# Inscribed angle and intercepted arc relationship test

### Inscribed Angles in Circles ( Read ) | Geometry | CK Foundation Lesson The Inscribed Angle Alternate a Tangent Angle. Classwork Develop a conjecture, or guess, about the relationship between a and b. Test your conjecture by using a protractor to measure a and b. Diagram 1 If a = m_BAC and b is the angle measure of the arc intercepted by ZBAC, then a = b. In this free video, you will learn about central and inscribed angles learn how to identify central & inscribed angles and their relationships with intercepted arcs. Label the following circle with diameter, radius, chord, tangent, and secant. ( When these lines Inscribed Angles: ˚ This will be graded as a quiz! Supplies What is the relationship between the intercepted arc of a chord and the angle.

The vertex S represents the point from which Chris makes his kick, the bisecting ray represents the path to the target at which Chris is aiming, and the sides of the 30 -angle form the outer boundaries of Chris s possible kicks, given that his margin of error is up to 15 on either side of the target. The angle diagram will be used as a tool in estimating the outer boundaries of Chris s kicks when he aims at point X from various locations on the soccer field. On the soccer field diagram, you will notice three points labeled Points 1, 2, and 3. These three points represent the different positions on the field from which Chris will attempt his shot at the goal. Place the vertex of your angle at Point 1 and make certain that point X lies on the angle bisector. Will Chris s shot be guaranteed to end up between the goalposts from this position on the field?

One at a time, place the vertex of the angle on Point 2 and then on Point 3. Each time, make certain that point X lies on the angle bisector. Determine whether Chris s shots are guaranteed to end up between the goalposts from these positions on the field. Which, if either, position, is a sure shot at the goal for Chris? With experimentation, you should find that there is a region of the playing field from which Chris is certain to have a shot into the goal zone, despite his margin of error.

Use the soccer field diagram and the angle diagram to test points on the field until you can make an informed conjecture as to the shape of this region. Write a description of the region. On the soccer field diagram, clearly identify at least eight points on the outer boundary of this region.

O and O are called. The smaller arc is known as the minor arc, and the larger arc is known as major arc Q. The angle formed by the two radii, O, is called a central angle of this circle. In general, a central angle is an angle whose vertex is at the center of a circle and whose sides contain radii of the circle. Notice that the major arc associated with points and lies outside O, while the minor arc lies in the interior of O. The notation for the measure of is m. The notation for a minor arc requires the endpoints of the arc.

The notation for a major arc requires a point on the arc included between the endpoints of the arc, Q. Semicircles are named as major arcs. The measure of a minor arc must be between 0 and The measure of a major arc must be at least and less than The measure of a semicircle is Use the definitions for central angle and intercepted arc along with triangle properties to find each of the following. Write a definition for congruent arcs. Explain why PQ QR. Use a protractor to find the measure of central angle O. Choose any point on the major arc and label the point P. Draw P and P. P and P form an inscribed angle. Name the arc intercepted by P. List the characteristics of an inscribed angle. Use a protractor to find the measure of inscribed angle P.

## Activities

Draw a different point R on the circle. Then draw a new inscribed angle that has a vertex R and that intercepts. Find the measure of the new inscribed angle. Make a conjecture about the measure of any inscribed angle of this circle that intercepts.

Test your conjecture by creating and measuring three more inscribed angles that intercept. The Proof The proof for the conjecture that you wrote in Item 11 examines each of the three possible positions of the center of the circle relative to the inscribed angle: Case 1 the center lies on a side of the inscribed angle; Case 2 the center is in the interior of the inscribed angle; and Case 3 the center lies outside of the inscribed angle.

If we can prove the conjecture to be true for each of these three cases, we can form one theorem that covers all possible locations of the center of the circle relative to the inscribed angle. The figure below shows a circle with center O. The values a, b, c and d refer to the degree measure of each indicated angle.

Identify each of the following. OC is an triangle d. Use your responses for Parts a and d to verify your response to Part c in Item The figure below shows a circle with center O, which according to Case 2 for the proof of your conjecture in Item 11 is in the interior of the inscribed angle, P.

For the angles shown below, use the conjecture you made in Item 11 to complete the following equation. The diameter CP has been drawn as a dotted segment. Use the lower case variables in the circle above and the results from Items 12 and 13 Case 1 to complete the following: The figure below shows a circle with center O, which according to Case 3 for the proof of your conjecture in Item 11 is in the exterior of the inscribed angle, P.

Notice that once again the diameter PC has been drawn as a dotted line and the lower case variables represent the degree measures of the indicated angles. Write a complete and convincing argument for Case 3 that explains why the measure of the inscribed angle that intercepts is half the measure of the central angle that intercepts.

We have now proven the conjecture from Item 11 for each of the three possible locations of the center of the circle relative to the inscribed angle. Since the conjecture has been proven to be true for each, we can now formulate one theorem that includes all three cases. Complete the Inscribed ngle Measure Theorem below: Tangent Circles Two coplanar circles that intersect at exactly one point. Congruent An angle whose vertex is the center of a circle. Arc An unbroken part of a circle consisting of two points called the endpoints and all the points on the circle between them.

Minor Arc An arc whose points are on or in the interior of a central angle.

### Arcs and Inscribed Angles

Major Arc An arc whose points are on or in the exterior of a central angle. Semicircle An arc that's endpoints lie on a diameter. Adjacent Arcs Arcs of the same circle that intersect at exactly one point. Sector of a Circle A region bounded by two radii of the circle and their intercepted arc. Segment of a Circle A region bounded by an arc and its chord.

Arc Length The distance along an arc measured in linear units. What is the equation for arc length? Inscribed Angle An angle whose vertex is on a circle and whose sides contain chords of the circle. Intercepted Arc An arc consisting of endpoints that lie on the sides of an inscribed angle and all the points of the circle between them. What is the relationship between the inscribed angle and the intercepted arc?

The intercepted arc is two times bigger than the inscribed angle. Subtends When a chord or arc's endpoints lie on the sides of the angle.