Geometry Chapter 3 Parallel Lines and Perpendicular Lines Pages
3-1 PAIRS & LINES OF ANGELS What you will learn: Identify lines and planes Identify parallel and perpendicular lines Identify pairs of angles formed by transversals What you will learn: Identify lines and planes Identify parallel and perpendicular lines Identify pairs of angles formed by transversals
3-1 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 VOCABULARY Perpendicular lines
CORE VOCABULARY Parallel Lines Skew Lines Parallel Planes Transversal Corresponding Angles Alternate interior Angles Alternate Exterior Angles Same-Side (consecutive) interior angles Parallel Lines Skew Lines Parallel Planes Transversal Corresponding Angles Alternate interior Angles Alternate Exterior Angles Same-Side (consecutive) interior angles
PARALLEL LINES Two lines that do not intersect Go in same direction Coplanar Two lines that do not intersect Go in same direction Coplanar
SKEW LINES Two lines that do not intersect Are not coplanar Two lines that do not intersect Are not coplanar
PARALLEL PLANES Two planes that do not intersect
TRANSVERSAL A line that intersects two or more coplanar parallel lines
CORRESPONDING ANGLES Congruent Same position Different location Congruent Same position Different location
ALTERNATE INTERIOR ANGLES Congruent Inside Opposites sides Congruent Inside Opposites sides
ALTERNATE EXTERIOR ANGLES Congruent Outside Opposites sides Congruent Outside Opposites sides
SAME-SIDE (consecutive) INTERIOR ANGLES Supplementary Inside Same side Supplementary Inside Same side
PARALLEL LINES Two coplanar lines that do not intersect
STRAIGHT ANGLE Exactly 180 degrees
VERTICAL ANGLES 2 angles directly across from each other congruent 2 angles directly across from each other congruent
SUPPLEMENTARY ANGLES Two angles whose measures add up to 180 degrees
3 – 2 PARALLEL LINES & TRANSVERSALS What you will learn: Use properties of parallel lines Prove theorems about parallel lines Solve real-life problems What you will learn: Use properties of parallel lines Prove theorems about parallel lines Solve real-life problems
3-2 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 VOCABULARY Transversal Corresponding Angles Alternate interior Angles Alternate Exterior Angles Same-Side (consecutive) interior angles Transversal Corresponding Angles Alternate interior Angles Alternate Exterior Angles Same-Side (consecutive) interior angles
TRANSVERSAL A line that intersects two or more coplanar parallel lines
CORRESPONDING ANGLES Congruent Same position Different location Congruent Same position Different location
ALTERNATE INTERIOR ANGLES Congruent Inside Opposites sides Congruent Inside Opposites sides
ALTERNATE EXTERIOR ANGLES Congruent Outside Opposites sides Congruent Outside Opposites sides
SAME-SIDE (consecutive) INTERIOR ANGLES Supplementary Inside Same side Supplementary Inside Same side
3 – 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 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 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 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 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 LINES 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 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 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 Lines Essential Question: What conjectures can you make about perpendicular lines? Essential Question: What conjectures can you make about perpendicular lines?
VOCABULARY Distance from a point to a line Perpendicular bisector Distance from a point to a line Perpendicular bisector
Distance from a point to a line The length of the perpendicular segment from the point to the line
Perpendicular Bisector A perpendicular bisector of a line segment is a line segment that is perpendicular to the segment at its midpoint
PARALLEL LINES 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 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
TRIANGLE Three sides Interior angle sum is 180˚ Symbol: ∆ Sides are called segments Each point is a vertex Three sides Interior angle sum is 180˚ Symbol: ∆ Sides are called segments Each point is a vertex
EQUIANGULAR All angles are 60˚
ACUTE TRIANGLE Three angles less than 90 degrees
RIGHT TRIANGLE One right angle
OBTUSE TRIANGLE One obtuse angle
EQUILATERAL TRIANGLE All sides congruent
ISOSCELES TRIANGLE At least two congruent sides
SCALENE TRIANGLE No congruent sides
EXTERIOR 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 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 STANDARD: classify polygons find measures of interior and exterior angles of polygons STANDARD: classify polygons find measures of interior and exterior angles of polygons
VOCABULARY 1. Polygon 2. Concave Polygon 3. Convex Polygon 4. Diagonal 5. Polygon Angle Sum 6. Polygon Exterior Angle Sum 7. Equilateral Polygon 8. Equiangular Polygon 9. Regular Polygon 1. Polygon 2. Concave Polygon 3. Convex Polygon 4. Diagonal 5. Polygon Angle Sum 6. Polygon Exterior Angle Sum 7. Equilateral Polygon 8. Equiangular Polygon 9. Regular Polygon
POLYGON Closed plane figure At least 3 sides and angles Classified by the number of sides Closed plane figure At least 3 sides and angles Classified by the number of sides
CONVEX POLYGON Doesn’t cave in
CONCAVE POLYGON caves in
Diagonal Connects vertices
POLYGON ANGLE SUM (n-2)180
POLYGON EXTERIOR ANGLE SUM The exterior angles of a polygon = 360
EQUILATERAL POLYGON All sides are congruent
EQUIANGULAR POLYGON* All angles are congruent
REGULAR POLYGON Equiangular Equilateral Equiangular Equilateral
3 - 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
VOCABULARY 1. Slope 2. y-intercept 3. x-intercept 4. Graphing Using Intercepts 5. Standard Form 6. Slope Intercept Form 7. Point Slope Form 1. Slope 2. y-intercept 3. x-intercept 4. Graphing Using Intercepts 5. Standard Form 6. Slope Intercept Form 7. Point Slope Form
SLOPE
y-intercept Where the graph intersects the y-axis
x-intercept Where the graph intersects the x-axis
Graphing Using intercepts Substitute “0” for x and y to find the intercepts
STANDARD FORM Ax + By = C
SLOPE INTERCEPT FORM y = mx + b b = y-intercept m = slope y = mx + b b = y-intercept m = slope
POINT SLOPE FORM y - y 1 = m(x - x 1 )
3 - 7 SLOPES OF PARALLEL AND PERPENDICULAR LINES STANDARD: relate slope and parallel lines relate slope and perpendicular lines STANDARD: relate slope and parallel lines relate slope and perpendicular lines
PARALLEL LINES Have equal slopes Two lines that do not intersect Go in same direction Have equal slopes Two lines that do not intersect Go in same direction
PERPENDICULAR 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 FORM y = mx + b b = y-intercept m = slope y = mx + b b = y-intercept m = slope
INTERSECT To cut Divide by passing through To cut Divide by passing through
CONGRUENT equal The same equal The same