Quadrilaterals and Other Polygons Chapter 7

Angles of Polygons I can use the interior and exterior angle measures of polygons.

Angles of Polygons Vocabulary (page 197 in Student Journal) diagonal: a segment joining a vertex to a nonadjacent vertex equilateral polygon: a polygon with all sides congruent

Angles of Polygons equiangular polygon: a polygon with all angles congruent regular polygon: a polygon that is both equilateral and equiangular

Angles of Polygons Core Concepts (pages 197 and 198 in Student Journal) Polygon Interior Angles Theorem The sum of the measures of the interior angles of a convex n-gon is (n - 2)180.

Angles of Polygons Corrollary to the Polygon Interior Angles Theorem The sum of the measures of the interior angles of a quadrilateral is 360 degrees. Polygon Exterior Angles Theorem The sum of the measures of the exterior angles of a convex polygon, one at each vertex, is 360 degrees.

Angles of Polygons Examples (space on pages 197 and 198 in Student Journal) a) Find the sum of the measures of the interior angles of the polygon.

Angles of Polygons Solution a) 1440 degrees

Angles of Polygons b) The sum of the measures of the interior angles of a convex polygon is 1800 degrees. Classify the polygon by its number of sides.

Angles of Polygons Solution b) 12 sides, so dodecagon

Angles of Polygons c) Find the value of x in the diagram.

Angles of Polygons Solution c) 107

Angles of Polygons Use the polygon below to answer the following. d) Is the polygon regular? e) Determine the angle measures for angles B, D, E, and G.

Angles of Polygons Solutions d) no e) 125 degrees

Angles of Polygons f) Find the value of x in the diagram below.

Angles of Polygons Solution f) 19

Properties of Parallelograms I can use properties to find side lengths and angles of parallelograms.

Properties of Parallelograms Vocabulary (page 202 in Student Journal) parallelogram: a quadrilateral with both pairs of opposite sides parallel

Properties of Parallelograms Core Concepts (pages 202 and 203 in Student Journal) Parallelogram Opposite Sides Theorem If a quadrilateral is a parallelogram, then its opposite sides are congruent. Parallelogram Opposite Angles Theorem If a quadrilateral is a parallelogram, then its opposite angles are congruent.

Properties of Parallelograms Parallelogram Consecutive Angles Theorem If a quadrilateral is a parallelogram, then its consecutive angles are supplementary. Parallelogram Diagonals Theorem If a quadrilateral is a parallelogram, then its diagonals bisect each other.

Properties of Parallelograms Examples (space on pages 202 and 203 in Student Journal) a) Find the values of x and y in the diagram.

Properties of Parallelograms Solution a) x = 27, y = 7

Properties of Parallelograms b) Given ABCD and GDEF are parallelograms, prove angle C is congruent to angle G.

Properties of Parallelograms Solution b)

Properties of Parallelograms c) Find the coordinates of the intersection of the diagonals of parallelogram ABCD given vertices A(1, 0), B(6, 0), C(5, 3), and D(0, 3).

Properties of Parallelograms Solution c) (3, 3/2)

Properties of Parallelograms d) Three vertices of parallelogram DEFG are D(-1, 4), E(2, 3) and F(4, -2). Find the coordinates of vertex G.

Properties of Parallelograms Solution d) (1, -1)

Proving that a Quadrilateral is a Parallelogram I can identify and verify parallelograms.

Proving that a Quadrilateral is a Parallelogram Core Concepts (pages 207 and 208 in Student Journal) Parallelogram Opposite Sides Converse If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

Proving that a Quadrilateral is a Parallelogram Parallelogram Opposite Angles Converse If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

Proving that a Quadrilateral is a Parallelogram Opposite Sides Parallel and Congruent Theorem If one pair of opposite sides of a quadrilateral is both parallel and congruent, then the quadrilateral is a parallelogram. Parallelogram Diagonals Converse If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.

Proving that a Quadrilateral is a Parallelogram Examples (space on pages 207 and 208 in Student Journal) a) In quadrilateral ABCD, if AB = BC and CD = AD, is ABCD a parallelogram? Explain.

Proving that a Quadrilateral is a Parallelogram Solution a) It cannot be determined because 2 pair of adjacent sides are known to be congruent and we need 2 pair of opposite sides to be congruent.

Proving that a Quadrilateral is a Parallelogram b) For what values of x and y is quadrilateral STUV a parallelogram?

Proving that a Quadrilateral is a Parallelogram Solution b) x = 9, y = 21

Proving that a Quadrilateral is a Parallelogram c) For what value of x is quadrilateral CDEF a parallelogram?

Proving that a Quadrilateral is a Parallelogram Solution c) 14

Proving that a Quadrilateral is a Parallelogram d) Show that quadrilateral ABCD is a parallelogram.

Proving that a Quadrilateral is a Parallelogram Solution d)

Properties of Special Parallelograms I can use properties of special parallelograms.

Properties of Special Parallelograms Vocabulary (page 212 in Student Journal) rhombus: a parallelogram with four congruent sides rectangle: a parallelogram with four right angles square: a parallelogram with four congruent sides and four right angles

Properties of Special Parallelograms Core Concepts (pages 212 and 213 in Student Journal) Rhombus Corollary A quadrilateral is a rhombus if and only if it has 4 congruent sides. Rectangle Corollary A quadrilateral is a rectangle if and only if it has 4 right angles.

Properties of Special Parallelograms Square Corollary A quadrilateral is a square if and only if it is a rhombus and a rectangle. Rhombus Diagonals Theorem A parallelogram is a rhombus if and only if its diagonals are perpendicular.

Properties of Special Parallelograms Rhombus Opposite Angles Theorem A parallelogram is a rhombus if and only if its diagonal bisects a pair of opposite angles. Rectangle Diagonals Theorem A parallelogram is a rectangle if and only if its diagonals are congruent.

Properties of Special Parallelograms Examples (space on pages 212 and 213 in Student Journal) For any rectangle ABCD, determine whether the statement is always, sometimes, or never true. AB = BC AB = CD

Properties of Special Parallelograms Solutions sometimes always

Properties of Special Parallelograms c) Classify the quadrilateral in the diagram.

Properties of Special Parallelograms Solution c) rhombus

Properties of Special Parallelograms d) Find the measure of angle ABC and measure of angle ACB in rhombus ABCD.

Properties of Special Parallelograms Solution d) measure of angle ABC = 58 degrees, measure of angle ACB = 61 degrees

Properties of Special Parallelograms e) In rectangle ABCD, AC = 7x – 15 and BD = 2x + 25. Find the lengths of the diagonals.

Properties of Special Parallelograms Solution e) 41 units

Properties of Special Parallelograms f) Classify quadrilateral ABCD with vertices A(-2, 3), B(2, 2), C(1, -2), and D(-3, -1).

Properties of Special Parallelograms Solution f) square

Properties of Trapezoids and Kites I can use properties of trapezoids and kites.

Properties of Trapezoids and Kites Vocabulary (page 217 in Student Journal) trapezoid: a quadrilateral with exactly one pair of parallel sides bases of a trapezoid: the parallel sides

Properties of Trapezoids and Kites base angles of a trapezoid: a pair of angles that share the same base legs of a trapezoid: the nonparallel sides isosceles trapezoid: a trapezoid with legs that are congruent

Properties of Trapezoids and Kites midsegment of a trapezoid: the segment that joins the midpoints of its legs kite: a quadrilateral with 2 pairs of consecutive sides congruent and no opposite sides congruent

Properties of Trapezoids and Kites Core Concepts (pages 217 and 218 in Student Journal) Isosceles Trapezoid Base Angles Theorem If a quadrilateral is an isosceles trapezoid, then each pair of base angles is congruent.

Properties of Trapezoids and Kites Isosceles Trapezoid Base Angles Converse If a trapezoid has a pair of congruent base angles, then it is an isosceles trapezoid. Isosceles Trapezoid Diagonals Theorem If a quadrilateral is an isosceles trapezoid, then its diagonals are congruent.

Properties of Trapezoids and Kites Trapezoid Midsegment Theorem If a quadrilateral is a trapezoid, then the midsegment is parallel to the bases and its length is half the sum of the length of the bases. Kite Diagonals Theorem If a quadrilateral is a kite, then its diagonals are perpendicular.

Properties of Trapezoids and Kites Kite Opposite Angles Theorem If a quadrilateral is a kite, then exactly 1 pair of opposite angles are congruent.

Properties of Trapezoids and Kites Examples (space on pages 217 and 218 in Student Journal) Show quadrilateral ABCD is a trapezoid. Then determine if it is isosceles.

Properties of Trapezoids and Kites Solution a)

Properties of Trapezoids and Kites Solution a)

Properties of Trapezoids and Kites b) Find the measure of angles B, C, and D in the diagram.

Properties of Trapezoids and Kites Solution b) measure of angle B = 138 degrees, measure of angle C = 138 degrees, measure of angle D = 42 degrees.

Properties of Trapezoids and Kites c) Find MN in the diagram.

Properties of Trapezoids and Kites Solution c) 16.2 inches

Properties of Trapezoids and Kites d) Find the length of midsegment YZ in trapezoid PQRS.

Properties of Trapezoids and Kites Solution d) 4√2 units

Properties of Trapezoids and Kites e) Find the measure of angle C in the kite shown.

Properties of Trapezoids and Kites Solution e) 115 degrees

Properties of Trapezoids and Kites f) What is the most specific name for the quadrilateral in the diagram?

Properties of Trapezoids and Kites Solution f) quadrilateral