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Chapter 6
Section 1
Let’s look at geometry. There will be no proofs, so this will be a cookbook chapter.
Definition (Line Segment). A line segment is a piece of a line between two points.
Definition (Ray (Half-line)). Is “half ” a line. It consists of a point and a line emanating from that point.
Line segment Ray
Definition (Angle). An angle is formed by two rays with a common endpoint.
Definition (Vertex). The vertex of angle is the common endpoint of the rays that make the angle.
Definition (Side). The side of an angle is one of the rays that make the angle.
B Vertex
A
C
Side
Side
θ
The angle is designated by: angle ABC, angle CBA, angle B, angle θ, ∠B. The angle can be thought of as being generated by a ray rotating about its endpoint from some initial position to a terminal position.
initial side
terminal side θ
Definition (Units of Angular Measure). Units of angular measure are usually the degree and the radian
Definition (Measure). The measure of an angle is the number of units of measure it contains.
Definition (Equality of Angles). Two angles are said to be equal if they have the same measure.
Definition (Degree). A degree is a unit of measure of an angle. One degree is
of a full revolution.
Definition (Right Angle). A right angle is an angle whose measure is 90 ◦.
Definition (Acute Angle). An acute angle is an angle whose measure is less than 90 ◦.
Definition (Obtuse Angle). An obtuse angle is an angle whose measure is more than 90 ◦^ and less than 180 ◦.
Right angle Acute angle Obtuse angle
Definition (Perpendicular lines). Two lines that meet at a right angle are perpendicular.
Definition (Complementary Angles). Complementary angles are angles whose sum is 90 ◦.
Definition (Supplementary Angles). Supplementary angles are angles whose sum is 180 ◦.
Example 1.
- The angles 60 ◦^ and 30 ◦^ are complementary. 60 ◦^ + 30◦^ = 90◦
- The angles 63 ◦^ and 117 ◦^ are supplementary. 63 ◦^ + 117◦^ = 180◦
Definition (Opposite Angles). The angles A and B in the figure below are called opposite angles.
B A
Theorem. Opposite angles are equal.
Definition (Transversal). A transversal is a line that intersects a family of lines.
Let the lines L 1 and L 2 be parallel:
L 1
L 2
T
A B
C D
E F
G H (^) Exterior angles: A,B,G,H Interior angles: C,D,E,F Corresponding angles: A,E; C,G; B,F ; D,H Alternate interior angles: C,F ; D,E
Theorem. If two parallel lines are cut by a transversal, then corresponding angles are equal and alternate interior angles are equal.
Example 3. Find ∠A,∠B,∠C.
73 ◦
A
34 ◦ B
C
Definition (Corresponding Segments). Let a family of parallel lines be cut by two transversal lines. The segments of the transversals lying between the same parallel lines are called corresponding segments.
a (^) b
c (^) d
e (^) f
So the segments a,b; c,d; e,f are corresponding segments.
Theorem. When a family of parallel lines are cut by two transversal lines the ratios of corresponding segments on the transversals are equal. a b
c d
e f
Example 4. Example 3 page 154.
HW 6.1: 1-6 all
Definition (Scalene Triangle). A scalene triangle is a triangle that has no equal sides.
Definition (Isosceles Triangle). An isosceles triangle is a triangle that has 2 equal sides.
Definition (Equilateral Triangle). An equilateral triangle is a triangle that has 3 equal sides.
An acute triangle has 3 acute angles. An obtuse triangle has 1 obtuse angle and 2 acute angles. An right triangle has 1 right angle and 2 acute angles.
Definition (Altitude/Base). The altitude of a triangle is the perpendicular distance from one side, the base, to the farthest point.
altitude
base
altitude
base
Theorem (Area of a Triangle). If b is the base and h is the altitude of a triangle, then its area is
A =
bh.
Sometimes we do not know the altitude of a triangle, but we know the lengths of its sides. We can find the area by:
Theorem (Heron’s Formula). If the sides of a triangle are a, b and c, let
s = a + b + c 2
We call s the semiperimeter. Then the area is
A =
s(s − a)(s − b)(s − c).
Example 8. The sides of a triangle are 3, 4 and 5. Find the area.
From the last section we found that the sum of the angles of an n-gon is (n − 2)180◦. So,
Theorem (Sum of the Angles of a Triangle). If a triangle has interior angles A, B and C, then
A + B + C = 180◦.
Definition (Exterior Angle). An exterior angle of a triangle is an angle between a side and an extension of the adjacent side.
θ
B
A
Theorem (Exterior Angle of a Triangle). An exterior angle equals the sum of the 2 opposite interior angles.
θ = A + B.
Definition (Congruent Triangles). Congruent triangles are triangles whose sides and angles are equal.
Definition (Similiar Triangles). Similar triangles are triangles whose angles are equal.
Definition (Corresponding Sides). Corresponding sides of similar triangles are the sides that lie between the same pair of equal angles.
a
b
c
d
e
f
So the corresponding sides are: a,d; b,e; c,f.
Theorem.
- If 2 angles of a triangle are equal to 2 angles of another triangle, then their third angles are equal.
- Corresponding sides in similar triangles are in proportion.
Example 9. Example 8, page 158
Recall that the side opposite the right angle in a right triangle is called the hypotenuse and the other sides are called legs.
Theorem (Pythagorean Theorem). If the legs of a right triangle are a and b and the hypotenuse is c, then
a^2 + b^2 = c^2.
Section 4
Theorem. Let a circle have radius r and diameter d = 2r. Then,
- Circumference: C = 2πr = πd
- Area: A = πr^2 = πd^2 4
Definition (Arc). An arc is a portion of a circle.
Definition (Sector). A sector is the region bounded by two radii and one of the arcs they intersect.
Definition (Chord). A chord is a line segment that intersects the circle at two points.
Definition (Segment). A segment is the region bounded by a chord and one of the arcs it makes.
arc
sector
chord
segment
Theorem (Intersecting Chords). If two chords in a circle intersect, the product of the parts of one chord is equal to the product of the parts of the other chord. ab = cd
a
b c
d
Example 13. Example 14, page 165
Definition (Tangent). A tangent is a line that intersects a circle at exactly one point.
Definition (Secant). A secant is a line that intersects a circle at two points.
Theorem (Tangent to a Circe). A tangent is perpendicular to the radius drawn through the point of contact.
Example 14. Example 15, page 165
Theorem (Tangents to a Circle). Two tangents drawn to a circle from a point outside the circle are equal and they make equal angles with a line from the point to the center of the circle.
x
x
Theorem. Any angle inscribed in a semicircle is a right angle.
Example 15. HW 4,8,
HW 6.4: 1,5,9,11,13,
