Download Ch.6 Introduction to Geometry

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Ch.6 Introduction to Geometry
Chapter 6 Notes Page 1
A point, represented as a small dot, has no length, width, or thickness.
A line, connecting two points along the shortest possible path, has no thickness and extends
infinitely in both directions.
A plane is a flat surface with no thickness and no boundaries.
A ray is a half-line with its endpoint included.
A portion of a line joining two points and including the endpoints is called a line segment.
An angle is formed by the union of two rays that have a common endpoint. One ray is called the
initial side and the other the terminal side.
The common endpoint of an angle’s initial side and terminal side is the vertex of the angle.
One way to measure angles is in degrees, symbolized by a small, raised circle.
An acute angle measures less than 90°.
A right angle, one-quarter of complete rotation, measures 90°.
An obtuse angle measures more than 90°, but less than 180°.
A straight angle, one-half a complete rotation, measures 180°.
Two angles whose measures have a sum of 90° are called complementary angles.
Two angles whose measures have a sum of 180° are called supplementary angles.
When two lines intersect, the opposite angles formed are called vertical angles. (Vertical angles
have the same measure)
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Chapter 6 Notes Page 2
Parallel lines are lines that lie in the same plane and have no points in common. If two different
lines in the same plane are not parallel, they have a single point in common and are called
intersecting lines. If the lines intersect at an angle of 90°, they are called perpendicular lines.
Ex1. The hand of a clock moves from 12 to 2 o’clock. Through how many degrees does it move?
Ex2. The hand of a clock moves from 12 to 5 o’clock. Through how many degrees does it move?
Ex3. Find 𝑚∠𝐷𝐵𝐶.
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Ex4. Show that ∠ABD and ∠DBC are supplementary angles.
Chapter 6 Notes Page 3
a) If m∠ABD is 66° greater than m∠DBC, find the measure of each angle.
b) If m∠ABD is 88° greater than m∠DBC, find the measure of each angle.
Ex5. One of the angles has a measure of 35°. Find the measure of each of the other seven
angles.
Triangles
A triangle is a geometric figure that has three sides, all of which lie on a flat surface or plane.
(The sum of the measures of the triangle is 180°)
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Chapter 6 Notes Page 4
Figures that have the same shape, but not the same size, are used in scale drawings.
Figures that have the same shape, but not necessarily the same size, are called similar figures.
In similar triangles, the lengths of the corresponding sides are proportion.
Ex1. Find the measure of angle A for triangle ABC
Ex2. Find the measures of angle 1 through 5.
Ex3. Explain why the given triangles are similar. Then find the missing length, x.
a)
b)
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Chapter 6 Notes Page 5
Ex4. A man who is 6 feet tall is standing 10 feet from the base of a lamppost. The man’s shadow
has a length of 4 feet. How tall is the lamppost?
Ex5. Find the height of the lookout tower using the figure that lines up the top of the tower
with the top of a stick that is 2 yards long and 3.5 yards from the line to the top of the tower.
Ex6. Find the length of the hypotenuse c in the right triangle shown.
Polygons, Perimeter, and Tessellations
The distance around the outside edges of the figure is called a perimeter.
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Chapter 6 Notes Page 6
Any closed shape in the plane formed by three or more line segments that intersect only at
their endpoints is a polygon.
A four-side polygon is called a quadrilateral.
A polygon whose sides are all the same length and whose angles all have the same measure is
called a regular polygon.
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Chapter 6 Notes Page 7
Ex1. The rectangular field has a length of 42 yards and a width of 28 yards. If fencing costs
$5.25 per foot, find the cost to enclose the field with fencing.
Ex2.
a) Find the sum of the measures of the angles of an octagon.
b) Find the measure of angle A.
c) Find the measure of exterior angle B.
Ex3. Find the perimeter of the figure.
Area and Circumference
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Chapter 6 Notes Page 8
Ex1. Find the area of each figure. Where necessary, express answers in terms of 𝜋 and then
round to the nearest tenth.
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Chapter 6 Notes Page 9
Ex2. Find the circumference and area of each circle. Where necessary, express answers in
terms of 𝜋 and then round to the nearest tenth.
Volume and Surface Area
The volume refers to the amount of space occupied by a solid object, determined by the number
of cubic units it takes to fill the interior of that object.
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Chapter 6 Notes Page 10
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Ex1. Find the volume of each figure.
Ex2. Find the surface area of figure.
Chapter 6 Notes Page 11