## Math 11 Chapter 2 Lesson 2: Two diagonal lines and two parallel lines

## 1. Summary of theory

### 1.1. Relative position of two lines in space

Given two lines a and b in space. Then there are cases for a and b.

**– Case 1:** There is a plane containing a and b (we say a and b are coplanar)

+ a and b have only one thing in common M. We say a and b intersect at M and denoted \(a \cap b=M\).

+ a and b have nothing in common. We say a and b are parallel and denoted a // b.

+ a matches b, symbol \(a \equiv b\).

**– Case 2:** There is no plane containing a and b.

Then we say a and b cross each other or a cross with b.

### 1.2. Nature

**– Theorem 1:** In space, through a point not on a given line, there is one and only one line parallel to the given line.

+ Comment: Two parallel lines a and b define a plane, denoted mp(a,b) or (a,b).

**– Theorem 2:** If three double planes intersect each other according to three distinct intersections, then the three intersections are either concurrent or parallel to each other.

**– Consequences:** If two distinct planes respectively contain two parallel lines, their intersection (if any) is also parallel to those two lines or coincides with one of them.

**– Theorem 3:** Two distinct lines parallel to the third line are parallel to each other.

## 2. Illustrated exercise

**Lesson 1:** Let S.ABCD pyramid whose base is a parallelogram. Let I, J be the midpoints of sides AD and BC, and G be the centroids of triangle SCD, respectively. Find the intersection of two planes (SCD) and (IJG).

**Solution guide:**

We have ABCD as a parallelogram and I, J are midpoints of AD and BC, so IJ // AB.

Yes

\(\left\{ \begin{array}{l} G \in \left( {SAB} \right) \cap \left( {IJG} \right)\\ AB \subset \left( {SAB} \right )\\ IJ \subset \left( {IJG} \right)\\ IJ//AB \end{array} \right. \Rightarrow \left( {SAB} \right) \cap \left( {IJG} \right ) = MN\) with MN passing through G and parallel to AB.

**Lesson 2:** Let S.ABCD pyramid with base ABCD a trapezoid with small base CD. Let M and N be the midpoints of SA and SB respectively.

a) Prove MN // CD.

b) Let P be the intersection of SC and (AND), I be the intersection of AN and DP. Prove SI // CD.

**Solution guide:**

a) We have: M, N are midpoints of SA and SB \(\Rightarrow \) MN is the median of triangle SAB, so MN // AB (1).

Again there is AB // CD (ABCD is a parallelogram) (2).

From (1) and (2) deduce MN // CD.

b) In (ABCD) call \(E=AD \cap BC\).

In (SCD) call \(P=SC \cap EN\).

I have: E \(\print\) AD \( \subset \) (AND).

\(\Rightarrow \)EN \( \subset \) (AND) \(\Rightarrow \)P \(\print\) (AND).

So P = SC \(\cap \) (AND).

Again, I is the intersection of AN and DP.

\( \Rightarrow \left\{ \begin{array}{l} I \in AN\\ I \in DP \end{array} \right. \Rightarrow \left\{ \begin{array}{l} I \ print \left( {SAB} \right)\\ I \in \left( {SCD} \right) \end{array} \right \Rightarrow SI = \left( {SAB} \right) \cap \left( {SCD} \right)\).

Because the \(\left\{ \begin{array}{l} \left( {SAB} \right) \cap \left( {SCD} \right) = SI\\ AB \subset \left( {SAB} \right) \\ CD \subset \left( {SCD} \right)\\ AB//CD \end{array} \right. \Rightarrow \)SI // CD.

**Lesson 3:** Let the pyramid S.ABCD whose base ABCD is a convex quadrilateral. Let M, N, E, F be the midpoints of the sides SA, SB, SC and SD respectively.

a) Prove that ME, NF, SO are concurrent (O is the intersection of AC and BD).

b) Four points M, N, E, F are coplanar.

**Solution guide: **

a) In (SAC) call I = ME \(\cap \) SO \(\Rightarrow \) I is the midpoint of SO \(\Rightarrow \) FI is the median of the triangle SOD.

Hence FI // OD.

Similarly we have NI // OB so N, I, F are collinear or I \( \subset \) NF.

So ME, NF, SO are concurrent.

b) Due to ME \(\cap \) NF = I so ME and NF define a plane. It follows that M, N, E, F are coplanar.

## 3. Practice

### 3.1. Essay exercises

**Lesson 1:** Let the pyramid S.ABCD have base ABCD as a parallelogram.

a) Find the intersection of two planes (SAD) and (SBC).

b) Let M be a point on side SC. Determine the intersection N of SD with (ABM). What is quadrilateral ABMN?

**Lesson 2:** Let the pyramid S.ABCD have base ABCD as a parallelogram. Let M, N, P, Q be the midpoints of the sides SA, SB, SC, SD, respectively.

a) Prove that MNPQ is a parallelogram.

b) Let I be a point on side BC. Determine the cross-section of the pyramid with (IMN).

**Lesson 3:** Let ABCD be a regular tetrahedron of side a. Let M and N be the midpoints of CD and AB respectively.

a) Determine the points I \(\print\) AC and J \(\print\) DN such that IJ // BM.

b) Calculate IJ according to a.

### 3.2. Multiple choice exercises

**Lesson 1:** Let S.ABCD pyramid whose base is a trapezoid with base sides AB and CD. Let M, N be the midpoints of SA and SB respectively. Which of the following assertion is true?

A. MN // CD.

B. MN, CD cross each other.

C. MN cut the CD.

D. MN \( \equiv \) CD.

**Lesson 2:** Let S.ABCD pyramid with AD not parallel to BC. Let M, N, P, Q, R, T be the midpoints of AC, BD, BC, CD, SA, SD respectively. Which pair of lines are parallel to each other?

A. MP and RT.

B. MQ and RT.

C. MN and RT.

D. PQ and RT.

**Lesson 3:** Let the pyramid S.ABCD have base ABCD as a parallelogram. Let K be the mid point SA. The cross-section of the pyramid S.ABCD cut by the plane (KBC) is:

A. Triangle KBC.

B. KBCH trapezoid (H is the midpoint of SD).

C. Trapezoid KGBC (G is the midpoint of SB).

D. Quadrilateral KBCD.

**Lesson 4:** Let ABCD, M and N be the midpoints of AB and AC, respectively. Plane \(\left( \alpha \right)\) through MN intersect the tetrahedron ABCD in the cross section of polygon T. Which of the following statements is true?

A.T is a rectangle.

B.T is a triangle.

C.T is a rhombus.

D. T is a triangle or a trapezoid or a parallelogram.

**Lesson 5:** Let ABCD tetrahedron. Points P, Q are the midpoints of AB and CD, respectively; point R lies on side BC such that BR = 2RC. Let S be the intersection of plane (PQR) and side AD. Calculate the ratio?

A. 2.

B. 1.

C. \(\frac{1}{2}\).

D. \(\frac{1}{3}\).

## 4. Conclusion

Through this lesson, you should be able to understand the following:

– The concept of two lines: coincide, parallel, intersect, cross each other in space and theorems and consequences in the lesson.

– Know how to apply theorems to solve problems.

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