1. 15.5 Angle Relationships in Circles Essential Question: What are the relationships between angles formed by lines that intersect a circle?

2. 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.

3. 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.

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

5. 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.

6. 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?

7. Assignment

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