1. 10.3 Inscribed Angles
Unit IIIC Day 5

2. Do Now Find the measure of the blue arc. 105º
m arc ZWX = 2* m<ZYX = 2*105º = 210º

3. Vocab. Review
If all of the vertices of a polygon lie on a circle, the polygon is inscribed in the circle.
The circle is circumscribed about the polygon.

4. Theorem 10.10
If a right triangle is inscribed in a circle, then the hypotenuse is a diameter of the circle.
Conversely, if one side of an inscribed triangle is a diameter of the circle, then the triangle is a right triangle (the angle opposite the diameter is the right angle).
B is a right angle if and only if ______________
AC is a diameter

5. Theorem 10.11
A quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary.
D, E, F, and G lie on some circle, C, if and only if _____________ and _____________
mD + mF = 180° and mE + mG = 180°

6. Proof of Thm
Given: Quadrilateral ABCD is inscribed in a circle.
Prove: Opposite angles are supplementary.
Hint: m<B + m<D = ½ m arc ADC + ½ m arc ABC. Why? What follows from that?

7. Ex. 5: Using Theorems 10.10 and 10.11 Find the value of each variable.
AB is a diameter. So, C is a right angle and mC = 90. °2x° = 90°. x = 45
DEFG is inscribed in a circle, so opposite angles are supplementary. mD + mF = 180°. z + 80 = 180. z = 100. DEFG is inscribed in a circle, so opposite angles are supplementary. mE + mG = 180°. y = 180. y = 60

8. Ex. 6: Using an Inscribed Quadrilateral
In the diagram, ABCD is inscribed in circle P. Find the measure of each angle.
ABCD is inscribed in a circle, so opposite angles are supplementary.
3x + 3y = 180
5x + 2y = 180
To solve this system of linear equations, you can use linear combinations or substitution.
LC: Simplify equation 1 by dividing by 3 on both sides: x + y = 60
Multiply that through by -2, so that our y’s will cancel.
-2x – 2y = -120
Add the two equations vertically
3x = 60
X = 20
Plug back in to get y: 20 + y = 60; y = 40
Substitution: solve the first equation for y to get y = 60 – x. Substitute this expression into the second equation.
5x + 2y = 180.
5x + 2 (60 – x) = 180
5x – 2x = 180
x = 20
y = 60 – 20 = 40
x = 20 and y = 40, so mA = 80°, mB = 60°, mC = 100°, and mD = 120°

9. Closure
Explain why the diagonals of a rectangle inscribed in a circle must be diameters of the circle.
Since the angle opposite the chord of the circle (slash diagonal of the rectangle) is 90°, the diagonals are chords whose endpoints intercept an arc of 2 * 90º = 180°. The only chords that intercept arcs of 180° are diameters.