Section 6.1

Identify and classify polygons. Find angle measures of quadrilaterals.

Polygon Side of a polygon Vertex of a polygon Diagonal of a polygon

6.1 Quadrilateral Interior Angles Thoerem

Polygon – a closed figure whose sides are formed by a finite number of coplanar segments called sides. the sides that have a common endpoint are noncollinear each side intersects exactly two other sides, but only at their endpoints. The vertex of each angle is a vertex of the polygon. A polygon is named by using the letters of its vertices, written in consecutive order.

State whether the figure is a polygon. If it is not, explain why. Not D – has a side that isn’t a segment – it’s an arc. Not E– because two of the sides intersect only one other side. Not F because some of its sides intersect more than two sides. Figures A, B, and C are polygons.

Is the figure a polygon? Explain your reasoning. a.b.c. d. SOLUTION a.Yes. The figure is a polygon formed by four straight sides. b.No. The figure is not a polygon because it has a side that is not a segment. c.No. The figure is not a polygon because two of the sides intersect only one other side. Yes. The figure is a polygon formed by six straight sides.d.

If the lines of any segment of the polygon are drawn and any of the lines contain points that lie in the interior of the polygon, then it is concave. Otherwise, it is convex (no points of the lines are in the interior).

Identify the polygon and state whether it is convex or concave.

Polygons are classified by the number of sides they have. A polygon with n number of sides is an n-gon. PolygonNumber of Sides Triangle3 Quadrilateral 4 Pentagon5 Hexagon6 Heptagon7 Octagon8 Nonagon9 Decagon10 Hendecagon11 Dodecagon12

3 sides 6 sides 4 sides 7 sides 5 sides 8 sides TRIANGLE HEXAGON QUADRILATERAL PENTAGON OCTAGON HEPTAGON

SOLUTION a.The figure is a polygon with four sides, so it is a quadrilateral. b.The figure is not a polygon because it has some sides that are not segments. c.The figure is a polygon with five sides, so it is a pentagon. The figure is not a polygon because some of the sides intersect more than two other sides. d. Decide whether the figure is a polygon. If so, tell what type. If not, explain why. a. b. c. d.

Decide whether the figure is a polygon. If so, tell what type. If not, explain why ANSWER No; two of the sides intersect only one other side. ANSWER No; one side is not a segment. ANSWER yes; pentagon ANSWER yes; quadrilateral

A segment that connects any two nonconsecutive vertices is a diagonal.

Like triangles, quadrilaterals have both interior and exterior angles. If you draw a diagonal in a quadrilateral, you divide it into two triangles, each of which has interior angles with measures that add up to 180°. So you can conclude that the sum of the measures of the interior angles of a quadrilateral is 2(180°), or 360°.

The sum of the measures of the interior angles of a quadrilateral is 360°. m  1 + m  2 + m  3 + m  4 = 360°

Find the measure of  S. SOLUTION Use the fact that the sum of the measures of the interior angles of a quadrilateral is 360°. ANSWER The measure of  S is 140°. m  P + m  Q + m  R + m  S = 360° Quadrilateral Interior Angles Theorem 70° + 80° + 70° + m  S = 360° Substitute angle measures. 220° + m  S = 360° Simplify. m  S = 140° Subtract 220° from each side.

ANSWER 90° ANSWER 65° ANSWER 72° Find the measure of  A

Find m  Q and m  R. Find the value of x. Use the sum of the measures of the interior angles to write an equation involving x. Then, solve the equation. Substitute to find the value of R. 80° 70° 2x° x°x° x°+ 2x° + 70° + 80° = 360°

3x = 360 3x = 210 x = 70 80° 70° 2x° x°x° Sum of the measures of int.  s of a quadrilateral is 360° Combine like terms Subtract 150 from each side. Divide each side by 3. Find m  Q and m  R. m  Q = x° = 70° m  R = 2x°= 140° ►So, m  Q = 70° and m  R = 140°

Pg. 306 – 308 #1 – 27 odd,