Chapter 6
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Transcript of Chapter 6
Types of Polygons
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
Interior Angle Sum Theorem The sum of the
measures of the interior angles of a polygon is found by S=180(n-2)
Ex: Hexagon
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
Find the measure of an interior and an exterior angle for each polygon.
24-gon
3x-gon
Find the measure of an exterior angle given the number of sides of a polygon
260 sides
Lesson 6.1: Angles of Polygons
The measure of an interior angle of a polygon is given. Find the number of sides.
175
168.75
A pentagon has angles (4x+5), (5x-5), (6x+10), (4x+10), and 7x. Find x.
180-175=5360/5= 72
Lesson 6.2: Parallelograms
Properties of Parallelograms
Opposite sides of a parallelogram are congruent
Opposite angles in a parallelogram are congruent
Consecutive angles in a parallelogram are supplementary
If a parallelogram has 1 right angle, it has 4 right angles.
The diagonals of a parallelogram split it into 2 congruent triangles
The diagonals of a parallelogram bisect each other
A parallelogram is a quadrilateral with both pairs of opposite sides parallel
A. ABCD is a parallelogram. Find AB.
B. ABCD is a parallelogram. Find mC.
C. ABCD is a parallelogram. Find mD.
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 If…
Both pairs of opposite sides are parallelBoth pairs of opposite sides are congruentBoth pairs of opposite angles are congruentThe diagonals bisect each otherOne pair of opposite sides is congruent and
parallel Then 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.
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
Unfold the paper and hold width wise Fold the edges in to meet at the center
crease Cut the creases on the tabs to make 4
flaps
Lesson 6.4 : Rectangles
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
In a rectangle the diagonals are congruent.
If diagonals of a parallelogram are congruent, then it is a rectangle.
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.
6.5: Squares (special type of parallelogram)
A quadrilateral with 4 congruent sides
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 Diagonals are perpendicular
A square is a rhombus and a rectangle.
Lesson 6.5 : Rhombi (special type of parallelogram)
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
In a rhombus: Diagonals are perpendicular Diagonals bisect the pairs of
opposite angles
A. The diagonals of rhombus WXYZ intersect at V.If mWZX = 39.5, find mZYX.
B. The diagonals of rhombus WXYZ intersect at V. If WX = 8x – 5 and WZ = 6x + 3, find x.
A. ABCD is a rhombus. Find mCDB if mABC = 126.
B. ABCD is a rhombus. If BC = 4x – 5 and CD = 2x + 7, find x.
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
A quadrilateral with exactly 1 pair of opposite parallel sides (bases), 2 pairs of base angles, and 1 pair of non-parallel sides (legs)
Isosceles Trapezoid: A trapezoid with congruent legs
and congruent base angles
Diagonals of an isosceles trapezoid are congruent
Median (of a trapezoid): The segment that connects
the midpoints of the legs
The median is parallel to the bases
base
base
leg leg
Base angle Base angle
Median = ½ (base + base)
A B
CD
AC = BD
A. Each side of the basket shown is an isosceles trapezoid. If mJML = 130, KN = 6.7 feet, and LN = 3.6 feet, find mMJK.
B. Each side of the basket shown is an isosceles trapezoid. If mJML = 130, KN = 6.7 feet, and JL is 10.3 feet, find MN.