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Transcript
Algebra III Lesson 12 Angles and Diagonals in Polygons Proofs of the Chord-Tangent Theorem Angles & Diagonals in Polygons At every corner of a polygon there is an inside and an outside angle. O I -The inside angle can be greater than 180° if the polygon is concave. O -At any given corner the inside and outside angles add to 180°. I Interior angles are the same as inside angles in convex polygons. I E An exterior angle is the supplement to a given interior angle. E A diagonal connects two corners not already connected by a side. Finding the Sum of the Interior Angles for any Polygon 1st – a triangle Σ(Interior Angles) = 180° 2nd – a rectangle - Put a diagonal in to create two triangles. - Each triangle has 180°. Σ(Interior Angles) = 2(180°) = 360° 3rd – a pentagon - Draw all possible diagonals from one corner. - This makes three triangles. Σ(Interior Angles) = 3(180°) = 540° Finally, a n-gon (a polygon with n sides) - Draw all possible diagonals from one corner. How many diagonals is this? Find the pattern: 3 sides → 0 diagonals 4 sides → 1 diagonal 5 sides → 2 diagonals n sides → (n – 3) diagonals - How many triangles are made? Find the pattern: 3 sides → 1 triangle 4 sides → 2 triangles 5 sides → 3 triangles n sides → (n – 2) triangles Σ(Interior Angles) = (n – 2)180° For a convex polygon, the sum of the exterior angles always equals 360°. Σ(Exterior Angles) = 360° Finding the Total Number of Diagonals in a Polygon 1) In the polygon, pick a corner and draw all possible diagonals. 2) Go to the next corner, and again draw all possible diagonals. Don’t draw a diagonal if it is already there from a previous corner. 3) Repeat this procedure until all corners have been done. 4) Count the number of diagonals. A Formula # of diagonals = n (n - 3) 2 n = the number of sides Cross check with a rectangle Example 12.1 Find the sum of the measures of (a) the interior angles and (b) the exterior angles of a seven sided convex polygon. a) Σ(Interior Angles) = (n – 2)180° = (7 – 2)180° = (5)180° = 900° b) Σ(Exterior Angles) = 360° Proof of the Chord-Tangent Theorem Refers to the angle made by the intersection of a chord and a tangent. Postulate – An inscribed angle equals one-half the measure of its arc. 50° 25° 25° Three Cases to the Chord-Tangent Theorem B B B X O X X O A Case 1 C O A Case 2 C A Case 3 C Example 12.2 B Prove Case 1 of the Chord-Tangent Theorem X O 1) A tangent and a radii are perpendicular by postulate. 2) So, m∠OAC = 90°, since OA is a radii. 3) Since BOA is a diameter, mBXA = 180° 4) ∴, m∠BAC = 1 mBXA 2 A C Example 12.3 Y Prove Case 2 of the Chord-Tangent Theorem B X O 1) Draw a diameter to A 1 mBY 2 1 3) m∠YAC = mYXA 2 2) m∠BAY = A By Postulate By Case 1 1 1 Adding. 4) m∠BAY + m∠YAC = mBY + mYXA 2 2 1 5) m∠BAC = mBXA 2 C Example 12.4 Y Prove Case 3 of the Chord-Tangent Theorem B X O 1) Put in diameter YOA 1 m∠YXA 2 1 3) m∠YAB = mYB 2 2) m∠YAC = Subtract. 4) m∠YAC − m∠YAB = A By Case 1 By Postulate 1 (mYXA − mYB) 2 1 5) ∴ m∠BAC = mBXA 2 Practice a) Find the sum of the measures of the interior angles and the sum of the measures of the exterior angles of a six-sided convex polygon. 1) Σ(Interior Angles) = (n – 2)180° = (6 – 2)180° = (4)180° = 720° 2) Σ(Exterior Angles) = 360° b) How many diagonals can be drawn in a nine-sided convex polygon? # of diagonals = n (n - 3) 2 = 9(9 - 3) 2 = 9⋅6 2 = 27