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A parallelepiped is a prism whose bases are parallelograms. In this case, all edges will be parallelograms.
Each parallelepiped can be considered as a prism with three different ways, since every two opposite faces can be taken as bases (in Figure 5, faces ABCD and A"B"C"D", or ABA"B" and CDC"D", or VSV"C" and ADA"D") .
The body in question has twelve edges, four equal and parallel to each other.
Theorem 3 . The diagonals of a parallelepiped intersect at one point, coinciding with the middle of each of them.
The parallelepiped ABCDA"B"C"D" (Fig. 5) has four diagonals AC", BD", CA", DB". We must prove that the midpoints of any two of them, for example AC and BD", coincide. This follows from the fact that the figure ABC"D", having equal and parallel sides AB and C"D", is a parallelogram.
Definition 7 . A right parallelepiped is a parallelepiped that is also a straight prism, that is, a parallelepiped whose side edges are perpendicular to the plane of the base.
Definition 8 . A rectangular parallelepiped is a right parallelepiped whose base is a rectangle. In this case, all its faces will be rectangles.
A rectangular parallelepiped is a right prism, no matter which of its faces we take as the base, since each of its edges is perpendicular to the edges emerging from the same vertex, and will, therefore, be perpendicular to the planes of the faces defined by these edges. In contrast, a straight, but not rectangular, parallelepiped can be viewed as a straight prism in only one way.
Definition 9 . The lengths of three edges of a rectangular parallelepiped, of which no two are parallel to each other (for example, three edges emerging from the same vertex), are called its dimensions. Two rectangular parallelepipeds having correspondingly equal dimensions are obviously equal to each other.
Definition 10 .A cube is a rectangular parallelepiped, all three dimensions of which are equal to each other, so that all its faces are squares. Two cubes whose edges are equal are equal.
Definition 11 . An inclined parallelepiped in which all edges are equal to each other and the angles of all faces are equal or complementary is called a rhombohedron.
All faces of a rhombohedron - equal rhombuses. (Some crystals have a rhombohedron shape, having great importance, for example, Iceland spar crystals.) In a rhombohedron you can find a vertex (and even two opposite vertices) such that all angles adjacent to it are equal to each other.
Theorem 4 . The diagonals of a rectangular parallelepiped are equal to each other. Diagonal square equal to the sum squares of three dimensions.
IN rectangular parallelepiped ABCDA"B"C"D" (Fig. 6) diagonals AC" and BD" are equal, since the quadrilateral ABC"D" is a rectangle (the straight line AB is perpendicular to the plane ECB"C", in which BC lies").
In addition, AC" 2 =BD" 2 = AB2+AD" 2 based on the theorem about the square of the hypotenuse. But based on the same theorem AD" 2 = AA" 2 + +A"D" 2; hence we have:
AC" 2 = AB 2 + AA" 2 + A" D" 2 = AB 2 + AA" 2 + AD 2.

In this lesson, everyone will be able to study the topic “Rectangular parallelepiped”. At the beginning of the lesson, we will repeat what arbitrary and straight parallelepipeds are, remember the properties of their opposite faces and diagonals of the parallelepiped. Then we'll look at what a cuboid is and discuss its basic properties.

Topic: Perpendicularity of lines and planes

Lesson: Cuboid

A surface composed of two equal parallelograms ABCD and A 1 B 1 C 1 D 1 and four parallelograms ABV 1 A 1, BCC 1 B 1, CDD 1 C 1, DAA 1 D 1 is called parallelepiped(Fig. 1).

Rice. 1 Parallelepiped

That is: we have two equal parallelograms ABCD and A 1 B 1 C 1 D 1 (bases), they lie in parallel planes so that the side edges AA 1, BB 1, DD 1, CC 1 are parallel. Thus, a surface composed of parallelograms is called parallelepiped.

Thus, the surface of a parallelepiped is the sum of all the parallelograms that make up the parallelepiped.

1. The opposite faces of a parallelepiped are parallel and equal.

(the shapes are equal, that is, they can be combined by overlapping)

For example:

ABCD = A 1 B 1 C 1 D 1 (equal parallelograms by definition),

AA 1 B 1 B = DD 1 C 1 C (since AA 1 B 1 B and DD 1 C 1 C are opposite faces of the parallelepiped),

AA 1 D 1 D = BB 1 C 1 C (since AA 1 D 1 D and BB 1 C 1 C are opposite faces of the parallelepiped).

2. The diagonals of a parallelepiped intersect at one point and are bisected by this point.

The diagonals of the parallelepiped AC 1, B 1 D, A 1 C, D 1 B intersect at one point O, and each diagonal is divided in half by this point (Fig. 2).

Rice. 2 The diagonals of a parallelepiped intersect and are divided in half by the intersection point.

3. There are three quadruples of equal and parallel edges of a parallelepiped: 1 - AB, A 1 B 1, D 1 C 1, DC, 2 - AD, A 1 D 1, B 1 C 1, BC, 3 - AA 1, BB 1, CC 1, DD 1.

Definition. A parallelepiped is called straight if its lateral edges are perpendicular to the bases.

Let the side edge AA 1 be perpendicular to the base (Fig. 3). This means that straight line AA 1 is perpendicular to straight lines AD and AB, which lie in the plane of the base. This means that the side faces contain rectangles. And the bases contain arbitrary parallelograms. Let us denote ∠BAD = φ, the angle φ can be any.

Rice. 3 Right parallelepiped

So, a right parallelepiped is a parallelepiped in which the side edges are perpendicular to the bases of the parallelepiped.

Definition. The parallelepiped is called rectangular, if its lateral edges are perpendicular to the base. The bases are rectangles.

The parallelepiped ABCDA 1 B 1 C 1 D 1 is rectangular (Fig. 4), if:

1. AA 1 ⊥ ABCD (lateral edge perpendicular to the plane of the base, that is, a straight parallelepiped).

2. ∠BAD = 90°, i.e. the base is a rectangle.

Rice. 4 Rectangular parallelepiped

A rectangular parallelepiped has all the properties of an arbitrary parallelepiped. But there are additional properties that are derived from the definition of a cuboid.

So, cuboid is a parallelepiped whose side edges are perpendicular to the base. The base of a rectangular parallelepiped is a rectangle.

1. In a rectangular parallelepiped, all six faces are rectangles.

ABCD and A 1 B 1 C 1 D 1 are rectangles by definition.

2. Lateral ribs are perpendicular to the base. This means that all the lateral faces of a rectangular parallelepiped are rectangles.

3. All dihedral angles of a rectangular parallelepiped are right.

Let us consider, for example, the dihedral angle of a rectangular parallelepiped with edge AB, i.e., the dihedral angle between planes ABC 1 and ABC.

AB is an edge, point A 1 lies in one plane - in the plane ABB 1, and point D in the other - in the plane A 1 B 1 C 1 D 1. Then the dihedral angle under consideration can also be denoted as follows: ∠A 1 ABD.

Let's take point A on edge AB. AA 1 is perpendicular to edge AB in the plane АВВ-1, AD is perpendicular to edge AB in the plane ABC. So, ∠A 1 AD - linear angle given dihedral angle. ∠A 1 AD = 90°, which means that the dihedral angle at edge AB is 90°.

∠(ABB 1, ABC) = ∠(AB) = ∠A 1 ABD= ∠A 1 AD = 90°.

Similarly, it is proved that any dihedral angles of a rectangular parallelepiped are right.

The square of the diagonal of a rectangular parallelepiped is equal to the sum of the squares of its three dimensions.

Note. The lengths of the three edges emanating from one vertex of a cuboid are the measurements of the cuboid. They are sometimes called length, width, height.

Given: ABCDA 1 B 1 C 1 D 1 - rectangular parallelepiped (Fig. 5).

Prove: .

Rice. 5 Rectangular parallelepiped

Proof:

Straight line CC 1 is perpendicular to plane ABC, and therefore to straight line AC. This means that the triangle CC 1 A is right-angled. According to the Pythagorean theorem:

Let's consider right triangle ABC. According to the Pythagorean theorem:

But BC and AD are opposite sides of the rectangle. So BC = AD. Then:

Because , A , That. Since CC 1 = AA 1, this is what needed to be proven.

The diagonals of a rectangular parallelepiped are equal.

Let us denote the dimensions of the parallelepiped ABC as a, b, c (see Fig. 6), then AC 1 = CA 1 = B 1 D = DB 1 =

Theorem. In any parallelepiped, opposite faces are equal and parallel.

Thus, the faces (Fig.) BB 1 C 1 C and AA 1 D 1 D are parallel, because two intersecting lines BB 1 and B 1 C 1 of one face are parallel to two intersecting lines AA 1 and A 1 D 1 of the other. These faces are equal, since B 1 C 1 =A 1 D 1, B 1 B=A 1 A (as opposite sides of parallelograms) and ∠BB 1 C 1 = ∠AA 1 D 1.

Theorem. In any parallelepiped, all four diagonals intersect at one point and are bisected at it.

Let's take (Fig.) some two diagonals in the parallelepiped, for example, AC 1 and DB 1, and draw straight lines AB 1 and DC 1.

Since the edges AD and B 1 C 1 are respectively equal and parallel to the edge BC, then they are equal and parallel to each other.

As a result, the figure ADC 1 B 1 is a parallelogram in which C 1 A and DB 1 are diagonals, and in a parallelogram the diagonals intersect in half.

This proof can be repeated for every two diagonals.

Therefore, diagonal AC 1 intersects BD 1 in half, diagonal BD 1 intersects A 1 C in half.

Thus, all diagonals intersect in half and, therefore, at one point.

Theorem. In a rectangular parallelepiped, the square of any diagonal is equal to the sum of the squares of its three dimensions.

Let (Fig.) AC 1 be some diagonal of a rectangular parallelepiped.

Drawing AC, we get two triangles: AC 1 C and ACB. Both of them are rectangular:

the first because the parallelepiped is straight, and therefore edge CC 1 is perpendicular to the base,

the second because the parallelepiped is rectangular, which means that there is a rectangle at its base.

From these triangles we find:

AC 2 1 = AC 2 + CC 2 1 and AC 2 = AB 2 + BC 2

Therefore, AC 2 1 = AB 2 + BC 2 + CC 2 1 = AB 2 + AD 2 + AA 2 1

Consequence. In a rectangular parallelepiped all diagonals are equal.

In a position not before a vowel, [th] is indicated by its letter - th, and before vowels - [th] is designated by the letters E, Yo, Yu, Ya, I, which in this case denote two sounds: [th] + vowel (ya, lighthouse , will announce). Understanding the sound composition of words in which the letters E, E, Yu, I, I denote a combination of sounds [th] + vowel, develops phonemic hearing in students, is a necessary condition full differentiation in the child’s awareness of the sound and letter forms of the word. It is most difficult to hear the sequence of sounds [yi] in the position after the soft dividing sign (nightingale), since the sounds [y] and [i] are acoustically close to each other. This means that this combination should be considered last.

Methodologically, it is advisable to present material on ways to designate the sound [th] in a general way.

To do this, the teacher can, during the lesson, draw up diagrams with the students showing the dependence of the designation [th] on its position in the word. When choosing words for exercises, it is best to use those in which the letters E, E, Yu, I are in stressed syllables

, - in unstressed letters, these letters can denote a sound close to [and], for example, ad[yi]. Alphabet and sound composition of Russian. language In order to deepen the student’s understanding of the relationship between the phonemic composition of Russian. language and the alphabet, it is advisable to compare the tape of letters known to children from the time they studied literacy with a table on which the entire composition is given according to. Hanging next to the table according to sounds of a tape of letters, you can reflect with schoolchildren on the questions: why are the letters L, M, N, R, Y highlighted on the tape in?

How many voiced consonant sounds do not have voiceless pairs? (Answer: 9.) Which sound among the unpaired voiced ones does not have not only a voiceless but also a hard pair?

Why are the letters X, Ts, Ch, Shch on the tape in a separate group?

Questions that help to better understand the relationship between the alphabet and the composition of phonemes:

1. Which sounds are more common in the Russian language: voiced or voiceless? How many pairs of voicedness and deafness are there in total?

2. How many sounds are paired in softness and hardness?

3. Name voiced sounds that do not have voiceless pairs, and voiceless sounds that do not have voiced pairs. 4. Name soft sounds that do not have hard pairs, and hard sounds that do not have soft pairs. You can give your work with the table an entertaining form.

Offer to solve the word according to its characteristics: 1st sound - voiceless sound pair [b], 2nd - vowel sound [u], 3rd - voiceless sound pair [zh], 4th - voiceless sound pair [g' ], 5th – vowel sound [and], 6th – solid pair sound [n’].

Organization of phonetic-graphic analysis.

Phonetic-graphic analysis is one of the types of sound-letter analysis. Its goal is to find out the relationship between sounds and letters in a word. The task of phonetic-graphic analysis is for the student to observe the syllabic pr-p Russian on specific words. graphics without being distracted by other issues.

Experienced teachers explain the spelling rule for combinations ЖИ - ШИ in different ways. The first explanation is a grammatical tale in which the letters Zh and Sh quarreled with the letter Y, and Ch and Shch with the letters Ya and Yu. Since then, these letters have never coexisted in the syllable SG.” Second explanation: “In the word skis, the sound L is hard , so after the letter L we write the letter Y. The sound Zh is also hard, but after the letter Z you need to write the letter I: this is what people agreed among themselves. Once upon a time, the sound Zh in our language was soft, and since then the rule has remained: after the letter Zh the letter Y is not written.” Then the teacher attaches blackboard card, on which in capital letters it is written ZHI - SHI with the letter I underlined, and students begin to write down words with this letter combination, underlining the letter I. (Ramzaeva T.G. “Russian language lessons in first grade”).

In the roots of words after C, I (rather than Y) is predominantly written: circus, daffodil, compass, quote. Exceptions: gypsy, chick, tiptoe, chick, chick-chick (interjection), chicks and derivatives from them.

After C is written O: tsk, tsk, tsk. In foreign words, both o and e are written in unstressed position: Herzegovina, duke, duchess.