All About Angles.

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Presentation transcript:

All About Angles

Two rays that meet at a point and extend indefinitely form an Two rays that meet at a point and extend indefinitely form an . The two rays are the of the angle. The point at which they meet is the of the angle.

Two rays that meet at a point and extend indefinitely form an angle Two rays that meet at a point and extend indefinitely form an angle. The two rays are the sides of the angle. The point at which they meet is the vertex of the angle.

An angle separates the plane into the region, the region, and the .

An angle separates the plane into the interior region, the exterior region, and the angle itself.

Angles are measured in using a protractor.

Angles are measured in degrees using a protractor. 45 0

Angles with equal measures are called .

Angles with equal measures are called congruent angles.

Angles can be named by: The vertex point if there are no other angles at that vertex that could be confused. Three letters with the vertex as the center letter and the other letters representing points from each side. A small number if one is given in the angle.

Angles can be named by: <R R The vertex point if there are no other angles at that vertex that could be confused. Three letters with the vertex as the center letter and the other letters representing points from each side. A small number if one is given in the angle. <R R

Angles can be named by: <R R The vertex point if there are no other angles at that vertex that could be confused. Three letters with the vertex as the center letter and the other letters representing points from each side. A small number if one is given in the angle. <R R

Angles can be named by: <KRB K R B The vertex point if there are no other angles at that vertex that could be confused. Three letters with the vertex as the center letter and the other letters representing points from each side. A small number if one is given in the angle. <KRB K R B

<1 Angles can be named by: 1 The vertex point if there are no other angles at that vertex that could be confused. Three letters with the vertex as the center letter and the other letters representing points from each side. A small number if one is given in the angle. <1 1

Examples: Skip question 1. We will come back to it later. #2. Why can this angle not be named <P? #3. Name ALL the angles: Skip question 4.

Examples: #3. Name ALL the angles: <MPN <NPO <MPO

Angle Classification angles measure more than 0 degrees but less than 90 degrees

Angle Classification Acute angles measure more than 0 degrees but less than 90 degrees

Angle Classification angles measure exactly 90 degrees. lines form right angles.

Angle Classification Right angles measure exactly 90 degrees. Perpendicular lines form right angles.

Angle Classification angles measure greater than 90 degrees but less than 180 degrees.

Angle Classification Obtuse angles measure greater than 90 degrees but less than 180 degrees.

Angle Classification A angle (line) has a measure of 180 degrees.

Angle Classification A straight angle (line) has a measure of 180 degrees.

When we want to say what the measure of an angle is, for example <ABC, we write m<ABC = 45 degrees

Example #5: Classify the Angles

Example #5: Classify the Angles Right Angle

Example #5: Classify the Angles

Example #5: Classify the Angles <ABC is an obtuse angle <ABD is an acute angle

Protractor Postulate Given with point O between A and B. consider ray OA and ray OB and any other rays that can be drawn with O as the endpoint on one side of line AB. These rays can be paired with the numbers from zero degrees to 180 degrees such that Ray OA is paired with zero Ray OB is paired with 180 degrees If ray OC is paired with c degrees and ray OD is paired with d degrees, then the m<COD is the absolute value of the difference of c degrees and d degrees

Example #6: Find m<COD Positions: Ray OA at 0 degrees Ray OB at 180 degrees Ray OC at 75 degrees Ray OD at 140 degrees

Example #6: Find m<COD Positions: Ray OA at 0 degrees Ray OB at 180 degrees Ray OC at 75 degrees Ray OD at 140 degrees 75 140 180

Example #6: Find m<COD Positions: Ray OA at 0 degrees Ray OB at 180 degrees Ray OC at 75 degrees Ray OD at 140 degrees Big # - Small # = Total 75 140 180

Example #6: Find m<COD Positions: Ray OA at 0 degrees Ray OB at 180 degrees Ray OC at 75 degrees Ray OD at 140 degrees Big # - Small # = Total 140 – 75 = Total 75 140 180

Example #6: Find m<COD Positions: Ray OA at 0 degrees Ray OB at 180 degrees Ray OC at 75 degrees Ray OD at 140 degrees Big # - Small # = Total 140 – 75 = Total 65 = Total m<COD = 65 degrees 75 140 180

Angle Addition Postulate If D is in the interior of <ABC then m<ABD + m<DBC = m<ABC

Example #7: Find m<ABC m<ABD = 62 degrees m<DBC = 31 degrees

Example #7: Find m<ABC m<ABD = 62 degrees m<DBC = 31 degrees m<ABD + m<DBC = m<ABC 620 310

Example #7: Find m<ABC m<ABD = 62 degrees m<DBC = 31 degrees m<ABD + m<DBC = m<ABC 62 + 31 = x 620 310

Example #7: Find m<ABC m<ABD = 62 degrees m<DBC = 31 degrees m<ABD + m<DBC = m<ABC 62 + 31 = x 93 = x 620 310

Example #7: Find m<ABC m<ABD = 62 degrees m<DBC = 31 degrees m<ABD + m<DBC = m<ABC 62 + 31 = x 93 = x m<ABC = 93 620 310

Example #8: Find m<DBC m<ABC = 90 degrees m<ABD = 56 degrees 560 900

Example #8: Find m<DBC m<ABC = 90 degrees m<ABD = 56 degrees m<ABD + m<DBC = m<ABC 560 900

Example #8: Find m<DBC m<ABC = 90 degrees m<ABD = 56 degrees m<ABD + m<DBC = m<ABC 56 + x = 90 560 900

Example #8: Find m<DBC m<ABC = 90 degrees m<ABD = 56 degrees m<ABD + m<DBC = m<ABC 56 + x = 90 x= 90 -56 560 900

Example #8: Find m<DBC m<ABC = 90 degrees m<ABD = 56 degrees m<ABD + m<DBC = m<ABC 56 + x = 90 x= 90 -56 X = 34 560 900

Example #8: Find m<DBC m<ABC = 90 degrees m<ABD = 56 degrees m<ABD + m<DBC = m<ABC 56 + x = 90 x= 90 -56 X = 34 m<DBC = 34 560 900

Example #9: Find m<ABD & m<DBC m<ABC = 88 degrees 880

Example #9: Find m<ABD & m<DBC m<ABC = 88 degrees m<ABD + m<DBC = m<ABC x0 x0 880

Example #9: Find m<ABD & m<DBC m<ABC = 88 degrees m<ABD + m<DBC = m<ABC x + x = 88 x0 x0 880

Example #9: Find m<ABD & m<DBC m<ABC = 88 degrees m<ABD + m<DBC = m<ABC x + x = 88 2x= 88 x0 x0 880

Example #9: Find m<ABD & m<DBC m<ABC = 88 degrees m<ABD + m<DBC = m<ABC x + x = 88 2x= 88 X = 44 x0 x0 880

Example #9: Find m<ABD & m<DBC m<ABC = 88 degrees m<ABD + m<DBC = m<ABC x + x = 88 2x= 88 X = 44 m<DBC = 44 m<ABD = 44 x0 x0 880

Angle Relationships - coplanar angles that have a common vertex and one common side, but NO common interior points

Angle Relationships Adjacent Angles - coplanar angles that have a common vertex and one common side, but NO common interior points

Angle Relationships - Two non adjacent angles formed by two intersection lines

Angle Relationships <1 & <2 <3 & <4 Vertical Angles - Two non adjacent angles formed by two intersection lines <1 & <2 <3 & <4

Angle Relationships - Two adjacent angles whose non common sides are two rays going in opposite directions

Angle Relationships <1 & <2 Linear Pair - Two adjacent angles whose non common sides are two rays going in opposite directions <1 & <2

Angle Relationships - Sum of the measures of the two angles is 90 degrees

Angle Relationships Complementary Angles - Sum of the measures of the two angles is 90 degrees

Angle Relationships Complementary Angles - Sum of the measures of the two angles is 90 degrees 600 300

Angle Relationships - Sum of the measures of the two angles is 180 degrees. If two angles form a linear pair, they are .

Angle Relationships Supplementary Angles- Sum of the measures of the two angles is 180 degrees. If two angles form a linear pair, they are supplementary. 1200 600

Angle Relationships Supplementary Angles- Sum of the measures of the two angles is 180 degrees. If two angles form a linear pair, they are supplementary. 1200 600

Angle Theorems Vertical angles are . If two angles are supplementary to the same angle or to congruent angles, they are . If two angles are complementary to the same angle or to congruent angles, they are .

Angle Theorems Vertical angles are congruent. If two angles are supplementary to the same angle or to congruent angles, they are congruent. If two angles are complementary to the same angle or to congruent angles, they are congruent.

Example #10: Find x m<AOC = 16x – 20 m<BOD = 13x + 7

Example #10: Find x What type of angles are these? m<AOC = 16x – 20 m<BOD = 13x + 7 What type of angles are these? 16x - 20 13x + 7

Vertical angles are congruent Example #10: Find x Vertical angles are congruent m<AOC = 16x – 20 m<BOD = 13x + 7 16x - 20 13x + 7

Vertical angles are congruent Example #10: Find x Vertical angles are congruent m<AOC = 16x – 20 m<BOD = 13x + 7 m<AOC = m<BOD 16x - 20 13x + 7

Vertical angles are congruent Example #10: Find x Vertical angles are congruent m<AOC = 16x – 20 m<BOD = 13x + 7 m<AOC = m<BOD 16x – 20 = 13x +7 16x - 20 13x + 7

Vertical angles are congruent Example #10: Find x Vertical angles are congruent m<AOC = 16x – 20 m<BOD = 13x + 7 m<AOC = m<BOD 16x – 20 = 13x +7 3x – 20 = 7 16x - 20 13x + 7

Vertical angles are congruent Example #10: Find x Vertical angles are congruent m<AOC = 16x – 20 m<BOD = 13x + 7 m<AOC = m<BOD 16x – 20 = 13x +7 3x – 20 = 7 3x = 27 16x - 20 13x + 7

Vertical angles are congruent Example #10: Find x Vertical angles are congruent m<AOC = 16x – 20 m<BOD = 13x + 7 m<AOC = m<BOD 16x – 20 = 13x +7 3x – 20 = 7 3x = 27 X = 9 16x - 20 13x + 7

Example #11: Find m<AOB & m<AOC m<AOB = 4x + 15 m<AOC = 3x + 25

Example #11: Find m<AOB & m<AOC m<AOB = 4x + 15 m<AOC = 3x + 25 What type of angles are these? 3x + 25 4x + 15

Example #11: Find m<AOB & m<AOC Linear Pair m<AOB = 4x + 15 m<AOC = 3x + 25 3x + 25 4x + 15

Example #11: Find m<AOB & m<AOC Linear Pair m<AOB = 4x + 15 m<AOC = 3x + 25 m<AOB + m<AOC = 180 3x + 25 4x + 15

Example #11: Find m<AOB & m<AOC Linear Pair m<AOB = 4x + 15 m<AOC = 3x + 25 m<AOB + m<AOC = 180 4x + 15 + 3x + 25 = 180 3x + 25 4x + 15

Example #11: Find m<AOB & m<AOC Linear Pair m<AOB = 4x + 15 m<AOC = 3x + 25 m<AOB + m<AOC = 180 4x + 15 + 3x + 25 = 180 7x +40 = 180 3x + 25 4x + 15

Example #11: Find m<AOB & m<AOC Linear Pair m<AOB = 4x + 15 m<AOC = 3x + 25 m<AOB + m<AOC = 180 4x + 15 + 3x + 25 = 180 7x +40 = 180 7x = 140 3x + 25 4x + 15

Example #11: Find m<AOB & m<AOC Linear Pair m<AOB = 4x + 15 m<AOC = 3x + 25 m<AOB + m<AOC = 180 4x + 15 + 3x + 25 = 180 7x +40 = 180 7x = 140 X = 20 3x + 25 4x + 15

Example #11: Find m<AOB & m<AOC Linear Pair m<AOB = 4x + 15 m<AOC = 3x + 25 m<AOB + m<AOC = 180 4x + 15 + 3x + 25 = 180 7x +40 = 180 7x = 140 X = 20 m<AOB = 4x +15 3x + 25 4x + 15

Example #11: Find m<AOB & m<AOC Linear Pair m<AOB = 4x + 15 m<AOC = 3x + 25 m<AOB + m<AOC = 180 4x + 15 + 3x + 25 = 180 7x +40 = 180 7x = 140 X = 20 m<AOB = 4x +15 m<AOB = 4(20)+15 3x + 25 4x + 15

Example #11: Find m<AOB & m<AOC Linear Pair m<AOB = 4x + 15 m<AOC = 3x + 25 m<AOB + m<AOC = 180 4x + 15 + 3x + 25 = 180 7x +40 = 180 7x = 140 X = 20 m<AOB = 4x +15 m<AOB = 4(20)+15 m<AOB = 80+15 3x + 25 4x + 15

Example #11: Find m<AOB & m<AOC Linear Pair m<AOB = 4x + 15 m<AOC = 3x + 25 m<AOB + m<AOC = 180 4x + 15 + 3x + 25 = 180 7x +40 = 180 7x = 140 X = 20 m<AOB = 4x +15 m<AOB = 4(20)+15 m<AOB = 80+15 m<AOB=95 degrees 3x + 25 4x + 15

Example #11: Find m<AOB & m<AOC Linear Pair m<AOB = 4x + 15 m<AOC = 3x + 25 m<AOB + m<AOC = 180 4x + 15 + 3x + 25 = 180 7x +40 = 180 7x = 140 X = 20 m<AOB = 4x +15 m<AOB = 4(20)+15 m<AOB = 80+15 m<AOB=95 degrees 3x + 25 m<AOC = 3x +25 4x + 15

Example #11: Find m<AOB & m<AOC Linear Pair m<AOB = 4x + 15 m<AOC = 3x + 25 m<AOB + m<AOC = 180 4x + 15 + 3x + 25 = 180 7x +40 = 180 7x = 140 X = 20 m<AOB = 4x +15 m<AOB = 4(20)+15 m<AOB = 80+15 m<AOB=95 degrees 3x + 25 m<AOC = 3x +25 m<AOC = 3(20)+25 4x + 15

Example #11: Find m<AOB & m<AOC Linear Pair m<AOB = 4x + 15 m<AOC = 3x + 25 m<AOB + m<AOC = 180 4x + 15 + 3x + 25 = 180 7x +40 = 180 7x = 140 X = 20 m<AOB = 4x +15 m<AOB = 4(20)+15 m<AOB = 80+15 m<AOB=95 degrees 3x + 25 m<AOC = 3x +25 m<AOC = 3(20)+25 m<AOC = 60 + 25 4x + 15

Example #11: Find m<AOB & m<AOC Linear Pair m<AOB = 4x + 15 m<AOC = 3x + 25 m<AOB + m<AOC = 180 4x + 15 + 3x + 25 = 180 7x +40 = 180 7x = 140 X = 20 m<AOB = 4x +15 m<AOB = 4(20)+15 m<AOB = 80+15 m<AOB=95 degrees 3x + 25 m<AOC = 3x +25 m<AOC = 3(20)+25 m<AOC = 60 + 25 m<AOC=85 degrees 4x + 15

Example #11: Find m<AOB & m<AOC Linear Pair m<AOB = 4x + 15 m<AOC = 3x + 25 m<AOB + m<AOC = 180 4x + 15 + 3x + 25 = 180 7x +40 = 180 7x = 140 X = 20 m<AOB = 4x +15 m<AOB = 4(20)+15 m<AOB = 80+15 m<AOB=95 degrees 3x + 25 m<AOC = 3x +25 m<AOC = 3(20)+25 m<AOC = 60 + 25 m<AOC=85 degrees 4x + 15

Example #12: Find x, m<X, m<Y <X and <Y are complementary angles m<X = 3x + 7 m<Y = 6x +20

Example #12: Find x, m<X, m<Y <X and <Y are complementary angles m<X = 3x + 7 m<Y = 6x +20 m<X + m<Y = 90

Example #12: Find x, m<X, m<Y <X and <Y are complementary angles m<X = 3x + 7 m<Y = 6x +20 m<X + m<Y = 90 3x + 7 +6x +20 = 90

Example #12: Find x, m<X, m<Y <X and <Y are complementary angles m<X = 3x + 7 m<Y = 6x +20 m<X + m<Y = 90 3x + 7 +6x +20 = 90 9x + 27 = 90

Example #12: Find x, m<X, m<Y <X and <Y are complementary angles m<X = 3x + 7 m<Y = 6x +20 m<X + m<Y = 90 3x + 7 +6x +20 = 90 9x + 27 = 90 9x = 63

Example #12: Find x, m<X, m<Y <X and <Y are complementary angles m<X = 3x + 7 m<Y = 6x +20 m<X + m<Y = 90 3x + 7 +6x +20 = 90 9x + 27 = 90 9x = 63 X = 7

Example #12: Find x, m<X, m<Y <X and <Y are complementary angles m<X = 3x + 7 m<Y = 6x +20 m<X = 3x +7 m<X + m<Y = 90 3x + 7 +6x +20 = 90 9x + 27 = 90 9x = 63 x = 7

Example #12: Find x, m<X, m<Y <X and <Y are complementary angles m<X = 3x + 7 m<Y = 6x +20 m<X = 3x +7 m<X = 3(7) + 7 m<X + m<Y = 90 3x + 7 +6x +20 = 90 9x + 27 = 90 9x = 63 x = 7

Example #12: Find x, m<X, m<Y <X and <Y are complementary angles m<X = 3x + 7 m<Y = 6x +20 m<X = 3x +7 m<X = 3(7) + 7 m<X = 21 + 7 m<X + m<Y = 90 3x + 7 +6x +20 = 90 9x + 27 = 90 9x = 63 x = 7

Example #12: Find x, m<X, m<Y <X and <Y are complementary angles m<X = 3x + 7 m<Y = 6x +20 m<X = 3x +7 m<X = 3(7) + 7 m<X = 21 + 7 m<X = 28 degrees m<X + m<Y = 90 3x + 7 +6x +20 = 90 9x + 27 = 90 9x = 63 x = 7

Example #12: Find x, m<X, m<Y <X and <Y are complementary angles m<X = 3x + 7 m<Y = 6x +20 m<X = 3x +7 m<X = 3(7) + 7 m<X = 21 + 7 m<X = 28 degrees m<X + m<Y = 90 3x + 7 +6x +20 = 90 9x + 27 = 90 9x = 63 x = 7 m<Y = 6x +20

Example #12: Find x, m<X, m<Y <X and <Y are complementary angles m<X = 3x + 7 m<Y = 6x +20 m<X = 3x +7 m<X = 3(7) + 7 m<X = 21 + 7 m<X = 28 degrees m<X + m<Y = 90 3x + 7 +6x +20 = 90 9x + 27 = 90 9x = 63 x = 7 m<Y = 6x +20 m<Y = 6(7) + 20

Example #12: Find x, m<X, m<Y <X and <Y are complementary angles m<X = 3x + 7 m<Y = 6x +20 m<X = 3x +7 m<X = 3(7) + 7 m<X = 21 + 7 m<X = 28 degrees m<X + m<Y = 90 3x + 7 +6x +20 = 90 9x + 27 = 90 9x = 63 x = 7 m<Y = 6x +20 m<Y = 6(7) + 20 m<Y = 42 + 20

Example #12: Find x, m<X, m<Y <X and <Y are complementary angles m<X = 3x + 7 m<Y = 6x +20 m<X = 3x +7 m<X = 3(7) + 7 m<X = 21 + 7 m<X = 28 degrees m<X + m<Y = 90 3x + 7 +6x +20 = 90 9x + 27 = 90 9x = 63 x = 7 m<Y = 6x +20 m<Y = 6(7) + 20 m<Y = 42 + 20 m<Y = 62 degrees

Example #12: Find x, m<X, m<Y <X and <Y are complementary angles m<X = 3x + 7 m<Y = 6x +20 m<X = 3x +7 m<X = 3(7) + 7 m<X = 21 + 7 m<X = 28 degrees m<X + m<Y = 90 3x + 7 +6x +20 = 90 9x + 27 = 90 9x = 63 x = 7 m<Y = 6x +20 m<Y = 6(7) + 20 m<Y = 42 + 20 m<Y = 62 degrees