Download corresponding parts of the triangles are congruent

Chapter 4
Congruent Triangles
Chapter Objectives
•
•
•
•
•
•
•
Classification of Triangles by Sides
Classification of Triangles by Angles
Exterior Angle Theorem
Triangle Sum Theorem
Adjacent Sides and Angles
Parts of Specific Triangles
5 Congruence Theorems for Triangles
Lesson 4.1
Triangles and Angles
Lesson 4.1 Objectives
•
•
•
•
Identify the parts of a triangle
Classify triangles according to their sides
Classify triangles according to their angles
Calculate angle measures in triangles
Classification of Triangles by Sides
Name
Equilateral
Isosceles
Scalene
3 congruent sides
At least 2
congruent sides
No Congruent
Sides
Looks Like
Characteristics
Classification of Triangles by Angles
Name
Acute
Equiangular
Right
Obtuse
3 acute
angles
3 congruent
angles
1 right
angles
1 obtuse angle
Looks Like
Characteristics
Example 1
• You must classify the triangle as specific as
you possibly can.
• That means you must name
– Classification according to angles
– Classification according to sides
• In that order!
• Example
Obtuse isosceles
Vertex
• The vertex of a triangle is any point at which
two sides are joined.
– It is a corner of a triangle.
– There are 3 in every triangle
Adjacent Sides and Adjacent Angles
• Adjacent sides are those
sides that intersect at a
common vertex of a
polygon.
– These are said to be adjacent
to an angle.
• Adjacent angles are those
angles that are right next to
each other as you move
inside a polygon.
– These are said to be adjacent
to a specific side.
Special Parts in a Right Triangle
• Right triangles have special names that go with it
parts.
• For instance:
– The two sides that form the right angle are called the
legs of the right triangle.
– The side opposite the right angle is called the
hypotenuse.
• The hypotenuse is always the longest side of a right triangle.
hypotenuse
legs
Special Parts of an Isosceles Triangle
• An isosceles triangle has only two
congruent sides
– Those two congruent sides are called legs.
– The third side is called the base.
legs
base
More Parts of Triangles
• If you were to extend the sides you will see that more
angles would be formed.
• So we need to keep them separate
– The three original angles are called interior angles because they
are inside the triangle.
– The three new angles are called exterior angles because they lie
outside the triangle.
Example 2
Classify the following triangles by their sides and
their angles.
Scalene
Obtuse
Scalene
Right
Isosceles
Acute
Theorem 4.1:
Triangle Sum Theorem
• The sum of the measures of the interior
angles of a triangle is 180o.
B
mA + mB + mC = 180o
C
A
Example 3
Solve for x and then classify the triangle based
on its angles.
75
Acute
50
3x + 2x + 55 = 180
Triangle Sum Theorem
5x + 55 = 180
Simplify
5x = 125
SPOE
x = 25
DPOE
Theorem 4.2:
Exterior Angle Theorem
• The measure of an exterior angle of a triangle
is equal to the sum of the measures of the two
nonadjacent interior angles.
B
A
C
m A +m B = m C
Example 4
Solve for x
x = 50 + 70
Exterior Angle Theorem
x = 120
Simplify
Corollary to the
Triangle Sum Theorem
• A corollary to a theorem is a statement that can be
proved easily using the original theorem itself.
– This is treated just like a theorem or a postulate in proofs.
• The acute angles in a right triangle are
complementary.
A
mA + mB = 90o
B
C
Homework 4.1
• In Class
– 1-9
• p199-201
• In HW
– 10-26, 31-39, 41-47, 49, 50, 52-68
• Due Tomorrow
Lesson 4.2
Congruence and Triangles
Lesson 4.2 Objectives
• Identify congruent figures and their
corresponding parts.
• Prove two triangles are congruent.
• Apply the properties of congruence to
triangles.
Congruent Triangles
• When two triangles are congruent, then
– Corresponding angles are congruent.
– Corresponding sides are congruent.
• Corresponding, remember, means that objects are
in the same location.
– So you must verify that when the triangles are drawn
in the same way, what pieces match up?
Naming Congruent Parts
• Be sure to pay attention to the proper notation when
naming parts.
–  ABC   DEF
• This is called a congruence statement.
B
D
F
AD
BE
C F
and
A
C
E
AB  DE
BC  EF
AC  DF
Theorem 4.3:
Third Angles Theorem
• If two angles of one triangle are congruent to
two angles of another triangle, then the third
angles are congruent.
Prove Triangles are Congruent
• In order to prove that two triangles are
congruent, we must
– Show that ALL corresponding angles are
congruent, and
– Show that ALL corresponding sides are congruent.
• We must show all 6 are congruent!
Example 5
Complete the following statements.
a)
Segment EF  ___________
a)
segment OP
P  ________
b)
b)
F
G  ________
c)
c)
d)
Q
mO = ________
d)
e)
110o
QO = ________
e)
7 km
GFE  __________
f)
f)
QPO
•
Yes, the order is important!
Theorem 4.4:
Properties of Congruent Triangles
• Reflexive Property of Congruent Triangles
–  ABC   ABC
• Reflexive Property of 
• Symmetric Property of Congruent Triangles
– If  ABC   DEF, then  DEF   ABC.
• Symmetric Property of 
• Transitive Property of Congruent Triangles
– If  ABC   DEF and  DEF   JKL, then
 ABC   JKL.
• Transitive Property of 
Homework 4.2
• None!
Lesson 4.3
Proving Triangles are Congruent:
SSS
&
SAS
Lesson 4.3 Objectives
• Prove triangles are congruent using the SSS
Congruence Postulate
• Prove triangles are congruent using the SAS
Congruence Postulate
Postulate 19:
Side-Side-Side Congruence Postulate
• If three sides of one triangle are congruent to
three sides of a second triangle, then the two
triangles are congruent.
Postulate 20:
Side-Angle-Side Congruence Postulate
• If two sides and the included angle of one
triangle are congruent to two sides and the
included angle of a second triangle, then the
two triangles are congruent.
Which One Do I Use?
• Remember there are 6 parts to every triangle.
– Identify which parts of the triangle do you know (100%
sure) are congruent.
– Rotate around the triangle keeping one thing in mind.
• Cannot rotate so that 2 parts in a row are missed!
• That means as you rotate by counting angle, then side, then angle,
then side, then angle, and then side you cannot miss two pieces in
a row!
– You can skip 1, but not 2!!
– Be sure the pattern that you find fits the same pattern in
the same way from the other triangle.
• If it fits, they are congruent.
Example 6
Decide whether or not the congruence statement is true.
Explain your reasoning.
Reflexive Property of Congruence
Because the segment is shared between
two triangles, and yet it is the same
segment
The statement is true
because of
SSS Congruence
Reflexive Property of
Congruence
The statement is not
true because the
vertices are
out of order.
The statement is not
true because the
vertices are
out of order.
Example 7
Decide whether or not there is enough information to conclude
SAS Congruence.
Yes!
Reflexive Property of
Congruence
Yes!
No
Homework 4.3
• In Class
– 1-5
• p216-218
• HW
– 6-20
• Due Tomorrow
Lesson 4.4
Proving Triangles are Congruent:
ASA
&
AAS
Lesson 4.4 Objectives
• Prove that triangles are congruent using the
ASA Congruence Postulate
• Prove that triangles are congruent using the
AAS Congruence Theorem
Postulate 21:
Angle-Side-Angle Congruence
• If two angles and the included side of one
triangle are congruent to two angles and
the included side of a second triangle, then
the two triangles are congruent.
Theorem 4.5:
Angle-Angle-Side Congruence
• If two angles and a nonincluded side of one
triangle are congruent to two angles and the
corresponding nonincluded side of the second
triangle, then the two triangles are congruent.
Example 8
Complete the proof
Given
Given
Reflexive POC
SSS Congruence
Homework 4.4
• In Class
– 1-7
• p223-227
• HW
– 8-18 evens
• Due Tomorrow
Lesson 4.5
Using Congruent Triangles
Lesson 4.5 Objectives
• Observe that corresponding parts of
congruent triangles are congruent
Showing Triangles are Congruent
•
You only have 4 shortcuts right now to show that
two triangles are congruent to each other.
1.
2.
3.
4.
SSS Congruence
SAS Congruence
ASA Congruence
AAS Congruence
•
•
Otherwise you need to show all 6 parts of a triangle have
matching congruent parts to another triangle.
If you can use one of the above 4 shortcuts to show
triangle congruency, then we can assume that all
corresponding parts of the triangles are congruent
as well.
Surveying
•
MNP  MKL
–
Given
•
Segment NM  Segment KM
– Definition of a midpoint
•
LMK  PMN
– Vertical Angles Theorem
•
KLM  NPM
– ASA Congruence
•
Segment LK  Segment PN
– Corresponding Parts of Congruent Triangles
Example 9
Tell which triangles you show to be congruent in order to prove the statement
is true.
What postulate or theorem would help you show the triangles are congruent.
Show:
STV  UTV
Reflexive Property of
Congruence
STV  UTV
SSS Congruence
Corresponding Parts of
Congruent Triangles
Show:
Segment XY  Segment ZW
Reflexive Property of
Congruence
Alternate Interior Angles
Theorem (Parallel Lines)
WXZ  YZX
ASA Congruence
Corresponding Parts of
Congruent Triangles
Homework 4.5
• In Class
– 1-3
• p232-235
• HW
– 4-18, 25-36
• Due Tomorrow
Lesson 4.6
Isosceles,
Equilateral,
and
Right Triangles
Lesson 4.6 Objectives
• Use properties of isosceles and equilateral
triangles.
• Identify more properties based on the
definitions of isosceles and equilateral
triangles.
• Use properties of right triangles.
Isosceles Triangle Theorems
• Theorem 4.6: Base
Angles Theorem
– If two sides of a triangle
are congruent, then the
angles opposite them
are congruent.
• Theorem 4.7: Converse
of Base Angles
Theorem
– If two angles of a
triangle are congruent,
then the sides opposite
them are congruent.
Example 10
Solve for x
Theorem 4.7
Theorem 4.6
4x + 3 = 15
7x + 5 = x + 47
4x = 12
x=3
6x + 5 = 47
6x = 42
x=7
Equilateral Triangles
• Corollary to Theorem
4.6
– If a triangle is
equilateral, then it is
equiangular.
• Corollary to Theorem
4.7
– If a triangle is
equiangular, then it is
equilateral.
Example 11
Solve for x
Corollary to Theorem 4.6
Corollary to Theorem 4.6
In order for a triangle to be
equiangular, all angles must
equal…
It does not matter which two sides you
set equal to each other, just pick the
pair that looks the easiest!
2x + 3 = 4x - 5
3 = 2x - 5
5x = 60
8 = 2x
x = 12
x=4
Theorem 4.8:
Hypotenuse-Leg Congruence Theorem
• If the hypotenuse and a leg of a right triangle are
congruent to the hypotenuse and a leg of a second
right triangle, then the two triangles are congruent.
– Abbreviate using
• HL
Example 12
Determine if enough information is given to conclude the
triangles are congruent using HL Congruence
Reflexive Property of
Congruence
Yes they are congruent!
Reflexive Property of
Congruence
Neither triangle is a
right triangle, so…
Not congruent
Homework 4.6
• In Class
– 1-7
• p239-242
• HW
– 8-28 even, 33, 34
• Due Tomorrow
Lesson 4.7
Triangles
And
Coordinate Proof
Lesson 4.7 Objectives
• Place geometric figures in a coordinate plane.
• Use the Distance Formula to verify congruent
triangles.
Coordinate Proof
• A coordinate proof involves placing geometric figures in a
coordinate plane.
• Then you employ the following tools to prove concepts from
your picture
– Distance Formula
(x2 – x1)2 + (y2 – y1)2
– Midpoint Formula
(
(y1 + y2)
(x1 + x2)
2
,
2
)
Homework 4.7
• WS