Geometry
Unit 3
Parallel Lines
DAY / ACTIVITY/OBJECTIVE / ASSIGNMENTMonday
Day 1
3.1 / CC.9-12.G.CO.1
Identify parallel, perpendicular, and skew lines.
Identify the angles formed by two lines and a transversal. / Packet Page 1-4
Day 2
3.2 / CC.9-12.G.CO.9
Prove and use theorems about the angles formed by parallel lines and a transversal.
Constructions for Parallel and Perpendicular Lines / Packet Page 5-8
Day 3
3.3 / CC.9-12.G.CO.9
Use angles formed by a transversal to prove two lines are parallel.
/ Packet Page 9-11
Day 4
3.4 / CC.9-12.G.CO.9
Prove and apply theorems about perpendicular lines.
Quiz 3.1-3.3 / Packet Pages 12-14
Day 5
3.5 and 3.6 / CC.9-12.G.GPE.5
Fine the slope of a line.
Use slopes to identify parallel and perpendicular lines. / Packet Page 15-18
Day 6
3.1-3.6 / Review
Quiz 2 3.1-3.6
7 / Unit 3 Test
Define Transversal:
Name the obvious transversal(s):
1. 2. 3.
When 2 coplanar lines are cut by a transversal, 8 angles are formed:
INTERIOR Ð’s: EXTERIOR Ð’s:
Some of these angles have a relationship that we have previously studied.
LINEAR PAIRS: VERTICAL Ð’s:
TYPES OF ANGLES
Alternate Interior Ð’s: ______interior Ð’s on ______sides of the transversal.
Name the Alt. Int. Ð’s:
Same–Side Interior Ð’s (consecutive int. Ð’s): Two ______Ð’s on the same side of the transversal.
Name the S.S.int. Ð’s:
Corresponding Ð’s: Two angles in ______relative to the two lines.
Name the Corr. Ð’s:
Alternate Exterior Ð’s: ______exterior Ð’s on ______sides of the transversal.
Name the Alt. Ext. Ð’s:
Use the given line as a transversal:
1. Name alt. int Ð’s using line x:
2. Name s.s. int. Ð’s using line y:
3. Name corr. Ð’s using line z:
4. Name alt. ext. Ð’s using line y:
5. Name alt. int. Ð’s using line z:
6. Name s.s. int Ð’s using line z:
1
Tell whether the statement is true or false. If false, sketch a counterexample. Do not assume points are coplanar unless specified.
______1. If a line intersects one of two parallel lines, then it must intersect the other.
______2. If two lines are coplanar, then they must be parallel.
______3. Two coplanar line segments, which have no point in common, must be parallel.
______4. Two lines, which are parallel to the same line, must be parallel to each other.
______5. If a plane contains one of two parallel lines, then it must contain the other.
______6. If a line is parallel to a plane, then it is parallel to every line in the plane.
______7. If two lines are ^ to the same line, then they must be parallel to each other.
______8. If two planes are ^ to the same line, then they must be parallel to each other.
______9. If two lines are skew to a third line, then they must be skew to each other.
______10. Two planes, which are parallel to the same plane, must be parallel to each other.
Assume a ⁄⁄ b. Complete the chart.
ANGLES / TRANSVERSAL / TYPE / , SUPPL., OR NONE(relationship between angles)
1. 1 and 14
2. 2 and 15
3. 7 and 9
4. 9 and 16
5. 10 and 17
6. 16 and 14
7. 9 and 14
8. 18 and 19
9. 1 and 16
10. 3 and 8
11. 6 and 9
12. 12 and 13
13. 7 and 11
14. 6 and 8
15. 4 and 13
16. 9 and 12
Day 2 - If 2 lines are parallel and they are intersected by a transversal, then the following is true about each pair of angles:
:
Examples – Find the measures of the angles (or value of the variables(s)).
The Converse of each theorem also works:
Converse of the Alternate Interior Angles Theorem:
Converse of the Alternate Exterior Angles Theorem:
Converse of the Same-Side Interior Angles Theorem:
Converse of Corresponding Angles Postulate:
Is it possible to prove the lines are parallel or not parallel? If so, state the postulate or theorem you would use. If not, state cannot be determined.
1. 2. 3.
4. 5. 6.
7. 8. 9.
Find the value of x so that n || m. State the theorem or postulate that justifies your solution.
10. 11. 12.
x = x = x =
______
Can you prove that lines p and q are parallel? If so, state the theorem or postulate that you would use.
13. 14. 15.
______
Name the type for each pair of angles
16. Ð1 @ Ð 8
17. Ð4 @ Ð6
18. Ð10 @ Ð 7
19. mÐ3 + mÐ4 = 180
20. Ð5 @ Ð 3
21. Ð6 @ Ð7
Day 3: Parallel Proofs
t
l
m
1. Given: l // m; Ð1 @ Ð4
Prove: s // t
1. l // m ; Ð1 @ Ð4 1. ______
2. Ð3 @ Ð1 2. ______
3. Ð3 @ Ð4 3. ______
4. s // t 4. ______
2. Given: l // m ; Ð2 @ Ð5
Prove: s // t
1. l // m ; Ð2 @ Ð5 1. ______
2. Ð2 @ Ð3 2. ______
3. Ð3 @ Ð5 3. ______
4. s // t 4. ______
3. Given: l // m; s // t
Prove: Ð2 @ Ð4
1. l // m ; s // t 1. ______
2. Ð2 @ Ð 3 2. ______
Ð3 @ Ð4
3. Ð2 @ Ð4 3. ______
4. Given: l // m; s // t
Prove: Ð1 @ Ð5
1. l // m; s // t 1. ______
2. Ð1 @ Ð3 2. ______
Ð3 @ Ð5
3. Ð1 @ Ð5 3. ______
5. Given: Ð3 is supplementary to Ð5.
Prove:
1. Ð3 is supplementary to Ð5 1. ______
2. mÐ3 + mÐ5 = 180 2. ______
3. Ð3 @ Ð4 3. ______
4. mÐ3 = mÐ4 4. ______
5. mÐ4 + mÐ5 = 180 5. ______
6. Ð4 is supplementary to Ð5 6. ______
7. 7. ______
6. Given: Ð2 @ Ð5; bisects Ð CBD.
Prove:
1. Ð2 @ Ð5; bisects Ð CBD. 1.
2. Ð3 @ Ð2 2.
3. Ð3 @ Ð5 3.
4. 4.
7. Given: l // m ; s // t
Prove: Ð2 @ Ð4
8. Given: l // m ; s // t
Prove: Ð1 @ Ð5
9. Given: l // m; Ð1 @ Ð4
Prove: s // t
10. Given: l // m; Ð2 @ Ð5
Prove: s // t
11. Given: ;
Prove: ÐB @ ÐE
12. Given:
Prove: mÐACD = mÐBAC + mÐCDE
13. Given: g // h; g // j
Prove: Ð2 @ Ð3
14. Given: ;
Prove: ÐB @ ÐD
15. Given: a // c; Ð1 @ Ð2
Prove: b // c
16. Given: ÐC is a supplement of ÐD
Prove: ÐA is a supplement of ÐB
Day 4: Perpendicular Lines
Perpendicular Bisector is a line perpendicular to a segment at the segment's midpoint.
Distance from a point to a line the length of the perpendicular segment from the point to the line.
1- 6 Use the given diagram on the right, in which AM = MB.
1. Name a pair of ^ rays.
2. ____ is the ^ bisector of ____.
3. Name a linear pair of angles which are @.
4. If t in X is ^ to at M, what can you say about t and ? Why?
5. If in X is a ^ bisector of , then R is on . Why?
6. If p contains M and is ^ to the plane determined by and , then p ___ and p ___ . Why?
7. In a plane , how many lines can be ^ to a given line at a given point?
8. Would your answer be different if the words “in a plane” were omitted from the question?
Homework on Perpendicular Lines.
True or False. If false, give a counterexample.
_____1. If ^, then ÐQPR is a right angle. ______2. If ^, then ÐABC is a right angle.
_____3. If 2 lines intersect to form a right angle, ______4. There is exactly one line ^ to a given at a
then the lines are ^. given point on the line.
_____5. If 2 angles are a linear pair, then each ______6. A given segment has exactly one ^
is a right Ð. bisector.
_____7. If M is the midpoint of and if is ^ to ______8. If 2 adjacent angles are @, then each is a
plane X at M, there is exactly one line in X which right angle.
is a ^ bisector of .
In 9 – 13 refer to the diagram below and the given info. : mÐCAB = 90; ÐCDA @ ÐBDA; ; mÐECB = 90
**Mark the diagram with the given information**
9. What pairs of lines are ^?
10. ____ is a ^ bisector of ____. Why?
11. If in X is a ^ bisector of , then F is on . Why?
12. If t is a line in the plane of the diagram, and t ^ at D, how are t and related? Why?
13. If G is on and ÐEGA @ ÐCGA, how are and related? Why?
14. If l ^ m and m ^ n, is l ^ n? Explain.
15. If m ^ n, is n ^ m ? Explain.
1. The perpendicular bisector of a segment is a line ______to a segment at the segment’s ______.
2. The shortest segment from a point to a line is ______to the line.
For Exercises 3 and 4, name the shortest segment from the point to the line and write an inequality for x.
3. 4.
Fill in the blanks to complete these theorems about parallel and
perpendicular lines.
5. If two coplanar lines are perpendicular to the same line, then the two lines are ______to each other.
6. If two intersecting lines form a linear pair of ______angles,
then the lines are perpendicular.
7. In a plane, if a transversal is perpendicular to one of two parallel lines, then it is ______to the other line.
Use the drawing of a basketball goal for Exercises 8–10.
In each exercise, justify Esperanza’s conclusion with one
of the completed theorems from Exercises 5–7. Write the
number 5, 6, or 7 in each blank to tell which theorem you used.
8. Esperanza knows that the basketball pole intersects the
court to form a linear pair of angles that are congruent.
She concludes that the pole and the court are perpendicular. ______
9. Esperanza knows that the hoop and the court are both
perpendicular to the pole. She concludes that the hoop
and the court are parallel to each other. ______
10. Esperanza knows that the hoop and the court are parallel to
each other. She also knows that the hoop is perpendicular to
the pole. Esperanza concludes that the pole and the court
are perpendicular. ______
Day 5 Slopes of Lines
Graph each pair of lines. Use slopes to determine whether the lines are parallel, perpendicular, or neither.
2. and for I(1, 0), J(5, 3), K(6, -1), 3. and for P(5, 1), Q(-1, -1), R(2, 1),
and L(0, 2) ______and S(3, -2) ______
whether each pair of lines is parallel, perpendicular, or neither.
4. with slope = 3 and with slope = -1 ______
5. with and with ______
Match the letter of each example to the correct form of a line.
6. point-slope form ______
7. slope-intercept form ______
8. horizontal line ______
9. vertical line ______
Write the equation of each line in the given form.
10. the horizontal line through (3, 7) in 11. the line with slope through (1, -5) in
point-slope form point-slope form
12. the line through and (2, 14) in 13. the line with x-intercept -2 and y-intercept
slope-intercept form -1 in slope-intercept form
Write the equation of each line in the given form. Graph each line.
14. the line with slope -2 and y-intercept 1 15. the line with slope through (4, 4) in
in slope-intercept form point-slope form
16. the line through (0, 0) and (2, 2) in 17. the line through (-1, -1) and (0, 2) in point-slope form slope-intercept form
Graph each line.
18. 19.
Determine whether the lines are parallel, intersect, or coincide.
20. ______
21. ______
22. ______
Write the equation of each line in the given form.
1. the horizontal line through (3, 7) in 2. the line with slope through (1, -5) in
point-slope form point-slope form
3. the line through and (2, 14) in 4. the line with x-intercept -2 and y-intercept
slope-intercept form -1 in slope-intercept form
Graph each line.
5. 6.
Determine whether the lines are parallel, intersect, or coincide.
7. ______
8. ______
9. ______
1