Inscribed Angles

Challenge Problem F G I H E l

D F G I H E l

Objectives Use inscribed angles to solve problems. Use properties of inscribed polygons.

Using Inscribed Angles An inscribed angle is an angle whose vertex is on a circle and whose sides contain chords of the circle. The arc that lies in the interior of an inscribed angle and has endpoints on the angle is called the intercepted arc of the angle.

Measure of an Inscribed Angle

Finding Measures of Arcs and Inscribed Angles Find the measure of the blue arc or angle. m = 2m  QRS = 2(90°) = 180°

Finding Measures of Arcs and Inscribed Angles Find the measure of the blue arc or angle. m = 2m  ZYX = 2(115°) = 230°

Ex. 1: Finding Measures of Arcs and Inscribed Angles Find the measure of the blue arc or angle. m = ½ m ½ (100°) = 50° 100°

Ex. 2: Comparing Measures of Inscribed Angles Find m  ACB, m  ADB, and m  AEB. The measure of each angle is half the measure of m = 60°, so the measure of each angle is 30°

Theorem If two inscribed angles of a circle intercept the same arc, then the angles are congruent.  C   D

Ex. 3: Finding the Measure of an Angle It is given that m  E = 75 °. What is m  F?  E and  F both intercept, so  E   F. So, m  F = m  E = 75° 75°

Using Properties of Inscribed Polygons If all of the vertices of a polygon lie on a circle, the polygon is inscribed in the circle and the circle is circumscribed about the polygon. The polygon is an inscribed polygon and the circle is a circumscribed circle.

Theorem 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 and the angle opposite the diameter is the right angle.  B is a right angle if and only if AC is a diameter of the circle.

Theorem

Ex. 5: Using Theorems 2x°

Ex. 5: Using Theorems 120° 80° y°y° z°z°

Ex. 5: Using Theorems 120° 80° y°y° z°z°

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 3y° 2y° To solve this system of linear equations, you can solve the first equation for y to get y = 60 – x. Substitute this expression into the second equation. 3x° 2x°

Ex. 6: Using an Inscribed Quadrilateral 5x + 2y = x + 2 (60 – x) = 180 5x – 2x = 180 3x = 60 x = 20 y = 60 – 20 = 40 Write the second equation. Substitute 60 – x for y. Distributive Property. Subtract 120 from both sides. Divide each side by 3. Substitute and solve for y.  x = 20 and y = 40, so m  A = 80°, m  B = 60°, m  C = 100°, and m  D = 120°

Angle Measures and Segment Lengths in Circles Objectives: 1.To find the measures of  s formed by chords, secants, & tangents. 2.To find the lengths of segments associated with circles.

Secants E A B F

Theorem. The measure of an  formed by 2 lines that intersect inside a circle is Measure of intercepted arcs

Theorem. The measure of an  formed by 2 lines that intersect outside a circle is Smaller Arc Larger Arc x°x° y°y° 1 x°x° y°y° 1 2 Secants: x°x° y°y° 1 Tangent & a Secant 2 Tangents 3 cases:

Ex.1 & 2: 94° 112° 68° 104° 92° 268°

Lengths of Secants, Tangents, & Chords 2 Chords 2 Secants Tangent & Secant

Ex. 3 &

Ex.5: 2 Secants

Ex.6: A little bit of everything! ° 175°

What have we learned?? When dealing with angle measures formed by intersecting secants or tangents you either add or subtract the intercepted arcs depending on where the lines intersect. There are 3 formulas to solve for segments lengths inside of circles, it depends on which segments you are dealing with: Secants, Chords, or Tangents.

Q G F D E C A 100° 30° 25° Challenge #1

Find the value of the variable Challenge #2

Find the value of the variable Challenge #3

Q G F D E C A 100° 30° 25°

Find the value of the variable

Find the value of the variable to the nearest tenth