Download Lesson 2.4. Parallel Lines

Lesson 2.4. Parallel Lines
Getting Ready:
1. Fold the paper once to make a line. Label the line l.
2. Draw a point on l. Label the point as Q. Fold the paper through point Q so that line l overlaps
with itself. Label the new line as m.
3. Describe the relation between lines l and m.
4. Mark a new point on l. Label it P.
5. Fold the paper through point P so that the line overlaps with itself. Label the new crease as
line n.
6. What is the relation between line m and line n?
Remember:
We all know that in a plane, two lines will either intersect or be parallel.
From that idea we can say that,
(a) in a plane if two coplanar lines do not intersect, then they are parallel; and
(b) in a plane if two coplanar lines are not parallel, then they are intersecting lines.
Something to think about…
From the previous idea, is it possible that two lines are not parallel and do not intersect?
Answer:
Yes, it is possible if the lines are noncoplanar lines. Lines of this kind are called skew lines.
Definition 2.4.1.
Skew lines are lines that are noncoplanar and do not intersect.
Definition 2.4.2.
Parallel lines are lines that are coplanar and do not intersect.
Definition 2.4.3.
Intersecting lines are lines that are coplanar and are not parallel.
Note:
The term parallel and the notion || are used for segments, rays, lines, and planes. The
symbol not || means "is not parallel to."
Let's Practice 2.4.1.
Let's Practice 2.4.2.
Bon Hansen works as a designer in Geom Company. He is assigned to design a new toy box.
The shape of the new box is to be half of a cube as shown at the right. Identify each pair of edges as
perpendicular, oblique, parallel, or skew.
Lesson 2.5. Conditions that Guarantee Parallelism
Task: To construct a line parallel to a given line through a point not on the line.
Getting Ready:
1. Use a straightedge to draw line m through P (0,0) and Q (3,1). Locate R (5,6) not on m.
2. Draw a line through R that intersects line m at Q. Mark ∠1.
3. Construct an angle congruent to ∠1 using R as vertex and segment RQ as one side. Draw a
line through R to form an angle congruent to ∠1. Name the line n, the ∠2, and the point
where n intersects the y-axis at S.
4. Identify the special angle pair formed by ∠1 and ∠2.
5. Make a conjecture about m and n. (Note: ∠1 is congruent to ∠2)
6. Find the coordinates of point S and where line n crosses the y-axis. Use points R and S to
find the slope of n.
7. Use points P and Q to find the slope of line m.
8. Do the results in nos. 6 and 7 validate your conjecture about the relationship between m
and n?
Postulate 2.5.1. The Parallel Postulate
Given a point and a line not containing it, there is exactly one line through the given point
parallel to the given line.
Postulate 2.5.2. The Perpendicular Postulate
Given a point and a line not containing it, there is exactly one line through the given point
perpendicular to the given line.
Challenge:
Point A is contained in both lines m and n. If lines m and n are parallel to line l, what can you
conclude about m and n? (NOTE: Consider Postulate 2.5.1.)
Postulate 2.5.3. CACP Postulate
Given two lines cut by a transversal, if corresponding angles are congruent, then the two
lines are parallel.
Theorem 2.5.1. AICP Theorem
Given two lines cut by a transversal, if alternate interior angles are congruent, then the lines
are parallel.
Example 2.5.1.
Theorem 2.5.2. SSIAS Theorem
Given two lines cut by a transversal, if same side interior angles are supplementary, then
the lines are parallel.
Example 2.5.2.
Theorem 2.5.4. AI-CA Theorem
Given two lines cut by a transversal, of alternate interior angles are congruent, then
corresponding angles are congruent.
Example 2.5.3.
Given: ∠𝑎 ≅ ∠𝑏
Prove: ∠𝑏 ≅ ∠𝑐
Theorem 2.5.5. The Three Parallel Lines Theorem
In a plane, if two lines are both parallel to a third line, then they are parallel.
Theorem 2.5.6.
If two coplanar lines are perpendicular to a third line, then they are parallel to each other.
Let's Practice 2.5.
Direction: Prove the following.
1-10.
11-20.
Chapter 3: Congruent Triangles
Lesson 3.1. Congruent Figures
Getting Ready:
1. Construct quadrilateral QUAD using the following set of measures for the sides and angles.
2. Compare your quadrilateral with the quadrilaterals made by your classmates. Are they
congruent? Do some of them have to be turned over before they will match?
3. If all the sides and angles of two polygons are congruent, will the polygons fit exactly?
Would they match if one or more sides and angles fail to be congruent?
4. Complete the statement:
"Two polygons are congruent if and only if..."
Definition 3.1.1 Congruent Triangles
Triangles that are of the same size and same shape are called congruent triangles.
Something to think about…
If we cut triangle ABC and place it on top of triangle PQR, will they fit exactly?
Note:
The symbol ↔ is read as "corresponds to."
Do you know?
The correspondence of the vertices can be used to name the corresponding congruent sides
and angles of the two triangles.
Challenge:
Name the corresponding congruent sides and angles for the two triangles.
Definition 3.1.1 The Congruent Triangles (CPCTC)
Two triangles are congruent if and only if their corresponding parts are congruent.
Example 3.1.
Prove the following.
Let's Practice:
Prove the following figure.