geometry chapter 2 -...
TRANSCRIPT
GEOMETRY CHAPTER 22-4: Use Postulates and Diagrams
Warm-Up
■ State which postulate justifies each statement
1.) If 𝑫 is the interior of < 𝑨𝑩𝑪, then
𝒎 < 𝑨𝑩𝑪 = 𝒎 < 𝑨𝑩𝑫 +𝒎 < 𝑫𝑩𝑪
2.) A protractor was used to find the measure of an
angle to be 𝟏𝟐𝟓°
3.) If M is between X and Y, then
𝐗𝐌+𝐌𝐘 = 𝐗𝐘
Angle Addition Postulate
Protractor Postulate
Segment Addition Postulate
2-4: Use Postulates and Diagrams
■ Objective: Students will be able to know and
use postulates related to points, lines and
planes.
■ Agenda
– Postulates (Old and New)
– Practice
– Sketches
– Interpreting
Basic Terms
■ Postulate: A basic assumption that is accepted without proof.
■ Exists: There is at least one.
■ Unique: There is no more than one.
■ One and only one: There is exactly one.
■ Determine: To decline or specify.
■ Theorem: A statement that can be proved using postulates,
definitions, and previously used theorems.
Postulates
■ In chapter 1, we covered 4 postulates
– Postulate 1: Ruler Postulate
– Postulate 2: Segment Addition Postulate
– Postulate 3: Protractor Postulate
– Postulate 4: Angle Addition Postulate
■ Here are seven new postulates
Postulate #1
■ The Ruler Postulate:
– The Distance between points A and B, named AB, is equal
to the absolute value of the difference of the coordinates
of A and B.
Postulate #2
■ Segment Addition Postulate:
If B is between A and C, then 𝐴𝐵 + 𝐵𝐶 = 𝐴𝐶.
Postulate #3
■ The Protractor Postulate:
The measure of < 𝑨𝑶𝑩 is equal to the absolute value of the difference between the
real numbers for 𝑶𝑨 and 𝑶𝑩.
Postulate #4
■ The Angle Addition Postulate:
If 𝑷 is the interior of < 𝑹𝑺𝑻, then 𝒎 < 𝑹𝑺𝑻 = 𝒎 < 𝑹𝑺𝑷 +𝒎 < 𝑷𝑺𝑻
New Postulates – Pg 88 (Textbook)
■ Postulate #5:
Through any two points their exists exactly one line
■ Postulate 6:
A line contains at least two points
A B
A Bm
New Postulates – Pg 88 (Textbook)
■ Postulate #7:
If two lines intersect, then their intersection is exactly
one point.
■ Postulate #8:
Through any three noncollinear points their exists
exactly one plane.
m
n
A
A
B C
New Postulates – Pg 88 (Textbook)
■ Postulate #9:
A plane contains at least three noncollinear points.
■ Postulate 10:
If two points lie in a plane, then the line containing
them is also in the plane.
A
B C
D
AB
New Postulates – Pg 88 (Textbook)
Postulate 11:
If two planes intersect, then their intersection is a line.
State the Postulate is being illustrated by the diagram.
1.) 2.)
Practice With Postulates
Postulate 8Postulate 10
State the Postulate illustrated by the diagram.
3.)
Practice With Postulates
If
Postulate 7
Then
State the Postulate illustrated by the diagram.
4.)
Practice With Postulates
Postulate 11
Sketch a diagram showing 𝑻𝑽intersecting 𝑷𝑸 at point 𝑾, so that
𝑻𝑾 ≅ 𝑾𝑽
You can sketch the diagram using the
following steps:
Step 1: Draw 𝑻𝑽 and label points T
and V
Step 2: Draw point W at the midpoint
of 𝑻𝑽, and mark the congruent
segments
Step 3: Draw 𝑷𝑸 through 𝑾
Sketch a Diagram
T
V
W
P
Q
Perpendicular Figures
A line is a line perpendicular to a plane if and only if the line intersects
the plane in a point and is perpendicular to every line in the plane that
intersects it at that point.
A m
p
Interpreting a Diagram
Which of the following statements
cannot be assumed from the
diagram?
A, B, and F are collinear
E, B, and D are collinear
𝐴𝐵 ⊥ 𝑝𝑙𝑎𝑛𝑒 𝑆
𝐶𝐷 ⊥ 𝑝𝑙𝑎𝑛𝑒 𝑇
𝐴𝐹 intersects 𝐵𝐶 at point B
Cannot be assumed (Why?)
Cannot be assumed (Why?)
Extra Practice
■ Sketch Diagrams to Illustrate the following:
𝑳𝑵, 𝑨𝑩, and 𝑫𝑪 intersecting at point M, 𝑨𝑩 bisecting 𝑳𝑵,
and 𝑫𝑪 ⊥ 𝑳𝑵.
L NM
A
B
D
C
HW 2-4
■ Pg. 91-94
■ #’s 3 – 10, 14 – 24, 27, 30 – 32, 35 – 38
■ (EC) 11 – 13