Transcript Lesson
Lesson 6-1 Angles of Polygons 5-Minute Check on Chapter 5 1. State whether this sentence is always, sometimes, or never true. The three altitudes of a triangle intersect at a point inside the triangle. Sometimes – when all angles are acute 2. Find n and list the sides of ΔPQR in order from shortest to longest if mP = 12n – 15, mQ = 7n + 26, and mR = 8n – 47. all angle add to 180 so n = 8; PQ < QR < PR 3. State the assumption you would make to start an indirect proof of the statement. If –2x ≥ 18, then x ≤ –9. x > -9 4. Find the range for the measure of the third side of a triangle given that the measures of two sides are 43 and 29. 43 – 29 = 14 < n < 72 = 29 + 43 5. Write an inequality relating mABD and mCBD. Since 11 > 10, then mABD < mCBD 6. Write an equation that you can use to find the measures of the angles of the triangle. 111 = 3x + (x – 5) Click the mouse button or press the Space Bar to display the answers. Objectives • Find and use the sum of the measures of the interior angles of a polygon – Sum of Interior angles = (n-2) • 180 – One Interior angle = (n-2) • 180 / n • Find and use the sum of the measures of the exterior angles of a polygon – Sum of Exterior angles = 360 – One Exterior angle = 360/n – Exterior angle + Interior angle = 180 Vocabulary • Diagonal – a segment that connects any two nonconsecutive vertices in a polygon. Angles in a Polygon 3 2 4 1 5 Octagon n=8 6 8 7 8 triangles @ 180° - 360° (center angles) = (8-2) • 180 = 1080 Sum of Interior angles = (n-2) • 180 Angles in a Polygon Sum of Interior Angles: Octagon n=8 (n – 2) * 180 where n is number of sides so each interior angle is (n – 2) * 180 n Interior Angle Sum of Exterior Angles: 360 so each exterior angle is 360 n Exterior Angle Octagon Sum of Exterior Angles: Sum of Interior Angles: One Interior Angle: One Exterior Angle: 360 1080 135 45 Interior Angle + Exterior Angle = 180 Polygons Sides Name Sum of Interior Angles One Interior Angle 3 Triangle 180 60 360 120 4 Quadrilateral 360 90 360 90 5 Pentagon 540 108 360 72 6 Hexagon 720 120 360 60 7 Heptagon 900 129 360 51 8 Octagon 1080 135 360 45 9 Nonagon 1260 140 360 40 10 Decagon 1440 144 360 36 12 Dodecagon 1800 150 360 30 n N - gon 360 360 ∕ n = (n-2) * 180 180 – Ext Sum Of One Exterior Exterior Angles Angles Angle Theorems ARCHITECTURE A mall is designed so that five walkways meet at a food court that is in the shape of a regular pentagon. Find the sum of measures of the interior angles of the pentagon. Since a pentagon is a convex polygon, we can use the Angle Sum Theorem. Interior Angle Sum Theorem Simplify. Answer: The sum of the measures of the angles is 540. The measure of an interior angle of a regular polygon is 135. Find the number of sides in the polygon. Use the Interior Angle Sum Theorem to write an equation to solve for n, the number of sides. Interior Angle Sum Theorem Distributive Property Subtract 135n from each side. Add 360 to each side. Divide each side by 45. Answer: The polygon has 8 sides. SHORT CUT!! The measure of an interior angle of a regular polygon is 135. Find the number of sides in the polygon. Exterior angle = 180 – Interior angle = 45 360 360 n = --------- = ------- = 8 Ext 45 The measure of an interior angle of a regular polygon is 144. Find the number of sides in the polygon. Answer: The polygon has 10 sides. Find the measure of each interior angle. Since n = 4 the sum of the measures of the interior angles is 180(4 – 2) or 360°. Write an equation to express the sum of the measures of the interior angles of the polygon. Sum of interior angles Substitution Combine like terms. Subtract 8 from each side. Divide each side by 32. Use the value of x to find the measure of each angle. Answer: Find the measure of each interior angle. Answer: Find the measures of an exterior angle and an interior angle of convex regular nonagon ABCDEFGHJ. At each vertex, extend a side to form one exterior angle. The sum of the measures of the exterior angles is 360. A convex regular nonagon has 9 congruent exterior angles. 9e = 360 e = measure of each exterior angle e = 40 Divide each side by 9. Answer: Measure of each exterior angle is 40. Since each exterior angle and its corresponding interior angle form a linear pair, the measure of the interior angle is 180 – 40 or 140. Find the measures of an exterior angle and an interior angle of convex regular hexagon ABCDEF. Answer: 60; 120 Polygon Hierarchy Polygons Quadrilaterals Parallelograms Rectangles Rhombi Squares Kites Trapezoids Isosceles Trapezoids Quadrilaterals Venn Diagram Quadrilaterals Parallelograms Rhombi Squares Rectangles Trapezoids Isosceles Trapezoids Kites Quadrilateral Characteristics Summary Convex Quadrilaterals Parallelograms 4 sided polygon 4 interior angles sum to 360 4 exterior angles sum to 360 Opposite sides parallel and congruent Opposite angles congruent Consecutive angles supplementary Diagonals bisect each other Rectangles Trapezoids Bases Parallel Legs are not Parallel Leg angles are supplementary Median is parallel to bases Median = ½ (base + base) Rhombi Angles all 90° Diagonals congruent All sides congruent Diagonals perpendicular Diagonals bisect opposite angles Squares Diagonals divide into 4 congruent triangles Isosceles Trapezoids Legs are congruent Base angle pairs congruent Diagonals are congruent Summary & Homework • Summary: – If a convex polygon has n sides and sum of the measures of its interior angles is S, then S = 180(n-2)° – The sum of the measures of the exterior angles of a convex polygon is 360° – Interior angle + Exterior angle = 180 (linear pair) • Homework: – pg 393-95; 1-3, 6-8, 13-15, 17, 22, 28