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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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 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
See how this crosses
a point on the inside?
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.
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.
A diagonal of a polygon is a segment that joins two nonconsecutive vertices. Polygon PQRST has 2 diagonals from point Q, QT and QSInterior angles of quadrilaterals
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
The sum of the measures of the interior angles of a quadrilateral is 360°.Theorem 6.1: Interior Angles of a Quadrilateral
m1 + m2 + m3 + m4 = 360°
Find m quadrilateral is 360Q and mR.
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
x°+ 2x° + 70° + 80° = 360°
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
Find m Q and mR.
mQ = x° = 70°
mR = 2x°= 140°
►So, mQ = 70° and mR = 140°