Download Geometry 8-1 Angles of Polygons

Geometry 8-1 Angles of Polygons
A. Sum of Measures of Interior Angles
1. Interior angles -The sum of the measures of the angles of each polygon can be
found by adding the measures of the angles of a triangle.
2. Theorem 8-1 (Interior angle sum Theorem) -If a convex polygon has n sides and
S is the sum of the measures of the interior angles, then S = 180(n - 2).
Convex Polygon
Number of
Number of
Sum of the angle
Sides
Triangles
measures
triangle
quadrilateral
pentagon
hexagon
heptagon
octagon
Ex 1: Find the measure of each interior angle of a regular pentagon.
-Use the interior angle sum theorem to find the sum of the angle measures.
S = 180(n - 2)
S = 180(___- 2)
S = ______ so each angle is ____/5 or ______.
Ex 2: Find the measure of each interior angle.
R
U
S
T
B. Sum of Measures of Exterior Angles
1. Theorem 8-2 Exterior Angle Sum Theorem -If a polygon is convex, then the sum
of the measures of the exterior angles, one at each vertex is 360.
____________________________________________
Ex 3: Find the measures of an exterior and interior angle for a regular hexagon.
Geometry 8-2 Parallelograms
A. Sides and angles of parallelograms
1. A quadrilateral with _______________ opposite sides is called a parallelogram.
B. Theorems
1. Theorem 8-3 Opposite sides of a parallelogram are congruent: Opp. sides of □
are≡.
2. Theorem 8-4 Opposite angles of a parallelogram are congruent: Opp. <’s of □
are ≡.
3. Theorem 8-5 Consecutive angles of a parallelogram are supplementary: Cons.
<’s of □ are suppl.
4. Theorem 8-6 If a parallelogram has one right angle, it has 4 right angles: If □
has 1 rt.>, it has 4 rt. <’s.
Ex. 1: Write a 2 column proof for Theorem 8-4
Given: □ABCD
Prove: <A______<C, <B______<D.
Statements
1. □ABCD
2. AB||DC , AD||BC
3. <A and <D are supplementary
<D and <C are supplementary
<C and <B are supplementary
4. <A <C , <B <D
Reasons
Ex. 2: RSTU is a parallelogram, find m<URT , m<RST , and y.
C. Diagonals of Parallelograms
1. Theorem 8-7: The diagonals of a parallelogram bisect each other. So: RQ
____QT and ____________
Ex: 3: What are the coordinates of the intersection of the diagonals of
parallelogram MNPR with vertices M(-3, 0), N(-1, 3), P(5, 4), R(3, 1).
2. Theorem 8-8 Each diagonal of a parallelogram separates the parallelogram into
two congruent triangles: Diag. separates □ into 2 congruent ▲.
Geometry 8-3 Tests for Parallelograms
A. Conditions for a parallelogram
1. Theorem 8-9 -If both pairs of opposite sides of a quadrilateral are congruent,
then the quadrilateral is a parallelogram: If both pairs of opp. sides are_____, then
quad. is _________ .
2. Theorem 8-10 -If both pairs of opposite angles of a quadrilateral are congruent,
then the quadrilateral is a parallelogram: If both pairs of opp. Ð's are . , then quad.
is Y .
3. Theorem 8-11 -If the diagonals of a quadrilateral bisect each other, then the
quadrilateral is a parallelogram. abbreviation: If diag. bisect each other, then quad.
is Y .
4. Theorem 8-12 -If one pair of opposite sides of a quadrilateral is both parallel and
congruent, then the quadrilateral is a parallelogram. abbreviation: If one pair of opp
sides is and ___________________________________________, then quad. is Y .
Ex 1: Determine whether the quadrilateral is a parallelogram.
A
20
B
12
12
C
20
D
Ex 2: Find x and y so that each quadrilateral is a parallelogram.
a.) E
8y – 15
F
b.) A
B
(5y + 28o)
6x – 12
2x + 16
(6y+14o)
D
H
7y
C
G
B. Parallelograms and the Coordinate Plane -You can use the Distance Formula
(or Pythagorean Theorem) to determine if a quadrilateral is a parallelogram in the
coordinate plane.
Ex 3: The coordinates of the vertices of a quadrilateral PQRS are P(-5,3), Q(-1,5),
R(6,1) and S(2,-1). Determine if quadrilateral PQRS is a parallelogram.
A quadrilateral is a parallelogram if and only if any one of the following is true:
1. Both pairs of opposite sides are parallel. (Definition)
2. Both pairs of opposite sides are congruent. (Theorem 8-9)
3. Diagonals bisect each other. (Theorem 8-10)
4. Both pairs of opposite angles are congruent (Theorem 8-11)
5. A pair of opposite sides is both parallel and congruent (Theorem 8-12)
Geometry 8-4 Rectangles
A. Properties of Rectangles
1. Definition -A rectangle is a quadrilateral with four right angles.
-if both pairs of opposite angles are congruent, then it is a parallelogram.
Thus a rectangle is a parallelogram.
A
B
M
D
C
2. Theorem 8-13: If a parallelogram is a rectangle, then its diagonals are
congruent.
abbreviation: if □ is a rectangle, diag. are .
Ex 1: Quadrilateral RSTU is a rectangle. If RT = 6x + 4, and SU = 7x - 4 , find x.
R
S
U
T
Ex 2: Find x and y if MNPL is a rectangle.
M
N
6y + 2
o
5x + 8o
3x + 2o
L
P
3. Theorem 8-14 If the diagonals of a parallelogram are congruent, then the
parallelogram is a rectangle.
Ex. 3: Determine whether parallelogram ABCD is a rectangle, given A(-2, 1), B(4,
3), C(5, 0), and D(-1, -2).
Geometry 8-5 Rhombi and Squares
A. Properties of Rhombi
1. A ________________ is a quadrilateral with all 4 sides congruent.
2. Theorem 8-15 -The diagonals of a rhombus are perpendicular.
AC BD and ___________
3. Theorem 8-16 -If the diagonals of a parallelogram are perpendicular, then the
parallelogram is a rhombus.
4. Theorem 8-17 -Each diagonal of a rhombus bisects a pair of opposite angles;
so <ABD  <CBD  <ADB _________,and <DAC  <BAC _________
_________
Ex 1: a.) Find y if m<1 = y2 – 10
b.) Find m<PNL if m<MPL = 64.
Ex 2: Determine whether parallelogram ABCD is a rhombus, a rectangle, or a
square for A(-4, -2), B(-2, 6) C(6, 4), D(4, -4). List all that apply.
Ex 3: A square table has four legs that are 2 feet apart. The table is placed over an
umbrella stand so that the hole in the center of the table lines up with the hole in
the stand. How far away from a leg is the center of the hole?
Properties of Rhombi and Squares Rhombi and Squares
1. A rhombus has all the properties of a
properties of a
parallelogram
2. All sides are congruent.
properties of a
3. Diagonals are perpendicular.
properties of a
1. A square has all the
parallelogram
2. A square has all the
rectangle.
3. A square has all the
rhombus.
4. Diagonals bisect the angles of the
rhombus.
Geometry 8-6 Trapezoids
A. Properties of Trapezoid
1. A ____________________ is a quadrilateral with exactly one pair of parallel
sides.
2. The parallel sides are called __________________.
3. The ________________ _______________ are formed by the base and one of
the legs.
4. The non-parallel sides are called _______________.
Base 1
A
B
Leg 1
Leg 2
D
C
Base 2
5. If the legs are congruent, then the trapezoid is an ___________________
__________________.
6. Theorem 8-18 - Both pairs of base angles of an isosceles trapezoid are
congruent.
7. Theorem 8-19 - The diagonals of an isosceles trapezoid are congruent.
Ex 1: Finish the flow proof of Theorem 8-19.
Given: MNOP is an isosceles trapezoid
Proof:
Prove: MO  NP
Ex 2: ABCD is a quadrilateral with vertices A(5, 1), B(-3, -1), C(-2, 3), and D(2,
4).
a.) Verify that ABCD is a trapezoid.
b.) Determine whether ABCD is an isosceles trapezoid. Explain.
B. Medians of Trapezoids
1. The segment that joins midpoints of the legs of a trapezoid is the median.
(sometimes called the ________________________)
2. Theorem 8-20 -The median of a trapezoid is parallel to the bases and its measure
is one-half the sum of the bases.
Ex: MN = ½ (AB + CD)
A
B
M
N
D
C
Ex 3: DEFG is an isosceles trapezoid with median MN.
a.) Find DG if EF = 20 and MN = 30.
b.) Find m<1, m<2, m<3 , and m<4 if m<1 = 3x + 5 and m<3 = 6x – 5.
A
B
3
M
N
1
D
4
2
C
Geometry 8-7 Coordinate Proof With Quadrilaterals
A. Position Figures
Ex 1: Position and label a square with sides a units long on the coordinate plane.
1.) Let A, B, C, and D be the vertices of the square.
2.) Place the square with vertex A at the origin, Place the square with vertex A at
the origin, AB along the positive x-axis, and AD along the y-axis. Label the
vertices A, B, C and D.
3.) The y-coordinate of B is 0 because the vertex is on the x-axis. Since the side
length is a, the x-coordinate is _______.
4.) D is on the y-axis so the x-coordinate is 0. The y-coordinate is 0 + a or ___.
5.) The x-coordinate of C is also ___. The y-coordinate is 0 + a, or ___ because
side BC is a units long.
Ex 2: Name the missing coordinate for the isosceles trapezoid.
B. Prove Theorems
1. Once we have place a figure on the coordinate plane, we can use slope formula,
distance formula and midpoint formula to prove theorems.
Ex 3: Place a square on the coordinate plane. Label the midpoints of the sides, M,
N, P, and Q. Write a coordinate proof to show MNPQ is a square.
a.) By midpoint formula, the coordinates of M, N, O, and P are as follows:
M (__ , ___), N (___ , ___), P (___ , ___), Q (___, ___)
b.) Find the slopes of QP, MN, QM, and PN .
Each pair of opposite sides is _______, therefore MNPQ is a __________, and a
____________ and a ____________
c.) Use the distance formula to find QP and QM.
MNPQ is a square because each pair of opposite sides is parallel, and consecutive
sides form _________ _________ and are ____________.