

In a semi-circle, the intercepted arc measures 180 ° and therefore any corresponding inscribed angle would measure half of it.

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Inscribed angle

In a given circle , an inscribed angle is an angle whose vertex lies on the circle and each of whose sides either intersects the circle in a second point (and so includes a chord of the circle) or is tangent to the circle. This latter case is sometimes referred to as a tangent-chord angle .

$[asy] defaultpen(fontsize(8)); int i, r=10;pair A=r*expi(-pi/2-pi/12), B=r*expi(-pi/2+pi/12), O=(0,0), P=B-5*expi(pi/12); draw(Circle((0,0),r));draw(A--O--B--A);draw(B--P);for(i=-1;i<6;++i){draw(A--r*expi(pi*i/4)--B);} dot(A^^B^^O^^P);label("A",A,(-1,-1));label("B",B,(1,-1));label("O",O,(0,1));label("P",P,(0,-1)); [/asy]$

The measure of an inscribed angle is equal to half of the measure of the arc it intercepts or subtends. Thus, in particular it does not depend on the location of the vertex on the circle.

Central angle

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COMMENTS

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Solve two challenging problems that apply the inscribed angle theorem to find an arc measure or an arc length. Problem 1. In the figure below
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The measure of an inscribed angle is equal to half of the measure of the arc it intercepts or subtends. Thus, in particular it does not depend on the location
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Here we have an inscribed angle intercepting an arc, so we can bust out the Inscribed Angle Theorem. We know the measure of arc mAB equals 2 × m∠ACB.
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Exercise 40 Page 729 - Practice and Problem Solving - 4. Inscribed Angles - For the first part, let's consider a circle and an inscribed angle ∠ FGH such
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All rights reserved. LESSON. Problem Solving. Inscribed Angles. 11-4. 1. Find m AB. z z. A. B. C. D. 160°. 2. Find the angle measures of RSTU. (8y J 4)°.