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Geometry Section 4-2

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Geometry Section 4-2 Geometry Section 4-2 Presentation Transcript

  • SECTION 4-2 Angles of TrianglesWednesday, February 1, 2012
  • ESSENTIAL QUESTIONS • How do you apply the Triangle Angle-Sum Theorem? • How do you apply the Exterior Angle Theorem?Wednesday, February 1, 2012
  • VOCABULARY 1. Auxiliary Line: 2. Exterior Angle: 3. Remote Interior Angles: 4. Flow Proof:Wednesday, February 1, 2012
  • VOCABULARY 1. Auxiliary Line: An extra line or segment that is added to a figure to help analyze geometric relationships 2. Exterior Angle: 3. Remote Interior Angles: 4. Flow Proof:Wednesday, February 1, 2012
  • VOCABULARY 1. Auxiliary Line: An extra line or segment that is added to a figure to help analyze geometric relationships 2. Exterior Angle: Formed outside a triangle when one side of the triangle is extended; The exterior angle is adjacent to the interior angle of the triangle 3. Remote Interior Angles: 4. Flow Proof:Wednesday, February 1, 2012
  • VOCABULARY 1. Auxiliary Line: An extra line or segment that is added to a figure to help analyze geometric relationships 2. Exterior Angle: Formed outside a triangle when one side of the triangle is extended; The exterior angle is adjacent to the interior angle of the triangle 3. Remote Interior Angles: The two interior angles that are not adjacent to a given exterior angle 4. Flow Proof:Wednesday, February 1, 2012
  • VOCABULARY 1. Auxiliary Line: An extra line or segment that is added to a figure to help analyze geometric relationships 2. Exterior Angle: Formed outside a triangle when one side of the triangle is extended; The exterior angle is adjacent to the interior angle of the triangle 3. Remote Interior Angles: The two interior angles that are not adjacent to a given exterior angle 4. Flow Proof: Uses statements written in boxes with arrows to show a logical progression of an argumentWednesday, February 1, 2012
  • THEOREMS & COROLLARIES 4.1 - Triangle Angle-Sum Theorem: 4.2 - Exterior Angle Theorem: 4.1 Corollary: 4.2 Corollary:Wednesday, February 1, 2012
  • THEOREMS & COROLLARIES 4.1 - Triangle Angle-Sum Theorem: The sum of the measures of the angles of any triangle is 180° 4.2 - Exterior Angle Theorem: 4.1 Corollary: 4.2 Corollary:Wednesday, February 1, 2012
  • THEOREMS & COROLLARIES 4.1 - Triangle Angle-Sum Theorem: The sum of the measures of the angles of any triangle is 180° 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 remote interior angles 4.1 Corollary: 4.2 Corollary:Wednesday, February 1, 2012
  • THEOREMS & COROLLARIES 4.1 - Triangle Angle-Sum Theorem: The sum of the measures of the angles of any triangle is 180° 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 remote interior angles 4.1 Corollary: The acute angles of a right triangle are complementary 4.2 Corollary:Wednesday, February 1, 2012
  • THEOREMS & COROLLARIES 4.1 - Triangle Angle-Sum Theorem: The sum of the measures of the angles of any triangle is 180° 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 remote interior angles 4.1 Corollary: The acute angles of a right triangle are complementary 4.2 Corollary: There can be at most one right or obtuse angle in a triangleWednesday, February 1, 2012
  • EXAMPLE 1 The diagram shows the paths a ball is thrown in a game played by kids. Find the measure of each numbered angle.Wednesday, February 1, 2012
  • EXAMPLE 1 The diagram shows the paths a ball is thrown in a game played by kids. Find the measure of each numbered angle. m∠1=180 − 43 − 74Wednesday, February 1, 2012
  • EXAMPLE 1 The diagram shows the paths a ball is thrown in a game played by kids. Find the measure of each numbered angle. m∠1=180 − 43 − 74 = 63°Wednesday, February 1, 2012
  • EXAMPLE 1 The diagram shows the paths a ball is thrown in a game played by kids. Find the measure of each numbered angle. m∠1=180 − 43 − 74 = 63° m∠2Wednesday, February 1, 2012
  • EXAMPLE 1 The diagram shows the paths a ball is thrown in a game played by kids. Find the measure of each numbered angle. m∠1=180 − 43 − 74 = 63° m∠2 = 63°Wednesday, February 1, 2012
  • EXAMPLE 1 The diagram shows the paths a ball is thrown in a game played by kids. Find the measure of each numbered angle. m∠1=180 − 43 − 74 = 63° m∠2 = 63° m∠3 =180 − 63 − 79Wednesday, February 1, 2012
  • EXAMPLE 1 The diagram shows the paths a ball is thrown in a game played by kids. Find the measure of each numbered angle. m∠1=180 − 43 − 74 = 63° m∠2 = 63° m∠3 =180 − 63 − 79 = 38°Wednesday, February 1, 2012
  • EXAMPLE 2 Find the measure of m∠FLW.Wednesday, February 1, 2012
  • EXAMPLE 2 Find the measure of m∠FLW. m∠FLW = m∠LOW + m∠OWLWednesday, February 1, 2012
  • EXAMPLE 2 Find the measure of m∠FLW. m∠FLW = m∠LOW + m∠OWL 2x − 48 = x + 32Wednesday, February 1, 2012
  • EXAMPLE 2 Find the measure of m∠FLW. m∠FLW = m∠LOW + m∠OWL 2x − 48 = x + 32 x − 48 = 32Wednesday, February 1, 2012
  • EXAMPLE 2 Find the measure of m∠FLW. m∠FLW = m∠LOW + m∠OWL 2x − 48 = x + 32 x − 48 = 32 x = 80Wednesday, February 1, 2012
  • EXAMPLE 2 Find the measure of m∠FLW. m∠FLW = m∠LOW + m∠OWL 2x − 48 = x + 32 x − 48 = 32 x = 80 m∠FLW = 2(80) − 48Wednesday, February 1, 2012
  • EXAMPLE 2 Find the measure of m∠FLW. m∠FLW = m∠LOW + m∠OWL 2x − 48 = x + 32 x − 48 = 32 x = 80 m∠FLW = 2(80) − 48 =160 − 48Wednesday, February 1, 2012
  • EXAMPLE 2 Find the measure of m∠FLW. m∠FLW = m∠LOW + m∠OWL 2x − 48 = x + 32 x − 48 = 32 x = 80 m∠FLW = 2(80) − 48 =160 − 48 =112°Wednesday, February 1, 2012
  • EXAMPLE 3 Find the measure of each numbered angle.Wednesday, February 1, 2012
  • EXAMPLE 3 Find the measure of each numbered angle. m∠5 =180 − 90 − 41Wednesday, February 1, 2012
  • EXAMPLE 3 Find the measure of each numbered angle. m∠5 =180 − 90 − 41 = 49°Wednesday, February 1, 2012
  • EXAMPLE 3 Find the measure of each numbered angle. m∠5 =180 − 90 − 41 = 49° m∠3 = 90 − 48Wednesday, February 1, 2012
  • EXAMPLE 3 Find the measure of each numbered angle. m∠5 =180 − 90 − 41 = 49° m∠3 = 90 − 48 = 42°Wednesday, February 1, 2012
  • EXAMPLE 3 Find the measure of each numbered angle. m∠5 =180 − 90 − 41 = 49° m∠3 = 90 − 48 = 42° m∠4 =180 − 90 − 42Wednesday, February 1, 2012
  • EXAMPLE 3 Find the measure of each numbered angle. m∠5 =180 − 90 − 41 = 49° m∠3 = 90 − 48 = 42° m∠4 =180 − 90 − 42 = 48°Wednesday, February 1, 2012
  • EXAMPLE 3 Find the measure of each numbered angle. m∠5 =180 − 90 − 41 = 49° m∠3 = 90 − 48 = 42° m∠4 =180 − 90 − 42 = 48° 90 − 34Wednesday, February 1, 2012
  • EXAMPLE 3 Find the measure of each numbered angle. m∠5 =180 − 90 − 41 = 49° m∠3 = 90 − 48 = 42° m∠4 =180 − 90 − 42 = 48° 90 − 34 = 56Wednesday, February 1, 2012
  • EXAMPLE 3 Find the measure of each numbered angle. m∠5 =180 − 90 − 41 = 49° m∠3 = 90 − 48 = 42° m∠4 =180 − 90 − 42 = 48° 90 − 34 = 56 56°Wednesday, February 1, 2012
  • EXAMPLE 3 Find the measure of each numbered angle. m∠5 =180 − 90 − 41 = 49° m∠3 = 90 − 48 = 42° m∠4 =180 − 90 − 42 = 48° 90 − 34 = 56 m∠2 =180 − 56 − 48 56°Wednesday, February 1, 2012
  • EXAMPLE 3 Find the measure of each numbered angle. m∠5 =180 − 90 − 41 = 49° m∠3 = 90 − 48 = 42° m∠4 =180 − 90 − 42 = 48° 90 − 34 = 56 m∠2 =180 − 56 − 48 = 76° 56°Wednesday, February 1, 2012
  • EXAMPLE 3 Find the measure of each numbered angle. m∠5 =180 − 90 − 41 = 49° m∠3 = 90 − 48 = 42° m∠4 =180 − 90 − 42 = 48° 90 − 34 = 56 m∠2 =180 − 56 − 48 = 76° 56° m∠1=180 − 76Wednesday, February 1, 2012
  • EXAMPLE 3 Find the measure of each numbered angle. m∠5 =180 − 90 − 41 = 49° m∠3 = 90 − 48 = 42° m∠4 =180 − 90 − 42 = 48° 90 − 34 = 56 m∠2 =180 − 56 − 48 = 76° 56° m∠1=180 − 76 =104°Wednesday, February 1, 2012
  • CHECK YOUR UNDERSTANDING p. 248 #1-11Wednesday, February 1, 2012
  • PROBLEM SETWednesday, February 1, 2012
  • PROBLEM SET p. 248 #13-37 odd, 46, 57 “We rarely think people have good sense unless they agree with us.” - Francois de La RochefoucauldWednesday, February 1, 2012