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Geometry Chapter 3 Parallel Lines and Perpendicular Lines Pages 124 195 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 3-1 PROPERTIES OF PARALLEL LINES 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 Angles Same-Side (consecutive) interior angles PARALLEL LINES Two lines that do not intersect Go in same direction Coplanar SKEW LINES 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 ALTERNATE INTERIOR ANGLES Congruent Inside Opposites sides ALTERNATE EXTERIOR ANGLES Congruent Outside Opposites sides SAME-SIDE (consecutive) INTERIOR ANGLES 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 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 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? CORE VOCABULARY 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 ALTERNATE INTERIOR ANGLES Congruent Inside Opposites sides ALTERNATE EXTERIOR ANGLES Congruent Outside Opposites sides SAME-SIDE (consecutive) INTERIOR ANGLES 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 3 – 3 Proofs and Parallel Lines 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 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 Core Concept: Five Types of Angle Pairs 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 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 3 – 4 Proofs and Parallel 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 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 TRIANGLE 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 REMOTE INTERIOR ANGLES on the opposite side of the exterior angles equal the measure of the exterior angle 3 - 5 POLYGON ANGLESUM THEOREM 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 POLYGON 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 3 - 6 LINES IN THE COORDINATE PLANE 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 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 POINT SLOPE FORM y - y1 = m(x - x1) 3 - 7 SLOPES OF PARALLEL 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 PERPENDICULAR LINES 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 INTERSECT To cut Divide by passing through CONGRUENT equal The same