What is an inscribed angle of a circle that intercepts a semicircle?

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 ∠ABC . The other end points than the vertex, A and C define the intercepted arc AC ⌢ of the circle. The measure of AC ⌢  is the measure of its central angle. That is, the measure of ∠AOC .

Inscribed Angle Theorem:

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

That is, m∠ABC= 1 2 m∠AOC .

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

Here, ∠ADC≅∠ABC≅∠AFC .

Example 1:

Find the measure of the inscribed angle ∠PQR .

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 ∠POR  of the intercepted arc PR ⌢ is 90° .

Therefore,

m∠PQR= 1 2 m∠POR                    = 1 2 ( 90° )                     =45° .

Example 2:

Find m∠LPN .

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

Here, the inscribed angles ∠LMN  and ∠LPN  have the same intercepted arc LN ⌢ .

So, ∠LMN≅∠LPN .

Therefore, m∠LMN=m∠LPN=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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An arc is a smooth curve joining two points. The length of an arc is called the arc length. An intercepted arc is the portion of the circle that lies in the interior of the angle together with the endpoints of the arc. An inscribed angle is an angle formed by two chords with the vertex on the circle. It is an angle in a circle that is formed by two chords that have a common end point on the circle.

Answer:

a right angle

Proof:

Theorem 72: If an inscribed angle intercepts a semicircle, then its measure is 90°.

Things to Remember:

• Central angles are probably the angles most often associated with a circle.
• Angles may be inscribed in the circumference of the circle or formed by intersecting chords and other lines.
• Theorem 70: The measure of an inscribed angle in a circle equals half the measure of its intercepted arc.
• Theorem 71: If two inscribed angles of a circle intercept the same arc or arcs of equal measure, then the inscribed angles have equal measure.

What is an arc: brainly.ph/question/243330

What is an inscribed angle: brainly.ph/question/12476724

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