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6.1 Polygons. Geometry Mrs. Spitz Spring 2005. Objectives:. Identify, name, and describe polygons such as the building shapes in Example 2. Use the sum of the measures of the interior angles of a quadrilateral. Assignments. pp. 325-327 # 4-46 all Definitions Postulates/Theorems.

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6 1 polygons

6.1 Polygons

Geometry

Mrs. Spitz

Spring 2005


Objectives
Objectives:

  • Identify, name, and describe polygons such as the building shapes in Example 2.

  • Use the sum of the measures of the interior angles of a quadrilateral.


Assignments
Assignments

  • pp. 325-327 # 4-46 all

  • Definitions

  • Postulates/Theorems


Definitions
Definitions:

SIDE

  • Polygon—a plane figure that meets the following conditions:

    • It is formed by 3 or more segments called sides, such that no two sides with a common endpoint are collinear.

    • Each side intersects exactly two other sides, one at each endpoint.

  • Vertex – each endpoint of a side. Plural is vertices. You can name a polygon by listing its vertices consecutively. For instance, PQRST and QPTSR are two correct names for the polygon above.


Example 1 identifying polygons

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/

Example 1: Identifying Polygons

Figures A, B, and C are polygons.




Convex or concave

Convex if no line that contains a side of the polygon contains a point in the interior of the polygon.

Concave or non-convex if a line does contain a side of the polygon containing a point on the interior of the polygon.

Convex or concave?

See how it doesn’t go on the

Inside-- convex

See how this crosses

a point on the inside?

Concave.


Convex or concave1

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

Convex or concave?

CONCAVE

A polygon is EQUILATERAL

If all of its sides are congruent.

A polygon is EQUIANGULAR

if all of its interior angles are congruent.

A polygon is REGULAR if it is

equilateral and equiangular.

CONVEX


Identifying regular polygons

Remember: Equiangular & equilateral concave.

Decide whether the following polygons are regular.

Identifying Regular Polygons

Heptagon is equilateral, but not equiangular, so it is NOT a regular polygon.

Pentagon is equilateral and equiangular, so it is a regular polygon.

Equilateral, but not equiangular, so it is NOT a regular polygon.


Interior angles of quadrilaterals

A diagonal of a polygon is a segment that joins two nonconsecutive vertices. Polygon PQRST has 2 diagonals from point Q, QT and QS

Interior angles of quadrilaterals

diagonals


Interior angles of quadrilaterals1

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°.

Interior angles of quadrilaterals


Theorem 6 1 interior angles of a quadrilateral

The sum of the measures of the interior angles of a quadrilateral is 360°.

Theorem 6.1: Interior Angles of a Quadrilateral

m1 + m2 + m3 + m4 = 360°


Ex 4 interior angles of a quadrilateral

Find m quadrilateral is 360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.

Ex. 4: Interior Angles of a Quadrilateral

80°

70°

2x°

x°+ 2x° + 70° + 80° = 360°


Ex 4 interior angles of a quadrilateral1

x°+ 2x° + 70° + 80° = 360° quadrilateral is 360

3x + 150 = 360

3x = 210

x = 70

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.

80°

Ex. 4: Interior Angles of a Quadrilateral

70°

2x°

Find m Q and mR.

mQ = x° = 70°

mR = 2x°= 140°

►So, mQ = 70° and mR = 140°


Reminder
Reminder: quadrilateral is 360

  • Quiz after 6.3 and 6.5.

  • Definitions – 20 point assignment due by Friday this week. (5th period – Thursday)

  • Postulates– the green boxes—20 point assignment due by Friday this week. (5th period – Thursday)


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