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.

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Properties of Polygons

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Properties of polygons

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 polygons1

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

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

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


Example 6 1c

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

a.

b.

Example 6-1c

Answer: triangle, convex, regular

Answer: quadrilateral, convex, irregular


Poly angle sum theorem

Poly Angle Sum Theorem

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


Polygon exterior angle sum theorem

Polygon Exterior Angle Sum Theorem

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


Example 1 3a

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 1-3a

Find the measure of each interior angle.


Example 1 3d

Answer:

Example 1-3d

Find the measure of each interior angle.


Example 1 4a

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

Example 1-4c

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

Answer: 60; 120


Properties of polygons

Turn and Talk p 407 4-11

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


Properties of parallelograms

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.


Example 2 2a

RSTU is a parallelogram. Find and y.

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

Example 2-2a

Definition of congruent angles

Substitution


Example 2 2d

ABCD is a parallelogram.

Answer:

Example 2-2d


Properties of polygons

Turn and Talk: p 414 6-12 all

Homework: p 414 16-31 all


Conditions for parallelograms

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 parallelograms1

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

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.


Example 3 4a

A

B

D

C

Example 3-4a

Find x so that the quadrilateral is a parallelogram.

Opposite sides of a parallelogram are congruent.


Example 3 4c

D

E

G

F

Example 3-4c

Find y so that the quadrilateral is a parallelogram.

Opposite angles of a parallelogram are congruent.


Example 3 4e

a.

b.

Answer:

Answer:

Example 3-4e

Find m and n so that each quadrilateral is a parallelogram.


Example 3 5a

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

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

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


Example 3 5g

Distance and Slope Formulas

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

Example 3-5g

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


Properties of polygons

Turn and Talk: p 421 4-10 all

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


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