3.1: Date: Geometry. Parallel Lines: are lines Symbols: Diagram: that do intersect. The symbol
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1 3.1: Date: Geometry Parallel Lines: are lines Symbols: Diagram: that do intersect. The symbol means is parallel to. Skew Lines: are ; they are not and do not intersect.. Parallel Planes are planes that do intersect. Ex 1). a) Which segments are parallel to RU? b) Which segments are skew to SW? c) Which plane is parallel to plane VWSR? d) Which segments are parallel to plane WXTS? A is a line that intersects two or more coplanar lines at distinct points.
2 Angle Pairs Formed by Transversals are nonadjacent interior angles that lie on opposite sides of the transversal. are interior angles that lie on the same side of the transversal. lie on the same side of the transversal and in corresponding positions. are nonadjacent exterior angles that lie on opposite sides of the transversal. Ex 2). Describe the relationship between the angles: 1. Ð2 and Ð7 2. Ð1 and Ð5 3. Ð4 and Ð8 4. Ð3 and Ð5 5. Ð4 and Ð5 Homework: pg. 152 #1 7, 10 25, 28 31
3 3.2: Date: Geometry Ex 1). Which angles measure 130? What do the other angles measure? Ex 2). What are the measures of 8 and 4? Explain.
4 Ex 3). Write a proof. Given: Prove: a b 1 and 3 are supplementary Ex 4). What is the value of y? Homework: pg. 161 #1 9, 11 17
5 3.3: Date: Geometry Postulate 3-2: Converse of the Corresponding Angles Postulate Theorem 3-4: Converse of the Alternate Interior Angles Theorem Theorem 3-5: Converse of the Same-Side Interior Angles Theorem Theorem 3-6: Converse of the Alternate Exterior Angles Theorem Ex 1). Which lines are parallel if 2 3? Justify your answer.
6 Ex 2). Use the diagram below. Write a two column proof. Given: 1 is supplementary to 5. Prove: a b Statements Reasons Ex 3). If m 1 = 65 and m 2 = 115, are lines a and b parallel? Explain. Ex 4). What is the value of x for which a b? Homework: pg. 169 #1, 2, 4 8, 10 20, 22, 24
7 3.4: Date: Geometry Theorem 3-7: If two lines are to the same line, then they are to each other. Theorem 3-8: In a plane, if two lines are to the same line, then they are to each other. Ex 1). What is the relationship between segments AB and CD? Explain. Theorem 3-9: Perpendicular Transversal Theorem In a plane, if a line is to one of two lines, then it is also to the other. Ex 2). In a plane, c b, b d, and d a. Explain why c d. Homework: pg. 176 #1, 2, 6-8, 10-14, 21, 22
8 3.5: Date: Geometry Postulate 3-3: Parallel Postulate Through a point on a line, there is one and only line to the given line. Theorem 3-10: Triangle Angle-Sum Theorem The of the measures of the angles of a triangle is. Ex 1). Solve for x, y, and z in the figure at the right. Theorem 3-11: Triangle Exterior Angle Theorem The measure of each exterior angle of a triangle equals the of the measures of its two angles.
9 Ex 2). What is the measure of <1? Ex 3). What is the measure of <2? Ex 4). What is the value of x? Homework: pg. 184 #1-6, 9-20, 28, 30
10 3.6: Constructing Parallel & Perpendicular Lines A. How to construct two parallel lines STEPS: 1. Use the given line, name it n. Use the given point, name it P. 2. Label a point H on line n. 3. Draw HP. 4. Use your compass to make an arc of any size (not too big) from point H. 5. Make this same size arc from point P. 6. Measure the distance between the two intersections made by the first arc. The intersection of the arc with line n and HP. 7. Keeping that measurement, place your compass on the intersection of the second arc and HP and make another arc that should intersect your second arc. 8. Make a point at this new intersection, call it X. 9. Draw a line through P and X, call it line m. 10. Line m should be parallel to line n.
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12 B. How to construct a line perpendicular to a line through a point not on the line STEPS: 1. Use the given line, name it n. Use the point P that is not on the line. 2. Place your compass on the point and draw an arc that intersects with the line in two places. 3. Measure the distance between the two intersections with your compass; from both intersections draw an arc. These arcs should intersect. 4. Draw a line through the point P and the intersection of these arcs. Name this line m. 5. Line m and line n should be perpendicular. 6. Measure the angle with your protractor to verify.
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14 C. How to construct two perpendicular lines STEPS: 1. Use the given line, name it n. 2. Plot a point on your line, name it P. 3. Place your compass on point P and draw an arc intersecting the line at two points. 4. Measure the distance between the two intersections with your compass and from both intersections draw an arc above and below the line. 5. Draw a line through the intersections of the arcs. Name this line m. 6. Line m and line n should be perpendicular. 7. Measure the angle with your protractor to verify.
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16 E. How to Construct a Special Quadrilateral STEPS: 1. Use the given line, name it n. Use the given point, name it P. 2. The given segments will be the lengths of two sides of the quadrilateral (the two parallel sides). 3. Begin by constructing a parallel line to line n through point P. (Make the line long, name it m). 4. Measure segment a as the top of your quadrilateral (on line m). Make an arc and mark the intersection as point Y. 5. Measure segment b as the bottom of your quadrilateral (on line n). Make an arc and mark the intersection as point X. 6. Connect point Y and point X. 7. Your quadrilateral is ABYX. a b
17 a b a b Homework: 3.6 Practice Worksheet #1
18 3.7 Day 1: Date: Geometry SLOPE: Formula: Diagram: The slope,, is the ratio of the change ( ) to the change ( ) between any two points. Ex 1). Find the slope between the two points. Give answers in simplest improper fraction form! a) (-2, 1) and (6, 5) b) (-3, 8) and (1, -3) c) d) (3, -5) and (3, 2)
19 Forms of Linear Equations: The of an equation of a nonvertical line is, where m is the and b is the. Symbols: The of an equation of a nonvertical line is, where m is the slope and (x1, y1) is a on the line. Ex 2). Graph each of the following: a) y = 1 x 1 b) y 2 = 2 (x + 4) 2 3 c) y + 3 = 3 (x 1) d) y = 2 e) x = 1 2 Homework: pg. 207 #1, 2, 5, 6-22(e), 25, 26 *Complete graphs on graph paper
20 3.7 Day 2: Date: Geometry Ex 1). Write the equation of the line with the given information: a) slope 2 and y-intercept 4 b) slope 1 and y-intercept -5 2 c) passes through(3, -7) and slope -1 d) passes through (1, 0) and slope 1 5 Ex 2). Write the equation of the line that passes through the given points: a) (-1, 6) and (-5, 3) b) Write the following equations in slope-intercept form: c) (1, 3) and (5, 5) d) (-6, -3) and (1, 2)
21 Ex 3). What are the equations of the horizontal and vertical lines through the following points: a) (-1, 3) b) (4, -2) c) (8, 1) d) ( 5, 3) Ex 4). Write each equation in slope-intercept form: a) y 5 = 1 (x + 4) b) 2x + y = 4 2 c) y + 3 = (x 5) d) 3x 2y = 4 Homework: pg. 211 #1, 2, 4-12, 15-18, 29-31
22 3.8 Day 1: Date: Geometry Slopes of Parallel Lines: If two nonvertical lines are parallel, then their slopes are. ex: Any two vertical lines or horizontal lines are. ex: Ex 1). Are the lines shown in the graph parallel? Explain. Ex 2). What is an equation in slope-intercept form for the line parallel to y = 4x 2 that contains (-2, -2) Ex 3). What is an equation in slope-intercept form for the line parallel to y = 1 3 x + 6 that contains (6, -3).
23 Ex 4). What is the equation in slope-intercept form for the line parallel to y = 2x + 5 that contains (-1, 3)? Ex 5). What is the equation in slope-intercept form for the line parallel to y = 2 x 2 that 5 contains (10, -1)? Ex 6). What is the equation in slope-intercept form for the line parallel to y = x that contains (-5, -2)? Homework: 3.8 Worksheet #1
24 3.8 Day 2: Date: Slopes of Perpendicular Lines: If two nonvertical lines are perpendicular, then their slopes are. ex: Any horizontal lines and vertical line are. ex: Ex 1). Are the lines shown in the graph perpendicular? Explain. Ex 2). What is an equation in slope-intercept form for the line perpendicular to y = 3x + 2 that contains (6, 2). Ex 3). What is an equation in slope-intercept form for the line perpendicular to y = 2 x 3 that 5 contains (-8, -2).
25 Ex 4). What is an equation in slope-intercept form for the line perpendicular to y = 2x 3 that contains (4, -6). Ex 5). What is an equation in slope-intercept form for the line perpendicular to y = 1 x 5 that 3 contains (-1, 2). Ex 6). What is an equation in slope-intercept form for the line perpendicular to y = 2 that contains (0, 8). Ex 7). What is an equation in slope-intercept form for the line perpendicular to x = 3 that contains (-5, 7). Homework: 3.8 Worksheet #2
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