Download 1 Classifying Triangles

4-1 Classifying Triangles
A. Definitions
1. A________ is a three sided polygon.
2. A____________ is a closed figure, in a plane, made up of segments, called
________, that intersect only at their endpoints, called ___________.
3. Triangle ABC, written
ABC, has the following parts.
sides: AB, BC,AC
vertices: A, B, C
angles: <ABC , <ACB , <BAC
A
B
C
B. Classifying triangles
1. By sides:
a. __________ – all 3 sides are different lengths.
b. __________ – at least two of the sides are the same length.
c. ____________ – All 3 sides are the same length
2. By angles
a. Acute triangle – All the angles are less than 90.
b. Right triangle – One of the angles is exactly 90.
c. Obtuse triangle – One of the angles is greater than 90.
C. Special Triangles
1. Right triangles
Hypotenuse
Leg
Leg
2. Isosceles triangles vertex angle
Vertex Angle
Leg
Leg
Base Angle
Base Angle
Base
Ex 1:
ABC is an equilateral triangle. Find x and the measure of each side if
AB = 4x -3 and BC = 3x + 4
Ex 2: Given with DAR vertices D(2, 6), A(4, -5), R(-3,0), use the distance
formula to show DAR is scalene.
Geometry 4-2 Measuring Angles in Triangles
A. Theorems
1. Theorem 4-1 – __________ _________ ____________ – The sum of the
measures of a triangle is 180.
Example 1: Find x in the triangle.
x
68o
33o
2. Theorem 4-2 – ______________ _____________ ________________ – If two
angles of one triangle are congruent to two angles of a second triangle, then the
3rd angles of the triangles are congruent.
3. Theorem 4-3 – _________________ _____________ _______________ -The
measure of an exterior angle of a triangle is equal to the sum of the measures of
the two remote interior angles.
B. Definitions
1. ______________ ____________ -<BCD is an exterior angle
2. _____________ ________________ ______________ - <A and <B are two
remote interior angles.
B
A
C
C. Explanation of Exterior Angle Theorem
Imagine <BCA = 70o, how big is <BCD =_______
Now, take a look at ABC , what is going to be the sum of <A and <B
?=_______
D
Example 2: Find the measure of each numbered angle.
65o
46o
82o
1
3
2
142o
4
5
3. A ___________ _____________ organizes a series of statements in logical
order, starting with the given statement. Each statement is written in a box with
the reason verifying the statement written below the box.
Example 3: Write a flow proof of the Exterior Angle Theorem.
ABC
Given
<CBD and <ABC form a Linear Pair
m<A + m<ABC + m<C = 180
<CBD and <ABC are Supplementary
m<A + m<ABC + m<C =
m<CBD + m<ABC
m<CBD + m<ABC = 180
m<A + m<C = m<CBD
4. A statement that can be easily proved using a theorem is called a
________________.
a.) Corollary 4.1 -The acute angles of a right triangle are complimentary.
Example: m<A + m<C = ___
b.) Corollary 4.2 -There can be at most one right or obtuse angle in a triangle.
Geometry 4-3 Congruent Triangles
A. Corresponding Parts of Congruent Triangles
1. Triangles that have the same size and shape are called ______________
______________.
ABC  XYZ -the order matters!
Vertex A corresponds to vertex X
2. Definition of Congruent Triangles (CPCTC) -Two triangles are congruent iff
their corresponding parts are congruent.
(CPCTC stands for Corresponding Parts of Congruent Triangles are Congruent it is the converse of the definition).
Example 1: If the two triangles are congruent.
Name all the congruent angles.
________ ________
________ ________
________ ________
Name all the congruent sides.
________ ________
________ ________
________ ________
3. Theorem 4.4 -Congruence of triangles is reflexive, symmetric, and transitive.
-meaning
ABC ABC for reflexive.
-meaning if ABC DEF, then DEF ABC
B. Identity Congruence Transformations
B
A
Y
C X
Z
1. In the picture, ABC  XYZ , if you turn, flip, or slide,
affect the congruence of the triangle.
2. These changes are called congruence transformations.
ABC  DEF
XYZ , it does not
Example 2: The vertices of RST are at R(-3, 0), S(-1, 5), and T(1, 1), and the
vertices of R’S’T’ are at R’(3, 0), T’(-1, -1), S’(1, -5).
a.) Prove that RST 
R’S’T’
b.) Name the congruence transformation for
RST and R’S’T’
Geometry 4-4 Proving Triangles Congruent
A. Postulate 4-1 SSS Postulate (Side-Side-Side Postulate)
-If the sides of one triangle are congruent to the sides of a second triangle, then
the triangles are congruent.
Ex 1: Given STU with vertices S(0, 5), T(0, 0), and U(-2, 0), and XYZ with
vertices X (4, 8), Y(4, 3) and Z(6, 3), determine if
STU XYZ .
ST=_____
TU=_____
SU=_____
XY=_____
YZ=_____
XZ=_____
B. Postulate 4-2 SAS Postulate (Side-Angle-Side Postulate)
-If two sides and the included angle of one triangle are congruent to two sides
and the included angle of another triangle, then the triangles are congruent.
Ex 2: Given: X is the midpoint of BD.
X is the midpoint of AC.
Prove:
DXC BXA
D
A
X
C
B
Method 1 Two-Column Proof
Statements
Reasons
1. X is the midpoint of BD
X is the mid point of AC
1.) Given
2.) DX XB
AX XD
2.)
3.) <DXC  <BXA
3.)
4.)
DXC 
BXA
4.)
Method 2 – Flow Proof
X is midpoint of BD
X is midpoint of AC
<DXC  <BXA
DX BX
CX  AX
DXC  BXA
Geometry 4-5 More Congruent Triangles
A. Current Congruence
1. We know three ways to prove triangles congruent so far. They are ______,
________, _________.
B. Postulate 4-3 ASA Postulate (Angle-Side-Angle Postulate)
-If two angles and the included side of one triangle are congruent to two angles
and the included side of another triangle, then the triangles are congruent.
Ex 1:
Given: BE bisects AD,
and <A D
Prove: AB DC
Statements
1) BE bisects AD, <A  <D
2)
Reasons
1) Given
2) Vertical <’s are 
3) AE ED
3)
4)
AEB DEC
4)
5) AB  DC
C. Theorem 4-5 AAS Postulate (Angle-Angle-Side Postulate)
-If two angles and a non-included side of one triangle are congruent to the
corresponding two angles and side of a second triangle, then the two triangles are
congruent.
Ex 2: Write a proof
Given: <PSU  <PTR
SU  TR
Prove: SP  TP
S
T
R
U
P
Statements
Reasons
1) < PSU  <PTR, SU  TR
2)
1) Given
2) Reflexive
3)
3)
4) SP  TP
Ex 3: Some of the measurements of
ABC and
determine if the two triangles are congruent?
DEF are given. Can you
Explore: Since we know <A  <D, and we know AB  DE, and CB  FE ; we
know that there are two congruent sides and one non-included angle that are
congruent. Our five methods of proving triangles congruent are:
1.) definition of congruent
2.) SSS congruence
3.) SAS congruence
4.) ASA congruence
5.) AAS congruence
’s (all 3 sides and all 3 angles)
-since none of these work, the triangles must not be congruent.
Counterexample: here is a case where the triangles are obviously not congruent,
but the same conditions hold true.
D. HL (Hypotenuse-Leg) – A special case of SSA that only works when trying
to prove 2 or more right triangles are congruent.
4-6 Analyzing Isosceles Triangles
A. Isosceles Triangles
1. Theorem 4-6 Isosceles Triangle Theorem - If two sides of a triangle are
congruent, then the angles opposite those sides are congruent
Ex 1: Proof of Theorem 4-6
Given: PQR, PQ  QR
Prove: <P  <R
Statements
Reasons
1) Let S be the midpoint of PR
2) Draw segment QS
3) PS  RS
1) Every segment has exactly one midpoint
2) Through any 2 points there is 1 line
3)
4) QS  QS
4)
5) PQ  QR
5)
6)
PQS  RQS
7) <P  <R
6)
7)
Ex 2: In isosceles
ISO with base SO. If m<S = 2x + 40 , and m<F = 3x + 22,
find the measure of each angle of the triangle.
2. Theorem 4-7 -If two angles of a triangle are congruent, then the sides opposite
those angles are congruent
3. Corollary 4-3 -A triangle is equilateral if and only if it is equiangular.
4. Each angle of an equilateral triangle measures 60o
.