Geometry Chapter 3 Parallel Lines and Perpendicular Lines Pages 124 - 195.
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Transcript of Geometry Chapter 3 Parallel Lines and Perpendicular Lines Pages 124 - 195.
![Page 1: Geometry Chapter 3 Parallel Lines and Perpendicular Lines Pages 124 - 195.](https://reader036.fdocuments.in/reader036/viewer/2022062322/5697bfb61a28abf838c9e397/html5/thumbnails/1.jpg)
Geometry Chapter 3 Parallel Lines and
Perpendicular Lines
Geometry Chapter 3 Parallel Lines and
Perpendicular Lines Pages 124 - 195
Pages 124 - 195
![Page 2: Geometry Chapter 3 Parallel Lines and Perpendicular Lines Pages 124 - 195.](https://reader036.fdocuments.in/reader036/viewer/2022062322/5697bfb61a28abf838c9e397/html5/thumbnails/2.jpg)
3-1 PAIRS & LINES OF ANGELS3-1 PAIRS & LINES OF ANGELS
What you will learn: Identify lines and planesIdentify parallel and
perpendicular linesIdentify pairs of angles
formed by transversals
What you will learn: Identify lines and planesIdentify parallel and
perpendicular linesIdentify pairs of angles
formed by transversals
![Page 3: Geometry Chapter 3 Parallel Lines and Perpendicular Lines Pages 124 - 195.](https://reader036.fdocuments.in/reader036/viewer/2022062322/5697bfb61a28abf838c9e397/html5/thumbnails/3.jpg)
3-1 PROPERTIES OF PARALLEL LINES3-1 PROPERTIES OF PARALLEL LINES
Essential Question: What does it mean when two
lines are parallel, intersecting, coincident, or skew?
Essential Question: What does it mean when two
lines are parallel, intersecting, coincident, or skew?
![Page 4: Geometry Chapter 3 Parallel Lines and Perpendicular Lines Pages 124 - 195.](https://reader036.fdocuments.in/reader036/viewer/2022062322/5697bfb61a28abf838c9e397/html5/thumbnails/4.jpg)
PREVIOUS VOCABULARYPREVIOUS VOCABULARYPerpendicular linesPerpendicular lines
![Page 5: Geometry Chapter 3 Parallel Lines and Perpendicular Lines Pages 124 - 195.](https://reader036.fdocuments.in/reader036/viewer/2022062322/5697bfb61a28abf838c9e397/html5/thumbnails/5.jpg)
CORE VOCABULARYCORE VOCABULARY
![Page 6: Geometry Chapter 3 Parallel Lines and Perpendicular Lines Pages 124 - 195.](https://reader036.fdocuments.in/reader036/viewer/2022062322/5697bfb61a28abf838c9e397/html5/thumbnails/6.jpg)
PARALLEL LINESPARALLEL LINESTwo lines
that do not intersect
Go in same direction
Coplanar
Two lines that do not intersect
Go in same direction
Coplanar
![Page 7: Geometry Chapter 3 Parallel Lines and Perpendicular Lines Pages 124 - 195.](https://reader036.fdocuments.in/reader036/viewer/2022062322/5697bfb61a28abf838c9e397/html5/thumbnails/7.jpg)
SKEW LINESSKEW LINESTwo lines
that do not intersect
Are not coplanar
Two lines that do not intersect
Are not coplanar
![Page 8: Geometry Chapter 3 Parallel Lines and Perpendicular Lines Pages 124 - 195.](https://reader036.fdocuments.in/reader036/viewer/2022062322/5697bfb61a28abf838c9e397/html5/thumbnails/8.jpg)
PARALLEL PLANESPARALLEL PLANESTwo planes
that do not intersect
Two planes that do not intersect
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TRANSVERSALTRANSVERSALA line that
intersects two or more coplanar parallel lines
A line that intersects two or more coplanar parallel lines
![Page 10: Geometry Chapter 3 Parallel Lines and Perpendicular Lines Pages 124 - 195.](https://reader036.fdocuments.in/reader036/viewer/2022062322/5697bfb61a28abf838c9e397/html5/thumbnails/10.jpg)
CORRESPONDING ANGLES
CORRESPONDING ANGLES
CongruentSame
positionDifferent
location
CongruentSame
positionDifferent
location
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ALTERNATE INTERIOR ANGLES
ALTERNATE INTERIOR ANGLES
CongruentInsideOpposites
sides
CongruentInsideOpposites
sides
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ALTERNATE EXTERIOR ANGLES
ALTERNATE EXTERIOR ANGLES
CongruentOutsideOpposites
sides
CongruentOutsideOpposites
sides
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SAME-SIDE (consecutive) INTERIOR ANGLES
SAME-SIDE (consecutive) INTERIOR ANGLES
SupplementaryInsideSame side
SupplementaryInsideSame side
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PARALLEL LINESPARALLEL LINES
Two coplanar lines that do not intersect
Two coplanar lines that do not intersect
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STRAIGHT ANGLESTRAIGHT ANGLE
Exactly 180 degreesExactly 180 degrees
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VERTICAL ANGLESVERTICAL ANGLES
2 angles directly across from each other
congruent
2 angles directly across from each other
congruent
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SUPPLEMENTARY ANGLES
SUPPLEMENTARY ANGLES
Two angles whose measures add up to 180 degrees
Two angles whose measures add up to 180 degrees
![Page 18: Geometry Chapter 3 Parallel Lines and Perpendicular Lines Pages 124 - 195.](https://reader036.fdocuments.in/reader036/viewer/2022062322/5697bfb61a28abf838c9e397/html5/thumbnails/18.jpg)
3 – 2 PARALLEL LINES & TRANSVERSALS
3 – 2 PARALLEL LINES & TRANSVERSALS
What you will learn:Use properties of parallel linesProve theorems about parallel
linesSolve real-life problems
What you will learn:Use properties of parallel linesProve theorems about parallel
linesSolve real-life problems
![Page 19: Geometry Chapter 3 Parallel Lines and Perpendicular Lines Pages 124 - 195.](https://reader036.fdocuments.in/reader036/viewer/2022062322/5697bfb61a28abf838c9e397/html5/thumbnails/19.jpg)
3-2 PARALLEL LINES & TRANSVERSALS3-2 PARALLEL LINES & TRANSVERSALS
Essential Question: When two parallel lines are
cut by a transversal, which of the resulting pairs of angles are congruent?
Essential Question: When two parallel lines are
cut by a transversal, which of the resulting pairs of angles are congruent?
![Page 20: Geometry Chapter 3 Parallel Lines and Perpendicular Lines Pages 124 - 195.](https://reader036.fdocuments.in/reader036/viewer/2022062322/5697bfb61a28abf838c9e397/html5/thumbnails/20.jpg)
CORE VOCABULARYCORE VOCABULARY
![Page 21: Geometry Chapter 3 Parallel Lines and Perpendicular Lines Pages 124 - 195.](https://reader036.fdocuments.in/reader036/viewer/2022062322/5697bfb61a28abf838c9e397/html5/thumbnails/21.jpg)
TRANSVERSALTRANSVERSALA line that
intersects two or more coplanar parallel lines
A line that intersects two or more coplanar parallel lines
![Page 22: Geometry Chapter 3 Parallel Lines and Perpendicular Lines Pages 124 - 195.](https://reader036.fdocuments.in/reader036/viewer/2022062322/5697bfb61a28abf838c9e397/html5/thumbnails/22.jpg)
CORRESPONDING ANGLES
CORRESPONDING ANGLES
CongruentSame
positionDifferent
location
CongruentSame
positionDifferent
location
![Page 23: Geometry Chapter 3 Parallel Lines and Perpendicular Lines Pages 124 - 195.](https://reader036.fdocuments.in/reader036/viewer/2022062322/5697bfb61a28abf838c9e397/html5/thumbnails/23.jpg)
ALTERNATE INTERIOR ANGLES
ALTERNATE INTERIOR ANGLES
CongruentInsideOpposites
sides
CongruentInsideOpposites
sides
![Page 24: Geometry Chapter 3 Parallel Lines and Perpendicular Lines Pages 124 - 195.](https://reader036.fdocuments.in/reader036/viewer/2022062322/5697bfb61a28abf838c9e397/html5/thumbnails/24.jpg)
ALTERNATE EXTERIOR ANGLES
ALTERNATE EXTERIOR ANGLES
CongruentOutsideOpposites
sides
CongruentOutsideOpposites
sides
![Page 25: Geometry Chapter 3 Parallel Lines and Perpendicular Lines Pages 124 - 195.](https://reader036.fdocuments.in/reader036/viewer/2022062322/5697bfb61a28abf838c9e397/html5/thumbnails/25.jpg)
SAME-SIDE (consecutive) INTERIOR ANGLES
SAME-SIDE (consecutive) INTERIOR ANGLES
SupplementaryInsideSame side
SupplementaryInsideSame side
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3 – 3 Proofs and Parallel Lines3 – 3 Proofs and Parallel Lines
What you will learn: Use the Corresponding Angles Converse Construct Parallel Lines Prove theorems about parallel lines Use Transitive Property of Parallel Lines
What you will learn: Use the Corresponding Angles Converse Construct Parallel Lines Prove theorems about parallel lines Use Transitive Property of Parallel Lines
![Page 27: Geometry Chapter 3 Parallel Lines and Perpendicular Lines Pages 124 - 195.](https://reader036.fdocuments.in/reader036/viewer/2022062322/5697bfb61a28abf838c9e397/html5/thumbnails/27.jpg)
3 – 3 Proofs and Parallel Lines3 – 3 Proofs and Parallel Lines
Essential Question: Name the two types of pairs
of angles that are supplementary
Essential Question: Name the two types of pairs
of angles that are supplementary
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WAYS TO PROVE TWO LINES PARALLEL
WAYS TO PROVE TWO LINES PARALLEL
Show that a pair of corresponding angles are congruent
Show that a pair of alternate interior or exterior angles are congruent
Show that a pair of same-side interior angles are supplementary
Show that a pair of corresponding angles are congruent
Show that a pair of alternate interior or exterior angles are congruent
Show that a pair of same-side interior angles are supplementary
![Page 29: Geometry Chapter 3 Parallel Lines and Perpendicular Lines Pages 124 - 195.](https://reader036.fdocuments.in/reader036/viewer/2022062322/5697bfb61a28abf838c9e397/html5/thumbnails/29.jpg)
WAYS TO PROVE TWO LINES PARALLEL
WAYS TO PROVE TWO LINES PARALLEL
Show that both lines are perpendicular to a third line
Show that both lines are parallel to a third line
Show that both lines are perpendicular to a third line
Show that both lines are parallel to a third line
![Page 30: Geometry Chapter 3 Parallel Lines and Perpendicular Lines Pages 124 - 195.](https://reader036.fdocuments.in/reader036/viewer/2022062322/5697bfb61a28abf838c9e397/html5/thumbnails/30.jpg)
Core Concept:Five Types of Angle Pairs
Core Concept:Five Types of Angle Pairs Corresponding ≅ Alternate Interior ≅ Alternate Exterior ≅ Same-Side Interior 180 Vertical ≅ Linear Pair 180
Corresponding ≅ Alternate Interior ≅ Alternate Exterior ≅ Same-Side Interior 180 Vertical ≅ Linear Pair 180
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PERPENDICULAR LINESPERPENDICULAR LINESTwo lines that intersect to form right angles
If a line is perpendicular to one of two parallel lines, it is also perpendicular to the other line
Two lines that intersect to form right angles
If a line is perpendicular to one of two parallel lines, it is also perpendicular to the other line
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3 - 4 PROOFS WITH PERPENDICULAR LINES
3 - 4 PROOFS WITH PERPENDICULAR LINES
What you will learn: Find the distance from a point to a line Construct Perpendicular lines Prove theorems about perpendicular
lines Solve real life problems involving
perpendicular lines
What you will learn: Find the distance from a point to a line Construct Perpendicular lines Prove theorems about perpendicular
lines Solve real life problems involving
perpendicular lines
![Page 33: Geometry Chapter 3 Parallel Lines and Perpendicular Lines Pages 124 - 195.](https://reader036.fdocuments.in/reader036/viewer/2022062322/5697bfb61a28abf838c9e397/html5/thumbnails/33.jpg)
3 – 4 Proofs and Parallel Lines3 – 4 Proofs and Parallel Lines
Essential Question: What conjectures can you
make about perpendicular lines?
Essential Question: What conjectures can you
make about perpendicular lines?
![Page 34: Geometry Chapter 3 Parallel Lines and Perpendicular Lines Pages 124 - 195.](https://reader036.fdocuments.in/reader036/viewer/2022062322/5697bfb61a28abf838c9e397/html5/thumbnails/34.jpg)
VOCABULARYVOCABULARYDistance from a point to a linePerpendicular bisector
Distance from a point to a linePerpendicular bisector
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Distance from a point to a lineDistance from a point to a line
The length of the perpendicular segment from the point to the line
The length of the perpendicular segment from the point to the line
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Perpendicular BisectorPerpendicular Bisector
A perpendicular bisector of a line segment is a line segment that is perpendicular to the segment at its midpoint
A perpendicular bisector of a line segment is a line segment that is perpendicular to the segment at its midpoint
![Page 37: Geometry Chapter 3 Parallel Lines and Perpendicular Lines Pages 124 - 195.](https://reader036.fdocuments.in/reader036/viewer/2022062322/5697bfb61a28abf838c9e397/html5/thumbnails/37.jpg)
PARALLEL LINESPARALLEL LINESTwo lines that do not
intersectGo in same direction If two lines are parallel to
the same line, they are parallel to each other
If two lines are perpendicular to the same line, then they are parallel to each other
Two lines that do not intersect
Go in same direction If two lines are parallel to
the same line, they are parallel to each other
If two lines are perpendicular to the same line, then they are parallel to each other
![Page 38: Geometry Chapter 3 Parallel Lines and Perpendicular Lines Pages 124 - 195.](https://reader036.fdocuments.in/reader036/viewer/2022062322/5697bfb61a28abf838c9e397/html5/thumbnails/38.jpg)
TRIANGLETRIANGLEThree sides Interior angle sum is 180˚ Symbol: ∆ Sides are called segmentsEach point is a vertex
Three sides Interior angle sum is 180˚ Symbol: ∆ Sides are called segmentsEach point is a vertex
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EQUIANGULAREQUIANGULARAll angles are 60˚All angles are 60˚
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ACUTE TRIANGLEACUTE TRIANGLE
Three angles less than 90 degrees
Three angles less than 90 degrees
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RIGHT TRIANGLERIGHT TRIANGLEOne right angle
One right angle
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OBTUSE TRIANGLEOBTUSE TRIANGLE
One obtuse angle
One obtuse angle
![Page 43: Geometry Chapter 3 Parallel Lines and Perpendicular Lines Pages 124 - 195.](https://reader036.fdocuments.in/reader036/viewer/2022062322/5697bfb61a28abf838c9e397/html5/thumbnails/43.jpg)
EQUILATERAL TRIANGLEEQUILATERAL TRIANGLEAll sides congruent
All sides congruent
![Page 44: Geometry Chapter 3 Parallel Lines and Perpendicular Lines Pages 124 - 195.](https://reader036.fdocuments.in/reader036/viewer/2022062322/5697bfb61a28abf838c9e397/html5/thumbnails/44.jpg)
ISOSCELES TRIANGLEISOSCELES TRIANGLEAt least two congruent sides
At least two congruent sides
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SCALENE TRIANGLESCALENE TRIANGLE
No congruent sides
No congruent sides
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EXTERIOR ANGLEEXTERIOR ANGLE Outside the triangle Equals the remote interior angles Supplementary to its adjacent angle
Outside the triangle Equals the remote interior angles Supplementary to its adjacent angle
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REMOTE INTERIOR ANGLESREMOTE INTERIOR ANGLES
on the opposite side of the exterior angles
equal the measure of the exterior angle
on the opposite side of the exterior angles
equal the measure of the exterior angle
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3 - 5 POLYGON ANGLE-SUM THEOREM
3 - 5 POLYGON ANGLE-SUM THEOREM
STANDARD: classify polygons find measures of interior and exterior angles of polygons
STANDARD: classify polygons find measures of interior and exterior angles of polygons
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VOCABULARYVOCABULARY1. Polygon2. Concave Polygon3. Convex Polygon4. Diagonal5. Polygon Angle Sum6. Polygon Exterior Angle Sum7. Equilateral Polygon8. Equiangular Polygon9. Regular Polygon
1. Polygon2. Concave Polygon3. Convex Polygon4. Diagonal5. Polygon Angle Sum6. Polygon Exterior Angle Sum7. Equilateral Polygon8. Equiangular Polygon9. Regular Polygon
![Page 50: Geometry Chapter 3 Parallel Lines and Perpendicular Lines Pages 124 - 195.](https://reader036.fdocuments.in/reader036/viewer/2022062322/5697bfb61a28abf838c9e397/html5/thumbnails/50.jpg)
POLYGONPOLYGONClosed plane figureAt least 3 sides and anglesClassified by the number of
sides
Closed plane figureAt least 3 sides and anglesClassified by the number of
sides
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CONVEX POLYGONCONVEX POLYGON
Doesn’t cave inDoesn’t cave in
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CONCAVE POLYGONCONCAVE POLYGON
caves incaves in
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DiagonalDiagonal
Connects verticesConnects vertices
![Page 54: Geometry Chapter 3 Parallel Lines and Perpendicular Lines Pages 124 - 195.](https://reader036.fdocuments.in/reader036/viewer/2022062322/5697bfb61a28abf838c9e397/html5/thumbnails/54.jpg)
POLYGON ANGLE SUM POLYGON ANGLE SUM
(n-2)180(n-2)180
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POLYGON EXTERIOR ANGLE SUM
POLYGON EXTERIOR ANGLE SUM
The exterior angles of a polygon = 360
The exterior angles of a polygon = 360
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EQUILATERAL POLYGONEQUILATERAL POLYGON
All sides are congruent
All sides are congruent
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EQUIANGULAR POLYGON*
EQUIANGULAR POLYGON*
All angles are congruent
All angles are congruent
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REGULAR POLYGON REGULAR POLYGON EquiangularEquilateral
EquiangularEquilateral
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3 - 6 LINES IN THE COORDINATE PLANE3 - 6 LINES IN THE
COORDINATE PLANE
STANDARD: graph lines given their equations
to write equations of lines
STANDARD: graph lines given their equations
to write equations of lines
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VOCABULARYVOCABULARY1. Slope2. y-intercept3. x-intercept4. Graphing Using Intercepts5. Standard Form6. Slope Intercept Form7. Point Slope Form
1. Slope2. y-intercept3. x-intercept4. Graphing Using Intercepts5. Standard Form6. Slope Intercept Form7. Point Slope Form
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SLOPESLOPE
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y-intercepty-intercept
Where the graph intersects the y-axis
Where the graph intersects the y-axis
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x-interceptx-intercept
Where the graph intersects the x-axis
Where the graph intersects the x-axis
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Graphing Using intercepts
Graphing Using intercepts
Substitute “0” for x and y to find the intercepts
Substitute “0” for x and y to find the intercepts
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STANDARD FORMSTANDARD FORM
Ax + By = CAx + By = C
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SLOPE INTERCEPT FORMSLOPE INTERCEPT FORM
y = mx + bb = y-interceptm = slope
y = mx + bb = y-interceptm = slope
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POINT SLOPE FORMPOINT SLOPE FORM
y - y1 = m(x - x1)y - y1 = m(x - x1)
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3 - 7 SLOPES OF PARALLEL AND
PERPENDICULAR LINES
3 - 7 SLOPES OF PARALLEL AND
PERPENDICULAR LINESSTANDARD: relate slope and parallel lines
relate slope and perpendicular lines
STANDARD: relate slope and parallel lines
relate slope and perpendicular lines
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PARALLEL LINESPARALLEL LINESHave equal
slopesTwo lines that
do not intersect
Go in same direction
Have equal slopes
Two lines that do not intersect
Go in same direction
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PERPENDICULAR LINESPERPENDICULAR LINES
The product of slopes is -1
Two lines that intersect to form right angles
The product of slopes is -1
Two lines that intersect to form right angles
![Page 71: Geometry Chapter 3 Parallel Lines and Perpendicular Lines Pages 124 - 195.](https://reader036.fdocuments.in/reader036/viewer/2022062322/5697bfb61a28abf838c9e397/html5/thumbnails/71.jpg)
SLOPE INTERCEPT FORMSLOPE INTERCEPT FORM
y = mx + bb = y-interceptm = slope
y = mx + bb = y-interceptm = slope
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INTERSECTINTERSECT
To cutDivide by passing through
To cutDivide by passing through
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CONGRUENTCONGRUENT
equalThe same
equalThe same