10.3 Inscribed Angles Unit IIIC Day 5.

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Presentation transcript:

10.3 Inscribed Angles Unit IIIC Day 5

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

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.

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

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°

Proof of Thm. 10.11 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?

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 + 120 = 180. y = 60

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 + 120 – 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°

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.