geometry section 4-2
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SECTION 4-2Angles of Triangles
Wednesday, 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
VOCABULARY1. Auxiliary Line:
2. Exterior Angle:
3. Remote Interior Angles:
4. Flow Proof:
Wednesday, February 1, 2012
VOCABULARY1. 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
VOCABULARY1. 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
VOCABULARY1. 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
VOCABULARY1. 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 argument
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THEOREMS & COROLLARIES4.1 - Triangle Angle-Sum Theorem:
4.2 - Exterior Angle Theorem:
4.1 Corollary:
4.2 Corollary:
Wednesday, February 1, 2012
THEOREMS & COROLLARIES4.1 - 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 & COROLLARIES4.1 - 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 & COROLLARIES4.1 - 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 & COROLLARIES4.1 - 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 triangle
Wednesday, February 1, 2012
EXAMPLE 1The 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 1The 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
Wednesday, February 1, 2012
EXAMPLE 1The 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 1The 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
Wednesday, February 1, 2012
EXAMPLE 1The 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 1The 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
Wednesday, February 1, 2012
EXAMPLE 1The 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 2Find the measure of m∠FLW .
Wednesday, February 1, 2012
EXAMPLE 2Find the measure of m∠FLW .
m∠FLW = m∠LOW + m∠OWL
Wednesday, February 1, 2012
EXAMPLE 2Find the measure of m∠FLW .
m∠FLW = m∠LOW + m∠OWL
2x − 48 = x + 32
Wednesday, February 1, 2012
EXAMPLE 2Find the measure of m∠FLW .
m∠FLW = m∠LOW + m∠OWL
2x − 48 = x + 32
x − 48 = 32
Wednesday, February 1, 2012
EXAMPLE 2Find the measure of m∠FLW .
m∠FLW = m∠LOW + m∠OWL
2x − 48 = x + 32
x − 48 = 32
x = 80
Wednesday, February 1, 2012
EXAMPLE 2Find 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
Wednesday, February 1, 2012
EXAMPLE 2Find 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
Wednesday, February 1, 2012
EXAMPLE 2Find 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 3Find the measure of each numbered angle.
Wednesday, February 1, 2012
EXAMPLE 3Find the measure of each numbered angle.
m∠5 =180 − 90 − 41
Wednesday, February 1, 2012
EXAMPLE 3Find the measure of each numbered angle.
m∠5 =180 − 90 − 41 = 49°
Wednesday, February 1, 2012
EXAMPLE 3Find the measure of each numbered angle.
m∠5 =180 − 90 − 41 = 49° m∠3 = 90 − 48
Wednesday, February 1, 2012
EXAMPLE 3Find the measure of each numbered angle.
m∠5 =180 − 90 − 41 = 49° m∠3 = 90 − 48 = 42°
Wednesday, February 1, 2012
EXAMPLE 3Find the measure of each numbered angle.
m∠5 =180 − 90 − 41 = 49° m∠3 = 90 − 48 = 42°
m∠4 =180 − 90 − 42
Wednesday, February 1, 2012
EXAMPLE 3Find 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 3Find 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
Wednesday, February 1, 2012
EXAMPLE 3Find 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
Wednesday, February 1, 2012
EXAMPLE 3Find 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 3Find 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° m∠2 =180 − 56 − 48
Wednesday, February 1, 2012
EXAMPLE 3Find 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° m∠2 =180 − 56 − 48 = 76°
Wednesday, February 1, 2012
EXAMPLE 3Find 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° m∠2 =180 − 56 − 48 = 76°
m∠1=180 − 76
Wednesday, February 1, 2012
EXAMPLE 3Find 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° m∠2 =180 − 56 − 48 = 76°
m∠1=180 − 76 =104°
Wednesday, February 1, 2012
CHECK YOUR UNDERSTANDING
p. 248 #1-11
Wednesday, February 1, 2012
PROBLEM SET
Wednesday, 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 Rochefoucauld
Wednesday, February 1, 2012