Sec 3 4 Parallel Lines the Triangle Angle

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Sec. 3 -4 Parallel Lines & the Triangle Angle – Sum Theorem Objectives: a)

Sec. 3 -4 Parallel Lines & the Triangle Angle – Sum Theorem Objectives: a) To classify triangles & find the measures of their angles. b) To use the exterior angles of a triangle.

Th(3 -7) Triangle Angle Sum Thm n Says that every triangle has 180° in

Th(3 -7) Triangle Angle Sum Thm n Says that every triangle has 180° in it. n Regardless of its size or classification. § Must construct the triangle such that AB // TS. T S 1 A 2 3 B

Proof: C T 1 Given: ΔABC & TS // AB 2 S 3 A

Proof: C T 1 Given: ΔABC & TS // AB 2 S 3 A B Prove: m A + m B +m 2 = 180 Statements Reasons 1. ΔABC & TS // AB 1. Given 2. m 1 + m 3 +m 2 = 180 2. Angle Add Post 3. Alt. Interior Angles are 3. m 1 = m A m 3 = m B 4. m A + m B +m 2 = 180 4. 4. Subs.

Example 1: Find the m Z Example 2: Find m a = m b

Example 1: Find the m Z Example 2: Find m a = m b = Z m c = C c 70 48 67 X m X + m Y +m Z = 180 m 48 + m 67 +m 2 = 180 m Z = 65 Y A a m b = 20 m c = 20 m a = 70 b D B

I. Classify Δs by their Angles 1) Right Triangle Ø One is Rt. (=

I. Classify Δs by their Angles 1) Right Triangle Ø One is Rt. (= 90) Ø The other 2 are Acute. 2) Obtuse Δ Ø One is Obtuse ( > 90) Ø Other 2 are Acute. 3) Acute Δ Ø All s are acute ( <90 ) > 90

4) Equiangular Δ Ø A special Acute Δ ØAll s must be acute ØAll

4) Equiangular Δ Ø A special Acute Δ ØAll s must be acute ØAll s must be Ø 180/3 = 60 Ø Each must be = 60

II. Classify Δs by their Sides 1) Scalene Δ Ø No Sides Ø No

II. Classify Δs by their Sides 1) Scalene Δ Ø No Sides Ø No s 2) Isosceles Δ Ø @ least 2 sides are Ø 2 s are , called base s 3) Equilateral Δ Ø All sides are Ø All s are Ø Equilateral Δs are also Equiangular Δs

III. Using Exterior s of a Δ A Exterior of a Δ B Remote

III. Using Exterior s of a Δ A Exterior of a Δ B Remote Interior s C

Th(3 -8) Δ Exterior Thm. n The measure of each Exterior of a triangle

Th(3 -8) Δ Exterior Thm. n The measure of each Exterior of a triangle equals the sum of the measures of its two interior s. 4 m 1 = m 2 + m 3 1 4 2 3 3

C Proof: 2 Given: 1 is an exterior of ΔABC Prove: m 1 =

C Proof: 2 Given: 1 is an exterior of ΔABC Prove: m 1 = m 2 + m 3 1 4 3 B A Statements Reasons 1. 1 is an exterior of ΔABC 1. Given 2. m 1 + m 4 = 180 2. add Postulate 3. m 4 + m 3 + m 2 = 180 3. Def of Δ 4. m 1 + m 4 = m 4 +m 3 + m 2 4. Subs 5. m 1 = m 3 + m 2 5. Subtr.

Example: n 125 Solve for the m 1 in the following trian Use the

Example: n 125 Solve for the m 1 in the following trian Use the exterior thm 125 = 90 + m 1 = 35 1

Example: n What type of triangle is ΔABC n Classify it by C its

Example: n What type of triangle is ΔABC n Classify it by C its sides and its angles m A = 45 90 B Angles Right 45 A Sides Isosceles

Example: n Whats wrong with this diagram? Exterior must be greater than the remote

Example: n Whats wrong with this diagram? Exterior must be greater than the remote interior . 85 80

What have we learned? ? n n n 180° in a Δ The exterior

What have we learned? ? n n n 180° in a Δ The exterior equals the sum of the two remote interior s. Classify Δs by their Sides – Scalene, Isosceles, or Equilateral n s – Right, Acute, Obtuse n