Chapter 3 Danny Ramsey, Ashley Krieg, Kyle Jacobs, and Chris Runion

What is Ch. 3 About? It is about lines and angles. It is about lines and angles. We learned : We learned : the properties of parallel and perpendicular lines. the properties of parallel and perpendicular lines. Six different ways to prove lines are parallel. Six different ways to prove lines are parallel. How to write an equation of a line with the given characteristics How to write an equation of a line with the given characteristics

3.1 Lines and Angles

3.1 Vocabulary Parallel lines: two lines that are coplanar and never intersect Parallel lines: two lines that are coplanar and never intersect Skew lines: two lines that are not coplanar and never intersect Skew lines: two lines that are not coplanar and never intersect Parallel planes: two planes that never intersect Parallel planes: two planes that never intersect

Review… RU || WZ RU || WZ WZ and TY are skew lines WZ and TY are skew lines Plane RUZW and plane STYX are parallel planes Plane RUZW and plane STYX are parallel planes

Postulates Parallel Postulate: If there is a line and a point not on the line, then there is exactly one line through the point parallel to the given line Parallel Postulate: If there is a line and a point not on the line, then there is exactly one line through the point parallel to the given line P l There is exactly one line through point P parallel to l.

Postulates Continued Perpendicular postulate: If there is a line and a point not on the line, then there is exactly one line through the point perpendicular to the given line. Perpendicular postulate: If there is a line and a point not on the line, then there is exactly one line through the point perpendicular to the given line. P l There is exactly one line through P perpendicular to l.

Construction Activity Perpendicular lines Draw a line ( l ) and a Point (P) off of the line. Put point of compass at P and open wide enough to intersect l twice. Label those intersections A and B. Using the same radius, draw an arc from A and B. Label the intersection Q. Use a straightedge to draw PQ. PQ l

More Vocabulary! Transversal: the line that intersects two or more coplanar lines at different points Transversal: the line that intersects two or more coplanar lines at different points Corresponding Angles: angles that occupy corresponding positions Corresponding Angles: angles that occupy corresponding positions Alternate Exterior Angles: two angles that are outside the two lines on opposite sides of the transversal Alternate Exterior Angles: two angles that are outside the two lines on opposite sides of the transversal Alternate Interior Angles: two angles between the two lines on opposite sides of the transversal Alternate Interior Angles: two angles between the two lines on opposite sides of the transversal Consecutive Interior Angles: two angles that lie between the two lines on the same side of the transversal Consecutive Interior Angles: two angles that lie between the two lines on the same side of the transversal

VOCAB PICTURES Transversal is RED Corresponding Angles: 1 & 5 Alternate Exterior Angles: 1 & 8 Alternate Interior Angles: 3 & 6 Consecutive Interior Angles: 3 & 5

3.2 Proof and Perpendicular Lines

3.2 Vocabulary Flow Proof: uses arrows and boxes to show the logical flow Flow Proof: uses arrows and boxes to show the logical flow Example Example

3.2 Theorems If two lines intersect to form a linear pair of congruent angles, then the lines are perpendicular. If two lines intersect to form a linear pair of congruent angles, then the lines are perpendicular. If two sides of two adjacent acute angels are perpendicular, then the angles are complementary. If two sides of two adjacent acute angels are perpendicular, then the angles are complementary. If two lines are perpendicular, then they intersect to form four right angles. If two lines are perpendicular, then they intersect to form four right angles.

3.3 Parallel lines and Transversals

Corresponding Angles Postulate If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent 1 2 <1 = <2

3.3 Theorems Alternate Interior Angles: if two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent Alternate Interior Angles: if two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent 3 4 <3 = <4

3.3 Theorems Consecutive Interior Angles: if two parallel lines are cut by a transversal, then the pairs of consecutive interior angels are supplementary. Consecutive Interior Angles: if two parallel lines are cut by a transversal, then the pairs of consecutive interior angels are supplementary. 5 6 M<5 + M<6 = 180˚

3.3 Theorems Alternate Exterior Angles: if two parallel lines are cut by a transversal, then the pairs of alternate exterior angles are congruent Alternate Exterior Angles: if two parallel lines are cut by a transversal, then the pairs of alternate exterior angles are congruent 7 8 <7 = <8

3.3 Theorems Perpendicular Transversal: if a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other Perpendicular Transversal: if a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other j h k J K

3.4 Proving Parallel Lines

POSTULATE Corresponding angles converse: if two lines are cut by a transversal so that corresponding angles are congruent, then the pairs of alternate interior angles are congruent Corresponding angles converse: if two lines are cut by a transversal so that corresponding angles are congruent, then the pairs of alternate interior angles are congruent j k J || k

Theorems about Transversals Alternate Interior Angles Converse: if two lines are cut by a transversal so that alternate interior angles are congruent, then the lines are parallel Alternate Interior Angles Converse: if two lines are cut by a transversal so that alternate interior angles are congruent, then the lines are parallel 3 1 j k If <1 = <3, then j || k

Theorems about Transversals Consecutive Interior Angles Converse: if two lines are cut by a transversal so that consecutive interior angles are supplementary, then the lines are parallel Consecutive Interior Angles Converse: if two lines are cut by a transversal so that consecutive interior angles are supplementary, then the lines are parallel 2 1 If m<1 + m<2 = 180°, j || k j k

Theorems about Transversals Alternate Exterior Angles Converse: if two lines are cut by a transversal sot that alternate exterior angles are congruent, then the lines are parallel. Alternate Exterior Angles Converse: if two lines are cut by a transversal sot that alternate exterior angles are congruent, then the lines are parallel. 4 5 If <4 = <5, then j || k j k

3.5 Using Properties of Parallel Lines

Theorems If two lines are parallel to the same line, then they are parallel to each other. If two lines are parallel to the same line, then they are parallel to each other. In a plane, if two lines are perpendicular to the same line, then they are parallel to each other In a plane, if two lines are perpendicular to the same line, then they are parallel to each other p q r If p || q and q || r, p || r

Construction Activity Copying an Angle Draw an acute angle with the vertex A Draw an acute angle with the vertex A Below the angle, draw a line using a straight edge put a point on the line and label it D Below the angle, draw a line using a straight edge put a point on the line and label it D Using a compass, put the point on A and open wide enough to intersect both rays. Label the intersections B and C Using a compass, put the point on A and open wide enough to intersect both rays. Label the intersections B and C Using the same radius on the compass, draw an arc with the center D, label the intersection E Using the same radius on the compass, draw an arc with the center D, label the intersection E Draw an arc with the radius BC and center E, label the intersection F Draw an arc with the radius BC and center E, label the intersection F Draw DF. <EDF = <BAC Draw DF. <EDF = <BAC

Construction Activity Parallel Lines Draw line M, using a straight edge, and point P off of the line. Draw line M, using a straight edge, and point P off of the line. Draw points Q and R on line M. Draw PQ Draw points Q and R on line M. Draw PQ Draw an arc with the center at Q so it crosses QP and QR Draw an arc with the center at Q so it crosses QP and QR Now copy <PQR, as shown in the previous construction activity, on QP. Be sure the angles are corresponding, Now copy <PQR, as shown in the previous construction activity, on QP. Be sure the angles are corresponding, Label the new angle <TPS Label the new angle <TPS Draw PS. Since <TPS and <PQR are congruent corresponding angles, PS || QR Draw PS. Since <TPS and <PQR are congruent corresponding angles, PS || QR

3.6 Parallel Lines in the Coordinate Plane

YOU MUST KNOW THIS!!! RISE RUN SLOPE = Y 2 – Y 1 X 2 – X 1 = M

Now, the picture y x (X 1, Y 1 ) (X 2, Y 1 ) Y 2 – Y 1 RISE X 2 – X 1 RUN

Slope of Parallel Lines Postulate In a coordinate plane, two nonvertical lines are parallel is and only if they have the same slope. Any two vertical lines are parallel. In a coordinate plane, two nonvertical lines are parallel is and only if they have the same slope. Any two vertical lines are parallel. Slope = -1

SLOPES Lines that have the same slope are parallel. Y = 2x + 3 ; Y = 2x – 6 Lines that have the same slope are parallel. Y = 2x + 3 ; Y = 2x – 6 Lines that are perpendicular have opposite reciprocal slopes. Y = -2x + 3 ; Y = 1/2x -9 Lines that are perpendicular have opposite reciprocal slopes. Y = -2x + 3 ; Y = 1/2x -9

3.7 Perpendicular lines in the coordinate plane

Slopes of Perpendicular Lines In a coordinate plane, two nonvertical lines are perpendicular if and only if the product of their slopes is -1. Vertical and horizontal lines are perpendicular. In a coordinate plane, two nonvertical lines are perpendicular if and only if the product of their slopes is -1. Vertical and horizontal lines are perpendicular. Product of Slopes: 2 ( - ½ ) = -1

But Wait!!! When Will I Ever Use This???

Sailing There are three basic sailing maneuvers - sailing into the wind, sailing across the wind, and sailing with the wind. These three maneuvers allow a sailboat to travel in almost any direction. A boat that is sailing into (or against) the wind is actually sailing at an angle of about 45° to the direction of the wind. A sailboat that is sailing into the wind must follow a zigzag course called tacking in order to avoid sailing directly into the wind. When a boat is pointed directly into the wind, the sails are rendered useless and the boat loses its ability to move. A boat can reach maximum speed by sailing across the wind or reaching. In this situation, the wind direction is perpendicular to the side of the boat. The third sailing technique is called sailing with the wind or running. Here, the sail is almost at right angles with the boat and the wind literally pushes the boat from the stern There are three basic sailing maneuvers - sailing into the wind, sailing across the wind, and sailing with the wind. These three maneuvers allow a sailboat to travel in almost any direction. A boat that is sailing into (or against) the wind is actually sailing at an angle of about 45° to the direction of the wind. A sailboat that is sailing into the wind must follow a zigzag course called tacking in order to avoid sailing directly into the wind. When a boat is pointed directly into the wind, the sails are rendered useless and the boat loses its ability to move. A boat can reach maximum speed by sailing across the wind or reaching. In this situation, the wind direction is perpendicular to the side of the boat. The third sailing technique is called sailing with the wind or running. Here, the sail is almost at right angles with the boat and the wind literally pushes the boat from the stern

Graphic Artists Graphic artists are creative, analytical, and detail-oriented. It is important to be able to create a visual image of an idea. This talent requires strong spatial reasoning skills. The use of various types of graphic design software involves an understanding of geometric ideas such as scaling and transformations, and an understanding of the use of percents in mixing colors. Graphic artists are creative, analytical, and detail-oriented. It is important to be able to create a visual image of an idea. This talent requires strong spatial reasoning skills. The use of various types of graphic design software involves an understanding of geometric ideas such as scaling and transformations, and an understanding of the use of percents in mixing colors.

REAL WORLD PROBLEM 90 degrees 85 degrees Apple St. Orange St. Watermelon Ave. Are Apple and Orange Streets Parallel? Are there any Perpendicular Intersections?