Chapter 4

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

• 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

• 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

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
4
Terminology
5
Theorems Regarding Congruent Triangles
Theorem 4.1 Triangle Sum Theorem The sum of
the measures of the interior angles of a triangle
is 180 mltA mltB mltC 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. mlt1 mltA mltB Corollary to the
Triangle Sum Theorem The acute angles of a
right triangle are complementary. mltA mltB
90
6
Proving Measures of a Triangle equal 180
2
1 3
4 5
Given ABC Prove mlt1 mlt2 mlt3
180 Statements
Reasons 1. 2. 3. 4. 5.
7
Finding Angle Measures
65
x
(2x 10)
2x
x