Lesson 3-4 Angles of a Triangle (page 93) Essential Question How can you apply parallel lines (planes) to make deductions?

Angles of a Triangle A . B. . C

TRIANGLE: . . C ∆ ACB ∆ BAC ∆ BCA ∆ CAB ∆ CBA ∆ ABC A B C three The figure formed by 3 segments joining 3 noncollinear points. ∆ ABC Symbol: ____________ A . three VERTEX: each of the ______ points. B. . C A B C Vertices: _____ _____ _____ Sides: _____ _____ _____ Angles: _____ _____ _____ ∠A ∠B ∠C

Classifications of Triangles by Sides Scalene Triangle No sides are congruent .

Classifications of Triangles by Sides Isosceles Triangle At least two sides are congruent .

Classifications of Triangles by Sides Equilateral Triangle All sides are congruent .

Classifications of Triangles by Angles Acute Triangle Three acute angles.

Classifications of Triangles by Angles Right Triangle One right (90º) angle.

Classifications of Triangles by Angles Obtuse Triangle One obtuse angle.

Classifications of Triangles by Angles Equiangular Triangle All angles are congruent .

Auxiliary Line: A line (ray or segment) added to a diagram to help in a proof . Please note that this is to HELP in a proof. This does not give you license to add lines to every diagram. There are times when this may be done, but please BEWARE!

The sum of the measures of the angles of a triangle is 180º . Theorem 3-11 The sum of the measures of the angles of a triangle is 180º . B Given: ∆ ABC Prove: m∠1 + m∠2 + m∠3 = 180º 2 1 3 A C

|| - lines ⇒ AIA ≅ See page 94! Substitution Property D  B Given: ∆ ABC Prove: m∠1 + m∠ 2 + m∠3 = 180º Proof: Statements Reasons Through B draw line BD parallel to line AC Through a point outside a line, there is exactly one line parallel to the given line. __________________________________________ _____________________________________________ 4 5 2 See page 94! 1 3 A C m∠ DBC +m∠5 = 180º m∠ DBC = m∠2 + m∠4 ∠ - Addition Postulate m∠2 + m∠4 + m∠5 = 180º Substitution Property || - lines ⇒ AIA ≅ ∠1 ≅ ∠4 OR m∠1 = m∠4 ∠3 ≅ ∠5 OR m∠3 = m∠ 5 m∠1 + m∠2 + m∠ 3 = 180º Substitution Property

50 Example # 1. Find the value of “x”. x + 40 + 90 = 180 x + 130 = 180 40º

45 Example # 2. Find the value of “x”. x + 100 + 35 = 180 100º 35º

55 Example # 3. Find the value of “x”. x + x + 70 = 180 2x + 70 = 180 70º x º

A statement that can be proved easily COROLLARY: A statement that can be proved easily by applying a theorem.

Corollary 1 If two angles of one triangle are congruent to two angles of another triangle, then the third angles are congruent . A X Y Z B C If ∠A ≅ ∠X and ∠B ≅ ∠Y, then ∠C ≅∠Z

Each angle of an equiangular triangle has a measure 60º . Corollary 2 Each angle of an equiangular triangle has a measure 60º . x º x º x º If 3 x = 180, then x = 60.

In a triangle, there can be at most one right angle or obtuse angle. Corollary 3 In a triangle, there can be at most one right angle or obtuse angle. Sum = 90º Sum < 90º m > 90º

The acute angles of a right triangle are complementary . Corollary 4 The acute angles of a right triangle are complementary . Sum = 90º If 2 ∠’s sum = 90º, then they are complementary.

Example # 4. Find the value of “x”. 40º 50º x º 40º 50 x = _____

Example # 5. Find the value of “x”. x º x º x º 60 x = _____

EXTERIOR ANGLE: (of a triangle) the angle formed when one side of a triangle is extended . Example: ∠4 4 3 Example: ∠1 & ∠2 2 2 1 1 REMOTE INTERIOR ANGLES: the angles of the triangle not adjacent to the exterior angle.

REMOTE INTERIOR ANGLES: EXTERIOR ANGLE: Example: ∠4 1 1 4 3 2 2 REMOTE INTERIOR ANGLES: Example: ∠1 & ∠2

Theorem 3-12 The measure of an exterior angle of a triangle equals the sum of the measures of the two remote interior angles. Given: ∆ ABC Prove: m∠1 + m∠2 = m∠ 4 B 2 1 3 4 A C

Substitution Property Given: ∆ ABC Prove: m∠1 + m∠2 = m∠4 Proof: Statements Reasons __________________________________________ _____________________________________________ 2 See page 96 C.E. #15! 1 3 4 A C m∠1 + m∠2+ m∠ 3 = 180º Sum of m. of∠‘s of ∆ = 180º m∠3 + m∠ 4 = 180º ∠- Addition Postulate Substitution Property m∠1+ m∠2 + m∠3 = m∠3 + m∠4 m∠3 = m∠3 Reflexive Property m∠1 + m∠ 2 = m∠4 Subtraction Property

40 Example # 6. Find the value of “x”. x + 80 = 120 x = _____ 80º 120º

50 Example # 7. Find the value of “x”. 3x = x + 100 2x = 100 x = _____ 100º 3 x º

Example # 8. Find the value of “x”. x = 90 + 60 60º 150 x = _____ x º

Do the Paper Triangle Proofs Assignment Written Exercises on pages 97 to 99 RECOMMENDED: 1 to 9 odd numbers REQUIRED: 10, 11, 13, 15, 17, 18, 19, 20, 25, 26 Do the Paper Triangle Proofs How can you apply parallel lines (planes) to make deductions?