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: 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 3 congruent angles 1 right angle 1 obtuse angle

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

Theorems Regarding Congruent Triangles B 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*

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

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