Triangles can be classified by angles and sides. Write a good definition of each term and provide a sketch: Classify triangles by angles:

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1 Chapter 4: Congruent Triangles A. 4-1 Classifying Triangles Identify and classify triangles by angles. Identify and classify triangles by sides. Triangles appear often in construction. Roofs sit atop a triangular shape called a truss. Why do you think that triangles are so important in construction? Triangles can be classified by angles and sides. Write a good definition of each term and provide a sketch: Classify triangles by angles: DEFINE: Acute Obtuse Right Equiangular An acute triangle with all angles congruent is Classify triangles by sides: DEFINE: Scalene Isosceles Equilateral *Are all equilateral triangles also isosceles triangles? *Are all isosceles triangles also equilateral triangles? Example 2:

2 Algebra Connection: (Ex.3) Coordinate Geometry: (Ex.4) Assignment p #1-4, 8, 13, 14, 22, 24, 27, 28, 34, 38 (paragraph proof) 40. B. 4.2 Angles of Triangles Activity Angle Sum Theorem: Proof: Given: Prove: Statements Reasons If we know the measure of two angles of a triangle, we can find the measure of the third. Example: Find the missing angle measures: Third Angle Theorem: Exterior Angle Theorem:

3 Example: Find the measure of each numbered angle in the figure: Corollary- A statement that is easily proved using a theorem. Corrollaries can be used in proofs if needed. Corollary 4.1: Corollary 4.2: Assignment: p #1, 2, 3, even, 39 (two-column proof) C. 4.3 Congruent Triangles (Trusses) DEFINE: Congruent triangles- **order of vertices is important! The vertices of the two triangles correspond in the same order as the letters naming the triangles. This correspondence of vertices can be used to name the corresponding parts of the two triangles: CPCTC: Examples: Congruence of triangles is,,

4 Study the proof on page 193. What is a transformation? And what does this have to do with congruent triangles? Proving Congruent Triangles in Coordinate Geometry: How would you prove two triangles are congruent in coordinate geometry? Example 2 page 194. Assignment: p #9-12, 13, 15, 24, 29-32, 36 Activity D. 4-4 Proving Congruence-SSS, SAS What does it mean for two triangles to be congruent? Would two congruent triangles have the same perimeter? Same Area? Explain. Is it always necessary to show that all of the corresponding parts of two triangles are congruent to show that the two triangles are congruent? POSTULATE: Side-Side-Side Congruence:(SSS) Coordinate Geometry: Determine whether triangle RTZ is congruent to triangle JKL. R(2,5) Z(1,1) T(5,2) L(-3, 0) K(-7,1) J(-4,4) Explain. POSTULATE: Side-Angle-Side Congruence: (SAS)

5 Included Angle- What about AAA? Does this prove two triangles are congruent? Study example 4 on page 203. Assignment: p #1, 2, 10, 11, 16, 17, 18, all, 28 E. 4-5 Proving Congruence- ASA, AAS Use the ASA Postulate to test for triangle congruence Use the AAS Theorem to test for triangle congruence. POSTULATE: Angle-Side-Angle Congruence (ASA)- Included Side- Paragraph Proof: Given: Prove: THEOREM: Angel-Angle-Side Congruence: (AAS) If two angles and a non-included side of one triangle are congruent to the corresponding two angles and side of a second triangle, then the two triangles are congruent. Proof: Given: Prove: Statements Reasons

6 3. 3. SSA? Does this prove two triangles are congruent? **At this point, it is very important to go back and study the proofs in this lesson and the previous lessons and notice the properties, theorems, and definitions that keep reoccurring in the proofs. This will be helpful when you have to complete a proof on your own! Concept Summary: Definition of Congruent Triangles SSS SAS ASA AAS All corresponding parts of one triangle are congruent to the corresponding parts of the other triangle. The three sides of one triangle must be congruent to the three sides of the other triangle. Two sides and the included angle of one triangle must be congruent to two sides and the included angle of the other triangle. Two angles and the included side of one triangle must be congruent to two angles and the included side of the other triangle. Two angles and a non-included side of one triangle must be congruent to two angles and a non-included side of the other triangle. Assignment: p # 6, 11, 14, 15, 18, 26 RIGHT TRIANGLE CONGRUENCE: Make a conjecture as to why you would only need two corresponding parts congruent of each triangle to prove triangle congruence. THEOREM: HA: LA: LL: HL:

7 p #10 **Sometimes these are the only ways to prove two triangles congruent** F. 4-6 Isosceles Triangles Use properties of isosceles triangles. Use properties of equilateral triangles. Architecture Connection Investigation PARTS OF AN ISOSCELES TRIANGLE: Vertex: Legs: Base: Base Angles: Isosceles Triangle Theorem: If two sides of a triangle are congruent, then the angles opposite those sides are congruent. Write a two-column proof of the isosceles triangle theorem. Proof: Given: Prove: Statements Reasons The converse of the Isosceles Triangle Theorem is also true.

8 THEOREM Properties of Equilateral Triangles: Corollaries 4.3 and 4.4 A triangle is equilateral iff it is. Each angle of an equilateral triangle measures. Example: Assignment p #9-27 odd, 29, 35, 36, 37 G. 4-7 Triangles and Coordinate Proof Position and label triangles for use in coordinate proofs. Write coordinate proofs. DEFINE: Coordinate Proof- A coordinate proof uses figures in the coordinate plane and Algebra to prove geometric concepts. Four Steps: Position and label a triangle: Find the missing coordinates:

9 Example of proof: use your graph paper Study Example 3 on page 223. Writing Coordinate Proofs. Assignment p even

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