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Chapter 1 Discovering Geometry

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**1.1 Basic Geometric Figures**

I. Point A. Geometric figure with no dimensions B. Used to identify a point in space C. Represented by a dot •

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**1.1 Basic Geometric Figures**

I. Point A. Geometric figure with no dimensions B. Used to identify a point in space C. Represented by a dot D. Labeled by a capital letter A •

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II. Line A. Geometric figure having infinite length B. No width or height C. Consists of points D. Represented by a double pointed arrow

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**• • II. Line A. Geometric figure having infinite length**

B. No width or height C. Consists of points D. Represented by a double pointed arrow A B • •

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**• • II. Line A. Geometric figure having infinite length**

B. No width or height C. Consists of points D. Represented by a double pointed arrow E. Labeled by any two point that it contains A B • •

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**• • • II. Line A. Geometric figure having infinite length**

B. No width or height C. Consists of points D. Represented by a double pointed arrow E. Labeled by any two point that it contains AB A B C • • •

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**• • • II. Line A. Geometric figure having infinite length**

B. No width or height C. Consists of points D. Represented by a double pointed arrow E. Labeled by any two point that it contains AB A B C • • • AC BC

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II. Line A. Geometric figure having infinite length B. No width or height C. Consists of points D. Represented by a double pointed arrow E. Labeled by any two point that it contains F. The intersection of two lines is a _______ point

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W • P J • •

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W • The intersection of WP and PJ is P. P J • •

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II. Line A. Geometric figure having infinite length B. No width or height C. Consists of points D. Represented by a double pointed arrow E. Labeled by any two point that it contains F. The intersection of two lines is a _______ point G. Through any one point there are infinitely many lines

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•

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II. Line A. Geometric figure having infinite length B. No width or height C. Consists of points D. Represented by a double pointed arrow E. Labeled by any two point that it contains F. The intersection of two lines is a _______ point G. Through any one point there are infinitely many lines H. Through any two points there is exactly one line

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Plane A. Geometric figure having infinite length and width but no height. B. Represented by a flat rectangular surface C. Planes consist of lines D. Labeled by any three points on the plane

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**P Q A L D K M B H C Are Points L, K, and M COPLANAR?**

Yes, they are COPLANAR because they LIE ON THE SAME PLANE P. Is point H, coplanar with points L, K, and M? No, because it lies on plane Q and points L, K, and M are in different plane, on plane P. NON-COPLANAR points are points that lie in different planes. PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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**P Q A D B C On what planes does point D lie?**

PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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**P Q A D B C On what planes does point D lie? It only lies on plane Q.**

PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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**E. Through any two points there are infinitely many planes **

A. Geometric figure having infinite length and width but no height. B. Represented by a flat rectangular surface C. Planes consist of lines D. Labeled by any three points on the plane E. Through any two points there are infinitely many planes F. Through any three points, there is exactly one plane

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Plane A. Geometric figure having infinite length and width but no height. B. Represented by a flat rectangular surface C. Planes consist of lines D. Labeled by any three points on the plane E. Through any two points there are infinitely many planes F. Through any three points, there is exactly one plane G. The intersection of two planes is a_______ line H. The intersection of three planes is a _____________ or ___________ point line

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**∙ ∙ Line Segment A. A piece of a line B. Has two endpoints**

C. Labeled by its endpoints ∙ ∙ S T ST

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**∙ ∙ Ray A. Geometric figure with one endpoint**

B. Labeled by it’s endpoint and one other point ∙ ∙ P Q PQ

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**1.2 Measuring Line Segments**

“Measure” of a Line Segment A. The distance between its endpoints B. Always positive

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**1.2 Measuring Line Segments**

B • • -5 -4 -3 -2 -1 1 2 3 4 5 a b coordinates

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**1.2 Measuring Line Segments**

AB = “the measure of AB” AB = _________ 7 units A B • AB • -5 -4 -3 -2 -1 1 2 3 4 5 a b coordinates

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**1.2 Measuring Line Segments**

“Measure” of a Line Segment A. The distance between its endpoints B. Always positive C. AB = b – a or a - b

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**1.2 Measuring Line Segments**

AB = 3 – (-4) AB = -4 – 3 AB = 3 + (+4) or AB = AB = -7 AB = 7 A B • AB • -5 -4 -3 -2 -1 1 2 3 4 5 a b coordinates

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**1.2 Measuring Line Segments**

AB = 3 – (-4) AB = -4 – 3 AB = 3 + (+4) or AB = AB = -7 = 7 units AB = 7 units A B • AB • -5 -4 -3 -2 -1 1 2 3 4 5 a b coordinates

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Examples PQ = ________________ 95 – 23 = 72 units P PQ Q • • 23 95

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**• • Examples EF = ________________ EF = ________________ E EF F -15 46**

46 – (-15) = 61 units OR EF = ________________ -15 – 46 = -61 = 61 units E EF F • • -15 46

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**• • Examples RS = ________________ RS = ________________ R RS S -92**

-18 – (-92) = 74 units | OR RS = ________________ -92 – (-18) = -74 = 74 units | R RS S • • -92 -18

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**1.2 Measuring Line Segments**

Segment Addition A. “collinear”= “on the same line” B. If A, B, & C are collinear and B is between A and C, then AB + BC = AC • • • A B C AB BC AC

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**Examples of Segment Addition**

A carpenter must cut a 54 inch board into two pieces so that one piece is twice as long as the other. What will be the length of the two board after the cut? AB + BC = AC x + 2x = 54 3x = 54 A B • • C • X 2x 54 in.

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**Examples of Segment Addition**

A carpenter must cut a 54 inch board into two pieces so that one piece is twice as long as the other. What will be the length of the two board after the cut? AB + BC = AC 1 8 3 54 x + 2x = 54 3 3x = 54 24 24 x = 18 in. A B • • C • X 2x = 2(18) 18 in. 36 in. 54 in.

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**Examples of Segment Addition**

A 45 foot piece of pipe must be cut so that the longer piece is 9 feet longer than the shorter. What will be the lengths of the two pieces? AB + BC = AC x + x + 9 = 45 2x = 45 - 9 = -9 2x = 36 A B • • • C X X + 9 45 ft.

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**Examples of Segment Addition**

A 45 foot piece of pipe must be cut so that the longer piece is 9 feet longer than the shorter. What will be the lengths of the two pieces? AB + BC = AC x + x + 9 = 45 1 8 2x = 45 2 36 - 9 = -9 2 2x = 36 16 16 x = 18 ft. A B • • • C X X + 9 = 18 ft. 27 ft. 45 ft.

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**1.2 Measuring Line Segments**

Midpoint of a Segment A. If A, B, and C are collinear and AC = CB, then C is the midpoint of AB.

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**1.2 Measuring Line Segments**

Midpoint of a Segment A. If A, B, and C are collinear and AC = CB, then C is the midpoint of AB. A • C • • B

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**1.2 Measuring Line Segments**

Midpoint of a Segment A. If A, B, and C are collinear and AC = CB, then C is the midpoint of AB. B. Midpoint Formula • A C • B • 12 58 a b

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**1.2 Measuring Line Segments**

Midpoint of a Segment A. If A, B, and C are collinear and AC = CB, then C is the midpoint of AB. B. Midpoint Formula a + b 2 The midpoint of AB = = 2 70 2 = = 35 A • C • • B 12 35 58 a b

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**1.2 Measuring Line Segments**

Midpoint of a Segment A. If A, B, and C are collinear and AC = CB, then C is the midpoint of AB. B. Midpoint Formula a + b 2 The midpoint of AB = = 2 20 2 = = 10 A • C • • B -15 10 35 a b

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**1.2 Measuring Line Segments**

Midpoint of a Segment A. If A, B, and C are collinear and AC = CB, then C is the midpoint of AB. B. Midpoint Formula a + b 2 The midpoint of AB = = 2 -96 2 -48 = = • A C • B • -84 -48 -12 a b

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**Examples of Segment Addition**

A carpenter must cut a 65 inch board into two pieces so that one piece is five inches more than twice the length of the other. What will be the length of the two board after the cut? AB + BC = AC x + 2x+5 = 65 A B • • • C X 2x+5 65 in.

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**1.3 Measuring Angles A. Using a Protractor ° 60**

ACUTE Angle less than 90

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1.3 Measuring Angles A. Using a Protractor RIGHT Angle 90

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**1.3 Measuring Angles A. Using a Protractor ° 140 OBTUSE Angle**

Greater than 90

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1.3 Measuring Angles A. Using a Protractor 120

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1.3 Measuring Angles A. Using a Protractor • • C • D B • E • A • F O

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**• • • • 1.3 Measuring Angles Using a Protractor B. Angle Addition A D**

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**• • • • 1.3 Measuring Angles Using a Protractor B. Angle Addition**

m ABD + m DBC = m ABC B. Angle Addition • A • D • B • C

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A 70 angle is divided into two smaller angles such that the larger angle is two more than three times the smaller. m ABD + m DBC = m ABC x + 3x +2 = 70 • A 4x + 2 = 70 –2 –2 ___ ___ 4x = 68 3(17) + 2 70 x = 17 3x + 2 • D • 53 B 17 x • C

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**1.3 Measuring Angles Using a Protractor B. Angle Addition**

C. Vertical Angle Conjecture “ the vertical angles formed by intersecting lines have equal measure”

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**1.4 Special Angles Complementary Angles**

A pair of angles whose sum is 90 1 2

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**1.4 Special Angles Complementary Angles**

A pair of angles whose sum is 90 1 2

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**1.4 Special Angles Supplementary Angles**

A pair of angles whose sum is 180° 1 2

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1.4 Special Angles Supplementary Angles 1 2

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**An angle is four times it’s compliment. Find both angles.**

x + 4x = 90 5x = 90 x = 18 4(18) = 72° 4x 18° x

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**1.5 Parallel and Perpendicular Lines**

Parallel Lines Lines on the same plane that do not intersect l m l || m

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**1.5 Parallel and Perpendicular Lines**

Parallel Lines Lines on the same plane that do not intersect Perpendicular Lines Two lines that intersect at a right angle

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**1.5 Parallel and Perpendicular Lines**

k k j j

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**1.5 Parallel and Perpendicular Lines**

Corresponding Angles 1 2 3 m<1 = m<2 = m<3

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**1.5 Parallel and Perpendicular Lines**

Corresponding Angles 45 3x + 20 = 5x – 10 m<1 = 3(15) +20 -3x x 20 = 2x – 10 +10 = 75 30 = 2x m<2 = 5(15) – 10 15 = x 65º 65º (3x+20) (5x -10) 1 2 m<1 = m<2 = m<3

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**1.5 Parallel and Perpendicular Lines**

Corresponding Angles 54 6x + 30 = 3x + 57 m<1 = 6(9) +30 -3x x 3x + 30 = - 30 = 27 3x = m<2 = 3(9) + 57 x = 9 84º 84º (6x+30) (3x +57) 1 2 m<1 = m<2 = m<3

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**1.5 Parallel and Perpendicular Lines**

Corresponding Angles 63 9x x + 39 = 180 m<1 = 9(7) +50 13x +89 = 180 13x = 91 28 m<2 = 4(7) + 39 x = 7 113º 67º (9x+50) (4x +39) 1 2 m<1 = m<2 = m<3

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