## Math 8 Chapter 4 Lesson 2: Rectangle (continued)

## 1. Theoretical Summary

### 1.1. Two parallel lines in space

- In space, two lines \(a\) and \(b\) are said to be parallel if they lie in the same plane and have no common point.
- With two distinct lines \(a\) and \(b\) in space they can: intersect; parallel; diagonal (not in the same plane)
- Two distinct lines parallel to a third line are parallel to each other.

### 1.2. The line is parallel to the plane. Two parallel planes

**a) A line parallel to the plane**

When the line \(a\) is not in the plane \((ABCD)\) but \(a\) is parallel to the line of this plane, we say the line \(a\) is parallel to the plane \((ABCD)\).

Symbol: \(a// mp (ABCD)\).

**b) Two parallel planes**

If the plane \((ABCD)\) contains two intersecting lines \(a\) and \(b\) that are parallel to the two lines \(a’\) and \(b’\) contained in the plane plane \((A’B’C’D’)\) then we say two planes \((ABCD)\) and \((A’B’C’D’)\) are parallel

Symbol: \(mp (ABCD)// mp (A’B’C’D’)\)

**Attention:**

- If a line is parallel to a plane, they have no point in common
- Two parallel planes have nothing in common.
- If two distinct planes have a point in common, then they share a line passing through that common point. We say these two planes intersect.

## 2. Illustrated exercise

### 2.1. Exercise 1

Observe the rectangular box in figure 75:

Name the faces of the box.

– Are \(BB’\) and \(AA’\) in the same plane?

– \(BB’\) and \( AA’\) have something in common or not?

**Solution guide**

– Faces: \(ABCD, A’B’C’D’, ABB’A’, CDD’C’,\)\(\, ADD’A’, BCC’B’\).

– \(BB’\) and \(AA’\) lie in the same plane as \(ABB’A’\).

– \(BB’\) and \(AA’\) have nothing in common.

### 2.2. Exercise 2

Find on figure 77 the lines parallel to the plane \((A’B’C’D’)\).

**Solution guide**

We have:

\(AB//A’B’\) so \(AB//mp(A’B’C’D’)\)

\(BC//B’C’\) so \(BC//mp(A’B’C’D’)\)

\(DC//D’C’\) so \(DC//mp(A’B’C’D’)\)

\(AD//A’D’\) so \(AD//mp(A’B’C’D’)\).

### 2.3. Exercise 3

A room is \(4.5m\) long, \(3.7m\) wide and \(3.0m\) high. They wanted to whitewash the ceiling and four walls. Given that the total area of the doors is \(5,8{m^2}\). Calculate the area to be whitewashed.

**Solution guide**

We have: The area to be whitewashed is equal to the area of the four walls + the ceiling area – the door area.

The area of 4 walls is the surrounding area of the box (or room).

So the area to be whitewashed = surrounding area + ceiling area – door area.

The ceiling area is:

\( 4.5 \times 3.7 = 16.65 (m^2)\)

The area around the room is:

\(2.(4,5 + 3.7).3= 49.2(m^2) \)

The area to be whitewashed is:

\(16.65 + 49.2 – 5.8 = 60.05 (m^2) \)

## 3. Practice

### 3.1. Essay exercises

**Question 1:** Find the total area of the rectangular box according to the dimensions given in the figure.

**Verse 2: **\(ABCD. {A_1}{B_1}{C_1}{D_1}\) is a cube (h.104)

a) When we connect \(A\) to \(C_1\) and \(B\) to \(D_1\), will the two lines \(AC_1\) and \(BD_1\) intersect or not?

b) Do \(AC_1\) and \(A_1C\) intersect or not?

c) Same question as b with \(BD_1\) and \(A_1A.\)

**Question 3: **Observe the cube \(ABCD. {A_1}{B_1}{C_1}{D_1}\) (h.105)

a) The line \(A_1B_1\) is parallel to which planes?

b) Is the line \(AC\) parallel to the plane \((A_1C_1B_1)\) or not?

**Question 4: **Find on the rectangular box \(ABCD. {A_1}{B_1}{C_1}{D_1}\) (h.106) a concrete example to prove the following statement is false:

Two lines lying in two parallel planes are parallel to each other.

### 3.2. Multiple choice exercises

**Question 1: **Let ABCD be a rectangular box. A’B’C’D’. How many lines are parallel to AA’.

A. 1

B. 2

C. 3

D. 4

**Verse 2: **Given cube ABCD.A’B’C’D’. How many planes are parallel to the line A’D’?

A. 1

B. 2

C. 3

D. 4

**Question 3:** Let ABCD.A’B’C’D’ rectangular box have AD = 6cm and DD’ = 8cm. Calculate BC’?

A. 10cm

B. 9cm

C. 8cm

D. 12cm

**Question 4: **Let ABCD.A’B’C’D’ rectangular box. How many lines are parallel to BC’?

A. 0

B.1

C.2

D.3

**Question 5: **Let ABCD.MNPQ cube with side length 2cm. Calculate the total area of the faces of a cube?

A. 8cm^{2}

B. 12cm^{2}

C. 20cm^{2}

D. 24cm^{2}

## 4. Conclusion

Through this lesson, you will learn some of the main topics as follows:

- Identify (through the model) the telltale signs of two parallel lines, a line parallel to a plane, and two parallel planes.
- Compare and contrast the similarities and differences in the parallel relationship between lines and faces, faces and faces.
- Recall and apply the formula for calculating the perimeter of a rectangular box.

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