Geometry CP Constructions Part I Page 1 of 4. Steps for copying a segment (TB 16): Copying a segment consists of making segments.

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1 Geometry CP Constructions Part I Page 1 of 4 Steps for copying a segment (TB 16): Copying a segment consists of making segments.

2 Geometry CP Constructions Part I Page 2 of 4 Steps for bisecting a segment (TB 25): Bisecting an angle consists of an angle in half. Both halves are.

3 Geometry CP Constructions Part I Page 3 of 4 Steps for copying an angle (TB 33) Copying an angle consists of making two angles.

4 Geometry CP Constructions Part I Page 4 of 4 Steps for bisecting an angle (TB 35): Bisecting an angle consists of an angle in half. Both halves are.

5 Geometry CP Constructions Part II Page 1 of 3 Steps for constructing a perpendicular bisector A perpendicular bisector a segment in, and forms angle with the segment at the point of intersection. 1. Place the compass point on one end of the line 2. Adjust the compass to just over half the line length 3. Without adjusting the compass width, draw an arc on each side of the line 4. Without changing the compass width, repeat for the other end of the line 5. Draw a straight line between the two arc intersections. Done. The line is the perpendicular bisector of PQ. Steps on constructing a perpendicular through a point not on the line This is not necessarily a bisector; the point could be anywhere in the exterior of the line. 1. Place the compass point on R 2. Adjust compass width to beyond the line 3. Draw two arcs across the line, creating points P and Q 4. From each point P,Q, draw an arc below the line so they cross. 5. Draw a line from R to where the arcs intersect. Done. The new line is perpendicular to PQ and passes through R

6 Geometry CP Constructions Part II Page 2 of 3 Steps to constructing a perpendicular through the given point to the given line The point could be anywhere on the line. It does not have to be the midpoint. Step 1. With the compass on K, set it to a medium width 2. Draw an arc on each side of K using that compass width, creating points P and Q. 3. With the compass on P, set its width to about half way between K and Q 4. Draw an arc on one side of the line. 5. Without changing the width, repeat from point Q, creating point R 6. Draw a line from K to R. Done. The line KR is perpendicular to PQ at K Steps to constructing a 45 angle (hint: construct a perpendicular bisector, then bisect) 1. Draw the line PQ, to be one leg of the angle 2. Set the compass on P and the width to just over half PQ 3. From P and Q, draw intersecting arcs above and below PQ 4. Draw a straight line between the two arc intersections. 5. From the midpoint of PQ, set the compass width to P. 6. Draw an arc across the perpendicular creating point C 7.Draw the line PR through the point C Done. The angle RPQ has a measure of 45 degrees

7 Geometry CP Constructions Part II Page 3 of 3 Construct a 135 angle below without using a protractor. (Hint: copy a 90 angle and a 45 angle)

8 Geometry CP Lesson 3-4: Equations of Lines Page 1 of 1 Main idea: Write an equation of a line given information about its graph. Equations of Vertical and Horizontal Lines 1. Vertical lines all have an equation of the form: a. The number is the x coordinate of ALL points on the line. 2. Horizontal lines all have an equation of the form: a. The number is the y coordinate of ALL points on the line. Example 1: Write the equation of the horizontal line that goes through (2, 8)? Example 2: Write the equation of the vertical line that goes through (-11, 5)? Example 3: Write the equation of the line perpendicular to x = 5 that goes through (3, 7)? Using slope-intercept form: y = mx + b 1. Given a slope and y-intercept: Plug em in! a. Example 4: Write the equation of the line with slope of -8 and y-intercept of Given a point and y-intercept: Plug in information, solve for m, then plug in m and b. a. Example 5: (2, 3) and y-int = 2 3. Given a point and the slope: Plug in information, solve for b, then plug in m and b. a. Example 6: (2, -5) and m = 3/2 4. Given 2 points: Find m, then continue as in Ex. 3. a. Example 7: (2, 2) and (5, -13) Given point Using Point-slope Form: y y1 = m(x x1) slope Example 8: Write the equation in point-slope form: (-10, 8) and m = -3/5 Use your answer in Ex. 8 to write the equation in slope-intercept form.

Geometry CP Lesson 3-1 and 3-2: Parallel Lines, Transversals, and Angles Page 1 of 2 Main ideas: Name angles formed by two lines cut by a transversal, use the properties of parallel lines to

9 Geometry CP Lesson 3-1 and 3-2: Parallel Lines, Transversals, and Angles Page 1 of 2 Main ideas: Name angles formed by two lines cut by a transversal, use the properties of parallel lines to determine angle relationships Definitions: Parallel lines are two lines that do not intersect and are coplanar. Skew lines are 2 lines that do not intersect and are not coplanar j To show lines are parallel, use small triangles. k o Notation: j k o Name 3 segments parallel to WZ o Name 2 segments skew to WZ : o What plane is parallel to plane DZY? Example: o Which line is to line TW and goes through V? P o Which line is to line TW and goes through V? o Which lines are skew to line TW and goes through V? o Which plane is parallel to plane TPQ? T Z Y W X A D B C Q U R S V W Definitions: Def. of a Transversal Def. of Exterior angles Def. of Interior angles Def. of corresponding angles (pair) Def. of alternate interior angles (pair) Def. of alternate exterior angles (pair) Def. of consecutive interior angles (pair) A line that intersects 2 or more coplanar lines at different points Angles that lie outside of the two lines Angles that lie inside of the two lines A pair of angles that are in corresponding positions A pair of angles that are inside the 2 lines and on opposite sides of the transversal A pair of angles that are outside the 2 lines and on opposite sides of the transversal A pair of angles that lie inside the 2 lines and on the same side of the transversal

10 Geometry CP Lesson 3-1 and 3-2: Parallel Lines, Transversals, and Angles Page 2 of 2 When the two lines are parallel, you get some special angle relationships: Corresponding Angles Postulate Alternate Interior Angles Theorem Alternate Exterior Angles Theorem Consecutive Interior Angles Theorem Perpendicular Transversal Theorem If 2 parallel lines are cut by a transversal, then the pairs of corresponding angles are If 2 parallel lines are cut by a transversal, then the pairs of alternate interior angles are If 2 parallel lines are cut by a transversal, then the pairs of alternate exterior angles are If 2 parallel lines are cut by a transversal, then the pairs of same side interior angles are If a line is to one of the 2 parallel lines, then it is also to the other line. Examples: Identify the postulate or theorem that makes each statement true & 6 are supplementary & 8 are supplementary 7. If line p and m are and k m, then k p Examples: Find the values of x and y k p m 70 x y x y 85 x x x

11 Geometry CP Lesson 3-3: Slopes of Lines Page 1 of 2 Main ideas: Find slopes of lines and use them to identify parallel and perpendicular lines. The slope of a line is the ratio. The slope determines the steepness of the line. Finding Slopes From Two Points Formula: b. Example 1: (-5, 2) and (0, -3) Answer: m = c. Example 2: (-3, 2) and (9, -7) Answer: m = d. Example 3: (4, 6) and (4, -2) Answer: m = e. Example 4: (-1, 5) and (6, 5) Answer: m = The slope of ALL vertical lines is The slope of ALL horizontal lines is Using Slopes to Determine Parallel and Perpendicular Lines Postulate 3.2 Two nonvertical lines are parallel if and only if they have the same slope Postulate Two nonvertical lines are perpendicular if and only if their slopes are negative reciprocals (i.e. the product of their slopes is -1). Determine if the lines are parallel, perpendicular or neither. Example 5: y = ¼x + 5 and 3y = -12x + 9 Example 6: y = ½x 7 and 4y = 2x 20 AB CD Example 7: and where A(-1, 4), B(-4, -3), C(3, 1) and D(2, -1) EF GH Example 8: and where E(0, 3), F(3, 1), G(0, 3) and H(-4, -3) Example 9: x = 3 and y = -20

12 Geometry CP Lesson 3-3: Slopes of Lines Page 2 of 2 Using Slopes to Graph Lines Example 10: Graph the line that contains K(5, 1) and is parallel to Example 11: Graph the line that contains P(-2, 1) and is perpendicular to D(0, -2) y y y x CD with C(-5, -4) and x x Using Slopes as Rates of Change o The slope of a line can be used to identify the coordinates of any point on a line. It can also be used to represent a rate of change which describes how a quantity is changing over time. o Example 12: A small college had 2546 students in 2006 and 2702 students in What was the average rate of increase in students? If the number of students increases at the same rate over time, how many students will the college have in 2012? o Example 13: Between 1995 and 2005, a manufacturer of camping equipment had an increase in annual sales by $7.4 million per year. In 2005, the total sales were $85.9 million. If sales increase at the same rate, predict what the total sales will be in 2015.

13 Geometry CP Lesson 3-5: Proving Lines are Parallel Page 1 of 2 Main ideas: Prove that lines are parallel In the figure below (left), can we assume that j k? Why? In the figure below (right), if 1 2, then is a b? Why? j a 1 k b 2 Corresponding Angles Converse Postulate Alternate Interior Angles Converse Theorem Alternate Exterior Angles Converse Theorem Consecutive Interior Angles Converse Theorem Theorem 3.8 If corresponding angles are, then the lines are parallel. If alternate interior angles are, then the lines are parallel. If alternate exterior angles are, then the lines are parallel. If consecutive interior angles are supplementary, then the lines are parallel. If two lines are perpendicular to the same line, then they are parallel. IMPORTANT: Use these when you are trying to prove that lines are parallel! Examples: Which lines are parallel? State the reasons. t h a m i t t t o k j k Example: Find the value of x that will make m n. (2x+1) m (3x 5) n n 78 2x 8 m b c 5 4 6

14 Geometry CP Lesson 3-5: Proving Lines are Parallel Page 2 of 2 Given: 5 6 and 6 4 Prove: b c Statement Reason Given: 1 2, 1 3 Prove: o k o k Statement Reason m Given: m p and m q Prove: p q p q 1 2 Statement Reason Given: w z, m 1 = (8y + 2), m 2 = (25y 20) Prove: y = 6 Statement w z 1 2 Reason

15 Geometry CP Lesson 3-6: Perpendiculars and Distance Page 1 of 2 Main ideas: Find the distance between a point and a line Find the distance between parallel lines How do you find the distance between a point and a line? Ex. 1: Sketch the segment that would be used to Ex. 2: Sketch the segment that would represent find the distance between P and AB. the distance between D and UH. P D A B U H Ex. 3: Construct a line perpendicular to m through P. Ex. 4: Construct a line perpendicular to k through P. Then find the distance between k and P using the Distance Formula. P k P m Step 1: Step 2: Step 3: Step 4:

16 Geometry CP Lesson 3-6: Perpendiculars and Distance Page 2 of 2 How would you find the distance between two parallel lines? Ex. 5: Sketch y = 2 and y = -1. Ex. 6: Sketch x = -3 and x = 1 Then find the distance between them. Then find the distance between them. Ex. 7: Graph (line j) y = 2x + 2 and (line k) y = 2x 3. Then find the distance between them. Step 1: Graph lines j and k. Step 2: Find the slope of a line perpendicular to lines j and k Step 3: Write the equation of the line perpendicular to line j, having the same y-intercept as line j. Let this be line p. Graph it. Step 4: Label segment AB which represents the distance between j and k. Option 4a: If you can tell precisely what the x and y coordinates are of A and B, then use the distance formula to find AB. Option 4b: If you can t, then you ll have to use algebra to find the intersection point of lines p and k. Do this by setting their equations equal to each other and solving for x and y. Then use the Distance formula to find AB. y Ex 8: Find the distance between (line j) y = ½x + 3 y and (line k) y = ½x 2 x x

2 and 6 4 and 8 1 and 5 3 and 7

2 and 6 4 and 8 1 and 5 3 and 7 Geo Ch 3 Angles formed by Lines Parallel lines are two coplanar lines that do not intersect. Skew lines are that are not coplanar and do not intersect. Transversal is a line that two or more lines at different