Transcript Geometry Chapter 1 – The Basics of Geometry

Chapter 4 – Congruent Traingles
The Bigger Picture
-Properties of Triangles and their classification based on their
sides and angles
-Extension of the angle theorems to help solve problems
regarding angle measures
-The applicability of the properties of triangles in the areas of
art, architecture, and engineering
The “What” and the “Why”
Classifying triangles by their sides
and their angles
Finding angle measures in triangles
-Laying the foundation understanding the
angles that underlie the design of objects
-Identify congruent figures and
corresponding parts
- Analyzing patterns in order to make
conjectures regarding future or repeating
patterns
-Use congruent triangles to plan and
write proofs
- Prove triangular parts of the framework of a
bridge or other engineering design are
congruent
Prove that Triangles are Congruent
-Using corresponding sides and angles
-Using the SSS and SAS Congruence Postulates
-Using the ASA Congruence Postulate
-Using the AAS Congruence Theorem
-Using the HL Congruency Theorem
-Using Coordinate Geometry
Use properties of Isosceles, equilateral, and
right triangles
-Applying the laws of physics such as the law of reflection
-Identifying and using triangle relationships in architectural
and engineering design
Congruent Triangles
On a cable stayed bridge the cables
attached to each tower transfer
the weight of the roadway to the
tower.
You can see from the smaller diagram
that the cables balance the weight
of the roadway on both sides of
each tower.
In the diagrams what type of angles
are formed by each individual
cable with the tower and roadway?
What do you notice about the triangles
on opposite sides of the towers?
Why is that so important?
Names of Triangles
Classification by Sides:
Isosceles Triangle
Equilateral Triangle
3 Congruent Sides
Scalene Triangle
At least 2 congruent sides
No Congruent sides
Classification by Angles:
Acute Triangle
3 acute angles
Equiangular Triangle
Right Triangle
3 congruent angles
1 right angle
Obtuse Triangle
1 obtuse angle
Terminology
Vertex: Point where two segments meet
C
Adjacent Sides: Two sides sharing a common
vertex
Opposite Side – Non Adjacent
Opposite Side
<A
Adjacent
Sides
A
B
Right and Isosceles Triangles:
Legs – In a Right Triangle, the sides
that form the right angle; In an
Isosceles Triangle, the two congruent
sides.
Hypotenuse – In a Right Triangle, the
side opposite the right angle
Base – In an Isosceles Triangle, the
third side.
Leg
Hypotenuse
Leg
Leg
Leg
Base
Theorems Regarding Congruent Triangles
B
Theorem 4.1: Triangle Sum Theorem
The sum of the measures of the interior angles
of a triangle is 180*
m<A + m<B + m<C = 180*
A
C
B
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
non-adjacent interior angles.
m<1 = m<A + m<B
Corollary to the Triangle Sum Theorem:
1
A
C
A
The acute angles of a right triangle are
complementary.
m<A + m<B = 90*
B
C
Proving Measures of a Triangle equal 180*
4
2
1
5
3
Given: ABC
Prove: m<1 + m<2 + m<3 = 180*
Statements
1.
2.
3.
4.
5.
Reasons
Finding Angle Measures
65*
x*
(2x + 10)*
2x*
x*