15.5 Angle Relationships in Circles Essential Question: What are the relationships between angles formed by lines that intersect a circle?

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15.5 Angle Relationships in Circles Essential Question: What are the relationships between angles formed by lines that intersect a circle?

Exploring angle measures in Circles A. Using geometry software, construct a circle with two secants CD and EF that intersect inside the circle at G, as shown in the figure. B. Create two new points H and I that are on the circle as shown. These will be used to measure the arcs. Hide B if desired. C. Measure / DGF formed by the secant lines, and measure arcs CHE and DIF. Record angle and arc measurements in the first column of the table.

Drag F around the circle and record the changes in measures in the table. Try to create acute, right, and obtuse angles. Be sure to keep H between C and E and I between D and F for accurate arc measurement. Move them if necessary.

Reflections Can you make a conjecture about the relationship between the angle measure and the two arc measures?

The tangent-secant Interior angle Measure Theorem Construct a circle with center A. Construct a tangent at point of tangency C. Construct a chord or secant that passes through C and label the other endpoint E. Measure angle DCE and arc EBC. Repeat by moving point E or C.

Reflections m/ DCE m arc EBC ½ m arc EBC What do you notice ? What is the relationship between the angle and arc formed by a tangent and secant intersecting on a circle?

Assignment