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The complete solution of seventh grade mathematics in People's Education Press.
1. The vertex angles are equal.

2. Definition of isosceles angle

As shown in the figure, both of them are on the same side of the section line and on the same side as the other two straight lines. Diagonal lines with this positional relationship are called isosurface.

3. Definition of internal angle

Two straight lines AB and CD are cut into eight angles by a third straight line EF. If both angles are on the inside of two straight lines and on both sides of the third straight line, then such a diagonal is called an inscribed angle.

4. Definition of ipsilateral internal angle

The inner angle of the same side, "the same side" refers to the same side of the third straight line; "Inside" means between two straight lines cut.

Among the eight angles formed by two straight lines being cut by the third straight line, there are four pairs of congruent angles, two pairs of internal staggered angles and two pairs of internal angles on the same side.

Characteristics of parallel lines

1. Two straight lines are parallel and complementary.

2. The two straight lines are parallel and the internal dislocation angles are equal.

These two straight lines are parallel, and the included angle is equal.

Determination of parallel lines

1. The inner angles on the same side are complementary, and the two straight lines are parallel.

2. The internal dislocation angles are equal and the two straight lines are parallel.

3. The same angle is equal and two straight lines are parallel.

If two straight lines are parallel to the third straight line at the same time, then the two straight lines are parallel to each other.

Chapter VII

triangle

A closed figure consisting of three end-to-end line segments is called a triangle.

Properties of triangles

1. The sum of any two sides of the triangle must be greater than the third side, which also proves that the difference between any two sides of the triangle must be less than the third side.

2. The sum of the internal angles of the triangle is equal to 180 degrees.

3. The bisector of the vertex, the midline of the bottom and the height of the bottom of the isosceles triangle coincide, that is, the three lines are one.

The three heights of a triangle intersect at a point.

The bisectors of the three internal angles of a triangle intersect at one point.

The bisector of the inner corner of a triangle intersects the bisector of the outer corner at the other two vertices.

isosceles triangle

The nature of isosceles triangle;

(1) The two base angles are equal;

(2) The bisector of the vertex angle, the median line on the bottom edge and the height on the bottom edge coincide with each other;

(3) All angles of an equilateral triangle are equal and equal to 60.

Right triangle (RT triangle for short):

(1) The two acute angles of a right triangle are complementary;

(2) The median line on the hypotenuse of the right triangle is equal to half of the hypotenuse;

(3) In a right triangle, if there is an acute angle equal to 30, then the right side it faces is equal to half of the hypotenuse;

(4) In a right triangle, if a right-angled side is equal to half of the hypotenuse, then the acute angle of this right-angled side is equal to 30;

Typical example:

Please add "+,-,-"and brackets to the four numbers "5,5, 1" to make the calculation result 24. The formula is.

If the front view, left view and top view of a geometry are all circles, then the shape of this geometry should be. (Just write one)

Definition: a*b=ab+a+b, if 3*x=27, the value of x is _ _ _ _ _.

A beam of light shines vertically on a horizontal ground with a flat mirror on it. If this beam of light becomes horizontal light after being reflected by the plane mirror, the degree of the acute angle formed by the plane mirror and the ground is.

There are two charging methods for mobile phones in a province: PHS monthly call charge 10 yuan, plus 0.4 yuan call charge every minute; The monthly phone bill of "Shenzhouxing" is 25 yuan per month, plus the phone bill of 0.2 yuan per minute. If a mobile phone user estimates that the monthly call time is about 150 minutes, he should choose the mode.

Party A, Party B and Party C go to Teacher Li to learn the piano, with Party A going once every six days, Party B going once every eight days and Party C going once every nine days. If they meet at Teacher Li's place when they are learning the piano on July 10, then they will meet at Teacher Li's place when they are learning the piano next time.