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2. 2 Definition of Congruent Triangles ABCPQR Δ ABC Δ PQR AP B Q C R If then the corresponding sides and corresponding angles are congruent ABCPQR, Δ ABC Δ PQR,

3. 3 Types of Triangles classified by the sides 1.Equilateral triangle has 3 congruent sides 2.Isosceles triangle has at least 2 congruent sides 3.Scalene triangle has no sides congruent Types of Triangles classified by the angles 1.Acute triangle has 3 acute angles. If these angles are congruent, then the triangle is also equiangular. 2.Right triangle has exactly one right angle 3.Obtuse triangle has exactly one obtuse angle Acute Equiangular Right Obtuse

4. 4 Sides of Right triangles and Isosceles triangles are given special names A C B Opposite side of < A Adjacent sides of < A Legs Hypotenuse BaseLegs Isosceles Triangle, when only 2 sides =, these sides called “legs” and the remaining side is called “base” Right Triangle, the side opposite the right angle is called “hypotenuse”, the adjacent sides called “legs”

5. 5 Theorem: Properties of Congruent Triangles 1.Every triangle is congruent to itself [ reflexive property ] 2.If then [ symmetric property ] 3.If andthen ABCPQR Δ ABC Δ PQR PQRABC Δ PQR Δ ABC PQRTUV Δ PQR Δ TUV ABCTUV Δ ABC Δ TUV [ transitive property ]

6. 6 Interior and Exterior Angles of a Triangle Exterior angle Interior angle Extend the angles Original angle 1 23 1 23 2 1 3 If you rotate each angle of a triangle and line them up, they add up to a straight line. < 1 + < 2 + < 3 = line 23 When an auxiliary line is drawn parallel to the opposite side of the triangle, then alternate interior angles are equal. Since 1 = 1, 2 = 2, 3 = 3, then angles of a triangle add up to a line or 180 °

7. 7 Theorem: Triangle Sum The sum of the measures of the interior angles of a triangle is 180 ° Theorem: Triangle Sum The sum of the measures of the interior angles of a triangle is 180 ° PROOF: Triangles add up to 180 DIAGRAM: GIVEN: Δ ABC ° PROVE: m < 1 + m < B + < C = 180 ° STATEMENTREASON A line is constructed through A parallel to BC. Parallel postulate < B < 2 and < C < 3Alternate Interior Angles Theorem ° m < 1 + m < 2 + < 3 = 180 °Because < 1 + < 2 + < 3 FORM A LINE ° m < 1 + m < B + < C = 180 °Substitution 2 3 3 2 1 B A C

8. 8 Example 1: Using the Triangle Sum Theorem ° ° ° ° In this triangle, find m < 1, m < 2, m < 3 by using the Triangle Sum Theorem m < 3 = 180 ° ─ ( 51 ° + 42 ° ) = 87 ° ° ° ° With m < 3, use Linear Pair postulate to get m < 2 = 180 ° ─ 87 ° = 93 ° ° ° ° ° Again, Using the Triangle Sum Theorem m < 1 = 180 ° ─ ( 28 ° + 93 ° ) = 59 ° 1 3 2 28 42 51 Third Angles Theorem: If 2 angles of 1 st triangle are congruent to 2 angles of a 2 nd triangle, then the 3 rd angles are congruent Theorem: Acute angles of a right triangle are complementary

9. 9 Exterior Angles Theorem: The measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote (non-adjacent) interior angles A B C Exterior angle of < C = sum of the 2 remote interior angles < A + < B

10. 10 A B C B A C A B Exterior Angles Theorem: As you can see by moving duplicate copies of the triangle, it clearly shows how the exterior angle of angle C = the 2 remote angles of the triangle, < A + < B.

11. 11 Proving that Triangles are Congruent (same shape and size) Side – Side – Side ( SSS ) Congruence Postulate: If 3 sides of one triangle are congruent to 3 sides of a second triangle, then the two triangles are congruent. Side – Side – Side ( SSS ) Congruence Postulate: If 3 sides of one triangle are congruent to 3 sides of a second triangle, then the two triangles are congruent. Side – Angle – Side ( SAS ) Congruence Postulate: If 2 sides and the included angle of one triangle are congruent to 2 sides and the included angle of a second triangle, then the two triangles are congruent. Side – Angle – Side ( SAS ) Congruence Postulate: If 2 sides and the included angle of one triangle are congruent to 2 sides and the included angle of a second triangle, then the two triangles are congruent.

12. 12 Proving that Triangles are Congruent (same shape and size) Angle – Side – Angle ( ASA ) Congruence Postulate: If 2 angles and the included side of one triangle are congruent to 2 angles and the included side of a second triangle, then the two triangles are congruent. Angle – Side – Angle ( ASA ) Congruence Postulate: If 2 angles and the included side of one triangle are congruent to 2 angles and the included side of a second triangle, then the two triangles are congruent. Angle – Angle – Side ( AAS ) Congruence Postulate: If 2 angles and a non-included side of one triangle are congruent to 2 angles and the corresponding non-included side of a second triangle, then the two triangles are congruent. Angle – Angle – Side ( AAS ) Congruence Postulate: If 2 angles and a non-included side of one triangle are congruent to 2 angles and the corresponding non-included side of a second triangle, then the two triangles are congruent.

13. 13 Examples where AAA and SSA don’t work thus Triangles are NOT Congruent (same shape and size) F D C B A As you can see, all 3 angles of both triangles are congruent to corresponding angles of other triangle, yet the triangles are NOT congruent. A B D C As you can see, triangles ABC and ABD are NOT congruent even though they have 2 congruent sides, BC BD and AB AB and a non-included angle, < A < A

14. 14 CPCTC: Corresponding Parts of Congruent Triangles are Congruent CPCTC: By definition, 2 triangles are congruent if and only if their corresponding parts are congruent. GIVEN: A is the midpoint of MT A is the midpoint of SR PROVE: MS ║ TR STATEMENTREASON 1. A is the midpoint of MT1.Given 2.MA TA2. Definition of midpoint 3. A is the midpoint of SR3. Given 2.SA RA4. Definition of midpoint 5. < MAS < TAR5. Vertical angles are congruent 6. Δ MAS Δ TAR6. SAS Congruent Postulate 7. < M < T7.CPCTC 8. MS ║ TR8. Alt. Interior <‘s 2 lines are ║ A R T M S

15. 15 Base Angles Theorem (Isosceles Triangle) If 2 sides of a triangle are congruent, then the angles opposite them are congruent then the angles opposite them are congruent Base Angles Theorem (Isosceles Triangle) If 2 sides of a triangle are congruent, then the angles opposite them are congruent then the angles opposite them are congruent GIVEN: NC NY PROVE: < C < Y STATEMENTREASON 1. Label H as the midpoint of CY1.Ruler Postulate 2.Draw NH2. 2 points determine a line 3. CH HY3. Definition of midpoint 4. NH NH4. Reflexive property of congruence 5. NC NY5. Given 6. Δ NHC Δ NHY6. SSS Congruence Postulate 7. < C < Y7. CPCTC H C Y N

16. 16 Theorem: If 2 angles of a triangle are congruent, then the sides opposite them are congruent. Corollary to Base Angles theorem: If a triangle is equilateral, then it is also equiangular. Corollary to Base Angles theorem: If a triangle is equilateral, then it is also equiangular. Corollary to theorem above: If a triangle is equiangular, then it is also equilateral. Corollary to theorem above: If a triangle is equiangular, then it is also equilateral. Definition of a Corollary: A corollary is a theorem that follows easily from a theorem that has been proven.

17. 17 Hypotenuse – Leg (HL) Congruence Theorem If the hypotenuse + a leg of a right are congruent to the hypotenuse + leg of a 2 nd right, the 2 triangles are congruent If the hypotenuse + a leg of a right Δ are congruent to the hypotenuse + leg of a 2 nd right Δ, the 2 triangles are congruent Hypotenuse – Leg (HL) Congruence Theorem If the hypotenuse + a leg of a right are congruent to the hypotenuse + leg of a 2 nd right, the 2 triangles are congruent If the hypotenuse + a leg of a right Δ are congruent to the hypotenuse + leg of a 2 nd right Δ, the 2 triangles are congruent GIVEN: AC DF, CB FE., AB EG < B and < FED are right <‘s PROVE: Δ ABC Δ DEF STATEMENT 1. < B is a right angle 9. FG AC 2.< FED is a right angle10. AC DF 3. FE DG11. FG DF 4. < FEG is a right angle12. < D < G 5. < B < FEG13. < FEG < FED 6. AB EG14. Δ FEG Δ FED 7. CB FE15. Δ ABC Δ DEF 8. Δ ABC Δ GEF E D G F C B A

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19. 19 m π  