Chapter 6 Quadrilaterals Types of Polygons n n

Chapter 6 Quadrilaterals

Types of Polygons n n n n n Triangle – 3 sides Quadrilateral – 4 sides Pentagon – 5 sides Hexagon – 6 sides Heptagon – 7 sides Octagon – 8 sides Nonagon – 9 sides Decagon – 10 sides Dodecagon – 12 sides All other polygons = n-gon

Lesson 6. 1 : Angles of Polygons n Interior Angle Sum Theorem ¨ The sum of the measures of the interior angles of a polygon is found by S=180(n-2) n Ex: Hexagon n Exterior Angle Sum Theorem ¨ The sum of the measures of the exterior angles of a polygon is 360 no matter how many sides.

Lesson 6. 1 : Angles of Polygons n Find the measure of an interior and an exterior angle for each polygon. ¨ 24 -gon ¨ 3 x-gon n Find the measure of an exterior angle given the number of sides of a polygon ¨ 260 sides

Lesson 6. 1: Angles of Polygons n The measure of an interior angle of a polygon is given. Find the number of sides. ¨ 175 ¨ 168. 75 n A pentagon has angles (4 x+5), (5 x-5), (6 x+10), (4 x+10), and 7 x. Find x.

A. Find the value of x in the diagram.

Lesson 6. 2: Parallelograms Opposite sides of a parallelogram are congruent A parallelogram is a quadrilateral with both pairs of opposite sides parallel Opposite angles in a parallelogram are congruent Consecutive angles in a parallelogram are supplementary Properties of Parallelograms If a parallelogram has 1 right angle, it has 4 right angles. The diagonals of a parallelogram bisect each other The diagonals of a parallelogram split it into 2 congruent triangles

A. ABCD is a parallelogram. Find AB. B. ABCD is a parallelogram. Find m C. C. ABCD is a parallelogram. Find m D.

A. If WXYZ is a parallelogram, find the value of r, s and t.

What are the coordinates of the intersection of the diagonals of parallelogram MNPR, with vertices M(– 3, 0), N(– 1, 3), P(5, 4), and R(3, 1)?

Lesson 6. 3 : Tests for Parallelograms n If… ¨ Both pairs of opposite sides are parallel ¨ Both pairs of opposite sides are congruent ¨ Both pairs of opposite angles are congruent ¨ The diagonals bisect each other ¨ One pair of opposite sides is congruent and parallel n Then the quadrilateral is a parallelogram

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

Which method would prove the quadrilateral is a parallelogram?

Determine whether the quadrilateral is a parallelogram.

Determine whether the quadrilateral is a parallelogram.

Find x and y so that the quadrilateral is a parallelogram.

COORDINATE GEOMETRY Graph quadrilateral QRST with vertices Q(– 1, 3), R(3, 1), S(2, – 3), and T(– 2, – 1). Determine whether the quadrilateral is a parallelogram. Justify your answer by using the Slope Formula.

n. Given quadrilateral EFGH with vertices E(– 2, 2), F(2, 0), G(1, – 5), and H( – 3, – 2). Determine whether the quadrilateral is a parallelogram. (The graph does not determine for you)

6. 4 -6. 6 Foldable Fold the construction paper in half both length and width wise n Unfold the paper and hold width wise n Fold the edges in to meet at the center crease n Cut the creases on the tabs to make 4 flaps n

Lesson 6. 4 : Rectangles n Characteristics of a rectangle: ¨ ¨ ¨ Both sets of opp. Sides are congruent and parallel Both sets opp. angles are congruent Diagonals bisect each other Diagonals split it into 2 congruent triangles Consecutive angles are supplementary If one angle is a right angle then all 4 are right angles n In a rectangle the diagonals are congruent. n If diagonals of a parallelogram are congruent, then it is a rectangle.

Quadrilateral EFGH is a rectangle. If GH = 6 feet and FH = 15 feet, find GJ.

Quadrilateral RSTU is a rectangle. If m RTU = 8 x + 4 and m SUR = 3 x – 2, find x.

Quadrilateral EFGH is a rectangle. If m FGE = 6 x – 5 and m HFE = 4 x – 5, find x.

Quadrilateral JKLM has vertices J(– 2, 3), K(1, 4), L(3, – 2), and M(0, – 3). Determine whether JKLM is a rectangle using the Distance Formula.

Lesson 6. 5 : Rhombi n n A quadrilateral with 4 congruent sides Characteristics of a rhombus: ¨ ¨ ¨ Both sets of opp. sides are congruent and parallel Both sets of opp. angles are congruent Diagonals bisect each other Diagonals split it into 2 congruent triangles Consecutive angles are supplementary If an angle is a right angle then all 4 angles are right angles n (special type of parallelogram) In a rhombus: ¨ ¨ Diagonals are perpendicular Diagonals bisect the pairs of opposite angles

6. 5: Squares (special type of parallelogram) n A quadrilateral with 4 congruent sides n Characteristics of a square: ¨ Both sets of opp. sides are congruent and parallel ¨ Both sets of opp. angles are congruent ¨ Diagonals bisect each other ¨ Diagonals split it into 2 congruent triangles ¨ Consecutive angles are supplementary ¨ If an angle is a right angle then all 4 angles are right angles ¨ Diagonals bisect the pairs of opposite angles n A square is a rhombus and a rectangle.

A. The diagonals of rhombus WXYZ intersect at V. If m WZX = 39. 5, find m ZYX. B. The diagonals of rhombus WXYZ intersect at V. If WX = 8 x – 5 and WZ = 6 x + 3, find x.

A. ABCD is a rhombus. Find m CDB if m ABC = 126. B. ABCD is a rhombus. If BC = 4 x – 5 and CD = 2 x + 7, find x.

QRST is a square. Find n if m TQR = 8 n + 8.

QRST is a square. Find QU if QS = 16 t – 14 and QU = 6 t + 11.

Determine whether parallelogram ABCD is a rhombus, a rectangle, or a square for A(– 2, – 1), B(– 1, 3), C(3, 2), and D(2, – 2). List all that apply. Explain.

6. 6: Trapezoids n A quadrilateral with exactly 1 pair of opposite parallel sides (bases), 2 pairs of base angles, and 1 pair of nonparallel sides (legs) n Diagonals of an isosceles trapezoid are congruent A B AC = BD base D leg Base angle n leg base n Base angle C Median (of a trapezoid): ¨ The segment that connects the midpoints of the legs Isosceles Trapezoid: ¨ A trapezoid with congruent legs and congruent base angles n The median is parallel to the bases Median = ½ (base + base)

A. Each side of the basket shown is an isosceles trapezoid. If m JML = 130, KN = 6. 7 feet, and LN = 3. 6 feet, find m MJK. B. Each side of the basket shown is an isosceles trapezoid. If m JML = 130, KN = 6. 7 feet, and JL is 10. 3 feet, find MN.

In the figure, MN is the midsegment of trapezoid FGJK. What is the value of x.

WXYZ is an isosceles trapezoid with median Find XY if JK = 18 and WZ = 25.

A. If WXYZ is a kite, find m XYZ.

A. If WXYZ is a kite, find m XYZ.