Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Metric tensor wikipedia , lookup
Lie sphere geometry wikipedia , lookup
Rational trigonometry wikipedia , lookup
Riemannian connection on a surface wikipedia , lookup
Line (geometry) wikipedia , lookup
Problem of Apollonius wikipedia , lookup
Trigonometric functions wikipedia , lookup
Euclidean geometry wikipedia , lookup
Lesson 10.4 Other Angle Relationships in Circles by Mrs. C. Henry Objectives: Goal 1: Use angles formed by tangents and chords to solve problems in geometry. Goal 2: Identify angles formed by lines that intersect a circle to solve problems. Goal 3: Utilize properties of tangent angles and chords to solve real-life problems, such as finding from how far away you can see fireworks. 10.4 Using Tangents and Chords You know that the measure of an angle inscribed in a circle is half the measure of its intercepted arc. This is true even if one side of the angle is tangent to the circle. Thm 10.12 Tangent Chord Angle m 1 = ½ mAB m 2 = ½ mBCA 10.4 Using Tangents and Chords Using the Postulates and Theorems Tangent Chord Angle Thm: Now it is your turn! Work the following problem. Sketch the diagram. Line m is tangent to the circle. Find the measure of the red angle or arc. 10.4 Using Tangents and Chords Using the Postulates and Theorems Tangent Chord Angle Thm: Now it is your turn! Work the following problem. Sketch the diagram. Line m is tangent to the circle. Find the measure of the red angle or arc. m 1 = ½ (150°) = 75° 10.4 Using Tangents and Chords If two lines intersect a circle, there are three places where the lines can intersect. 2 on the circle inside the circle outside the circle You know how to find angle and arc measures when lines intersect on the circle. Now let’s look at an example of inside the circle. Thm 10.13 Crossed Chords m 1 = ½ (mCD + mAB) m 2 = ½ (mBC + mAD) 10.4 Using Tangents and Chords Using the Postulates and Theorems Crossed Chords Thm: Now it is your turn! Work the following problem. Sketch the diagram. 10.4 Using Tangents and Chords Using the Postulates and Theorems Crossed Chords Thm: Now it is your turn! Work the following problem. Sketch the diagram. m RHW = ½ (mRW + mST) m RHW = ½ (110° + 40°) m RHW = ½ (150°) m RHW = 75° 10.4 Using Tangents and Chords Finally, let’s look at how to handle angle and arc measures when the lines intersect outside the circle. Do you remember what a secant is? Thm 10.14 Exterior Intersecting Tangents/Secants A tangent and a secant, 2 tangents, or 2 secants intersect in the exterior of a circle. m 1 = ½ (mBC – mAC) m 2 = ½ (mPQR – mPR) m 3 = ½ (mXY – mWZ) Secant line (or segment): a line (or segment) that intersects a circle at two points. 10.4 Using Tangents and Chords Using the Postulates and Theorems Exterior Intersecting Tangents & Secants Thm: Now it is your turn! Work the following problem. Sketch the diagram. A 10.4 Using Tangents and Chords Using the Postulates and Theorems Exterior Intersecting Tangents & Secants Thm: Now it is your turn! Work the following problem. Sketch the diagram. A m RST = ½ (mRAT – mRT) m RST = ½ ((360° – 80°) – 80°) m RST = ½ (280° – 80°) m RST = ½ (200°) m RST = 100°