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Properties of Polygons

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Properties of Polygons

Each segment that forms a polygon is a side of the polygon

The common endpoint of the two sides is a vertex of the polygon.

A segment that connects any two nonconsecutive vertices is a.

Page 404

Properties of Polygons

A regular polygon is one that is both equilateral and equiangular.

A polygon is concave if any part of a diagonal contains points in the exterior of the polygon.

If no diagonal contains points in the exterior, then the polygon is convex.

A regular polygon is always convex.

Example 6-1a

Name the polygon by its number of sides. Then classify it as convex or concave, regular or irregular.

There are 4 sides, so this is a quadrilateral.

No line containing any of the sides will pass through the interior of the quadrilateral, so it is convex.

The sides are not congruent, so it is irregular.

Answer: quadrilateral, convex, irregular

Example 6-1b

Name the polygon by its number of sides. Then classify it as convex or concave, regular or irregular.

There are 9 sides, so this is a nonagon.

A line containing some of the sides will pass through the interior of the nonagon, so it is concave.

The sides are not congruent, so it is irregular.

Answer: nonagon, concave, irregular

Name each polygon by the number of sides. Then classify it as convex or concave, regular or irregular.

a.

b.

Example 6-1cAnswer: triangle, convex, regular

Answer: quadrilateral, convex, irregular

Poly Angle Sum Theorem

The sum of the interior angle measrues of a convex polygon with n sides is (

Polygon Exterior Angle Sum Theorem

The sum of the exterior angle measures, one angle at each vertex, of a convex polygon is 360.

Since the sum of the measures of the interior angles is Write an equation to express the sum of the measures of the interior angles of the polygon.

Example 1-3aFind the measure of each interior angle.

Example 1-4a

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.

Example 1-4c

Find the measures of an exterior angle and an interior angle of convex regular hexagon ABCDEF.

Answer: 60; 120

Homework: p 407 14-18 e, 22-26 e, 28-32 e, 35-41 all.

Properties of Parallelograms

A quadrilateral with two pairs of parallel sides is a parallelogram.

If a quadrilateral is a parallelogram, then its opposite sides are congruent.

If a quadrilateral is a parallelogram, then its opposite angles are congruent.

If a quadrilateral is a parallelogram, then its consecutive angles are supplementary.

If a quadrilateral is a parallelogram, then its diagonals bisect each other.

RSTU is a parallelogram. Find and y.

If lines are cut by a transversal, alt. int.

Example 2-2aDefinition of congruent angles

Substitution

Homework: p 414 16-31 all

Conditions for Parallelograms

If one pair of opposite sides of a quadrilateral are parallel and congruent, then the quadrilateral a parallelogram.

If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

Conditions for Parallelograms

If an angle of a quadrilateral is supplementary to both of its consecutive angles, then the quadrilateral is a parallelogram.

If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.

Example 3-3a

Determine whether the quadrilateral is a parallelogram. Justify your answer.

Answer: Each pair of opposite sides have the same measure. Therefore, they are congruent. If both pairs of opposite sides of a quadrilateral are congruent, the quadrilateral is a parallelogram.

B

D

C

Example 3-4aFind x so that the quadrilateral is a parallelogram.

Opposite sides of a parallelogram are congruent.

E

G

F

Example 3-4cFind y so that the quadrilateral is a parallelogram.

Opposite angles of a parallelogram are congruent.

Example 3-5a

COORDINATE GEOMETRY Determine whether the figure with vertices A(–3, 0), B(–1, 3), C(3, 2), and

D(1, –1) is a parallelogram. Use the Slope Formula.

Example 3-5c

COORDINATE GEOMETRY Determine whether the figure with vertices P(–3, –1), Q(–1, 3), R(3, 1), and S(1, –3)is a parallelogram. Use the Distance and Slope Formulas.

Example 3-5f

Determine whether the figure with the given vertices is a parallelogram. Use the method indicated.

a. A(–1, –2), B(–3, 1), C(1, 2), D(3, –1);

Slope Formula

Distance and Slope Formulas

b. L(–6, –1), M(–1, 2), N(4, 1), O(–1, –2);

Example 3-5gDetermine whether the figure with the given vertices is a parallelogram. Use the method indicated.

Homework: p 421 13-18, 20-24 e, 26-32 e