Integrated Math 4 Day 41
10.4 Circles, Angles, and Arcs
· Daily Openers
· Go over and collect homework
10.4 Circles, Angles, and Arcs
central angle – an angle that has its vertex at the center of the circle.
§ The rays of the angle intercept an arc.
minor arc – ÐXYZ intercepts minor arc XZ and major arc XWZ.
semicircle – half a circle.
minor arc – less than a semicircle.
major arc – more than a semicircle.
Arc Addition Postulate
If C is a point on an arc with endpoints A and B, then
mAC + mBC = mACB
inscribed angle – an angle that has its vertex on the circle.
Inscribed Angle Theorem
The measure of an inscribed angle in a circle is ½ the measure of its intercepted arc.
Ex. Find x and y. Ex. Find z.
Ex. Find w. Ex. Find x.
chord – a segment with both endpoints on the circle.
secant – a line, ray, or segment that intersects a circle in two places.
tangent – a line that intersects a circle at exactly one point.
§ In circle O, is a chord.
and are both secants.
is a tangent.
Intersecting Secants Theorem
If 2 secants intersect inside a circle, then the measure of each Ð formed is equal to the mean of the
arcs intercepted by the angle and its vertical angle.
Ex. Find y. Ex. Find a and b.
Theorem
If 2 secants, 2 tangents, or a tangent and a secant intersect outside a circle, then the measure
of the Ð formed is equal to ½ the difference of the measures of the intercepted arcs.
Ex. Find f. Ex. Find r.
Homework – pages 442–443 #1–6 and page 694 E.P. 10.4 #1–9
Daily Openers – 1. Are the sides those of a right D? 9cm, 12cm, 15cm
2. Simplify:
3. Simplify:
4. A 45°– 45°– 90° right D has a leg of length , find the other 2 side lengths.
5. For a 30°– 60°– 90° right D whose shortest leg has a length of , find the other
2 side lengths.