Download parallelogram

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.
a) AB = BC
b) AB = CD
Properties of Special
Parallelograms
Solutions
a) sometimes
b) 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)
a) 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