1. 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

2. 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?

3. Names of Triangles Classification by Sides: Classification by Angles:
Equilateral Triangle Isosceles Triangle Scalene Triangle
3 Congruent Sides At least 2 congruent sides No Congruent sides
Classification by Angles:
Acute Triangle Equiangular Triangle Right Triangle Obtuse Triangle
3 acute angles congruent angles right angle obtuse angle

4. Terminology A B C Vertex: Point where two segments meet
Adjacent Sides: Two sides sharing a common
vertex
Opposite Side – Non Adjacent
Adjacent
Sides
Opposite Side
<A
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

5. Theorems Regarding Congruent TrianglesB
A C 1 A C
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*
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:
The acute angles of a right triangle are complementary.
m<A + m<B = 90*

6. Proving Measures of a Triangle equal 180*2
Given: ABC
Prove: m<1 + m<2 + m<3 = 180*
Statements Reasons 1. 2. 3. 4. 5.

7. Finding Angle Measures
65*
x* (2x + 10)*
2x* x*