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.

There are 4 sides, so this is a quadrilateral. 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 6-1a

There are 9 sides, so this is a nonagon. 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 6-1b

Answer: triangle, convex, regular Name each polygon by the number of sides. Then classify it as convex or concave, regular or irregular. a. b. Answer: triangle, convex, regular Answer: quadrilateral, convex, irregular Example 6-1c

Sum of Interior Angle Measures Poly Angle Sum Theorem The sum of the interior angle measrues of a convex polygon with n sides is (𝑛−2)180° Polygon Number of Sides Sum of Interior Angle Measures Triangle 3 (3−2)180° = 180 Quadrilateral 4 (4−2)180° = 360 Pentagon 5 (5−2)180° =540 Hexagon 6 (6−2)180° = 720 n-gon n (𝑛−2)180°

Polygon Exterior Angle Sum Theorem The sum of the exterior angle measures, one angle at each vertex, of a convex polygon is 360°.

Find the measure of each interior angle. 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. Answer: Example 1-3d

At each vertex, extend a side to form one exterior angle. 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 1-4a

Find the measures of an exterior angle and an interior angle of convex regular hexagon ABCDEF. Answer: 60; 120 Example 1-4c

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 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. Definition of congruent angles Substitution Example 2-2a

ABCD is a parallelogram. Answer: Example 2-2d

Turn and Talk: p 414 6-12 all 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.

Determine whether the quadrilateral is a parallelogram 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 3-3a

Find x so that the quadrilateral is a parallelogram. B C D Opposite sides of a parallelogram are congruent. Example 3-4a

Find y so that the quadrilateral is a parallelogram. Opposite angles of a parallelogram are congruent. Example 3-4c

Find m and n so that each quadrilateral is a parallelogram. b. Answer: Answer: Example 3-4e

D(1, –1) is a parallelogram. Use the Slope Formula. 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 3-5a

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 3-5c

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 3-5f

Distance and Slope Formulas Determine whether the figure with the given vertices is a parallelogram. Use the method indicated. Distance and Slope Formulas b. L(–6, –1), M(–1, 2), N(4, 1), O(–1, –2); Example 3-5g

Turn and Talk: p 421 4-10 all Homework: p 421 13-18, 20-24 e, 26-32 e