geometry chapter 3 parallel lines and perpendicular lines pages 124 - 195
TRANSCRIPT
Geometry Chapter 3 Parallel Lines and
Perpendicular Lines
Geometry Chapter 3 Parallel Lines and
Perpendicular Lines Pages 124 - 195
Pages 124 - 195
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
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?
PREVIOUS VOCABULARYPREVIOUS VOCABULARYPerpendicular linesPerpendicular lines
CORE VOCABULARYCORE VOCABULARY
PARALLEL LINESPARALLEL LINESTwo lines
that do not intersect
Go in same direction
Coplanar
Two lines that do not intersect
Go in same direction
Coplanar
SKEW LINESSKEW LINESTwo lines
that do not intersect
Are not coplanar
Two lines that do not intersect
Are not coplanar
PARALLEL PLANESPARALLEL PLANESTwo planes
that do not intersect
Two planes that do not intersect
TRANSVERSALTRANSVERSALA line that
intersects two or more coplanar parallel lines
A line that intersects two or more coplanar parallel lines
CORRESPONDING ANGLES
CORRESPONDING ANGLES
CongruentSame
positionDifferent
location
CongruentSame
positionDifferent
location
ALTERNATE INTERIOR ANGLES
ALTERNATE INTERIOR ANGLES
CongruentInsideOpposites
sides
CongruentInsideOpposites
sides
ALTERNATE EXTERIOR ANGLES
ALTERNATE EXTERIOR ANGLES
CongruentOutsideOpposites
sides
CongruentOutsideOpposites
sides
SAME-SIDE (consecutive) INTERIOR ANGLES
SAME-SIDE (consecutive) INTERIOR ANGLES
SupplementaryInsideSame side
SupplementaryInsideSame side
PARALLEL LINESPARALLEL LINES
Two coplanar lines that do not intersect
Two coplanar lines that do not intersect
STRAIGHT ANGLESTRAIGHT ANGLE
Exactly 180 degreesExactly 180 degrees
VERTICAL ANGLESVERTICAL ANGLES
2 angles directly across from each other
congruent
2 angles directly across from each other
congruent
SUPPLEMENTARY ANGLES
SUPPLEMENTARY ANGLES
Two angles whose measures add up to 180 degrees
Two angles whose measures add up to 180 degrees
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
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?
CORE VOCABULARYCORE VOCABULARY
TRANSVERSALTRANSVERSALA line that
intersects two or more coplanar parallel lines
A line that intersects two or more coplanar parallel lines
CORRESPONDING ANGLES
CORRESPONDING ANGLES
CongruentSame
positionDifferent
location
CongruentSame
positionDifferent
location
ALTERNATE INTERIOR ANGLES
ALTERNATE INTERIOR ANGLES
CongruentInsideOpposites
sides
CongruentInsideOpposites
sides
ALTERNATE EXTERIOR ANGLES
ALTERNATE EXTERIOR ANGLES
CongruentOutsideOpposites
sides
CongruentOutsideOpposites
sides
SAME-SIDE (consecutive) INTERIOR ANGLES
SAME-SIDE (consecutive) INTERIOR ANGLES
SupplementaryInsideSame side
SupplementaryInsideSame side
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
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
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
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
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
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
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
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?
VOCABULARYVOCABULARYDistance from a point to a linePerpendicular bisector
Distance from a point to a linePerpendicular bisector
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
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
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
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
EQUIANGULAREQUIANGULARAll angles are 60˚All angles are 60˚
ACUTE TRIANGLEACUTE TRIANGLE
Three angles less than 90 degrees
Three angles less than 90 degrees
RIGHT TRIANGLERIGHT TRIANGLEOne right angle
One right angle
OBTUSE TRIANGLEOBTUSE TRIANGLE
One obtuse angle
One obtuse angle
EQUILATERAL TRIANGLEEQUILATERAL TRIANGLEAll sides congruent
All sides congruent
ISOSCELES TRIANGLEISOSCELES TRIANGLEAt least two congruent sides
At least two congruent sides
SCALENE TRIANGLESCALENE TRIANGLE
No congruent sides
No congruent sides
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
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
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
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
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
CONVEX POLYGONCONVEX POLYGON
Doesn’t cave inDoesn’t cave in
CONCAVE POLYGONCONCAVE POLYGON
caves incaves in
DiagonalDiagonal
Connects verticesConnects vertices
POLYGON ANGLE SUM POLYGON ANGLE SUM
(n-2)180(n-2)180
POLYGON EXTERIOR ANGLE SUM
POLYGON EXTERIOR ANGLE SUM
The exterior angles of a polygon = 360
The exterior angles of a polygon = 360
EQUILATERAL POLYGONEQUILATERAL POLYGON
All sides are congruent
All sides are congruent
EQUIANGULAR POLYGON*
EQUIANGULAR POLYGON*
All angles are congruent
All angles are congruent
REGULAR POLYGON REGULAR POLYGON EquiangularEquilateral
EquiangularEquilateral
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
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
SLOPESLOPE
y-intercepty-intercept
Where the graph intersects the y-axis
Where the graph intersects the y-axis
x-interceptx-intercept
Where the graph intersects the x-axis
Where the graph intersects the x-axis
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
STANDARD FORMSTANDARD FORM
Ax + By = CAx + By = C
SLOPE INTERCEPT FORMSLOPE INTERCEPT FORM
y = mx + bb = y-interceptm = slope
y = mx + bb = y-interceptm = slope
POINT SLOPE FORMPOINT SLOPE FORM
y - y1 = m(x - x1)y - y1 = m(x - x1)
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
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
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
SLOPE INTERCEPT FORMSLOPE INTERCEPT FORM
y = mx + bb = y-interceptm = slope
y = mx + bb = y-interceptm = slope
INTERSECTINTERSECT
To cutDivide by passing through
To cutDivide by passing through
CONGRUENTCONGRUENT
equalThe same
equalThe same