1. The corresponding edges in two congruent triangles are equal.

2. Equiangles and equilateral sides of the same triangle.

3. The bisector of the vertex or the high bisector of the bottom of the isosceles triangle.

4. The opposite sides or diagonals of a parallelogram are equal to two line segments separated by intersection points.

5. The midpoint of the hypotenuse of a right triangle is equal to the distance between the three vertices.

6. Any point on the middle vertical line of a line segment is equal to the distance between two segments of the line segment.

7. The distance from any point on the bisector of an angle to both sides of the angle is equal.

8. A straight line passing through the midpoint of one side of a triangle and parallel to the third side is equal to the line segment formed by the bisection of the second side.

9. In the same circle (or equal circle), the chords opposite to the equal arc or two chords equidistant from the center of the circle or the chords opposite to the equal central angle and circumferential angle are equal.

10. At a point outside the circle, the tangent lengths of the two tangents leading to the circle are equal, or the chord perpendicular to the inner diameter of the circle is equal in two parts divided by the diameter.

1 1. The last two terms (or the first two terms) in the proportional formula are equal.

* 12. Appearance of the inner (outer) common tangent of two circles, etc.

13. Two line segments equal to the same line segment are equal.

2. Prove that two angles are equal.

1. The angles corresponding to two congruent triangles are equal.

2. Equiangular corners of the same triangle.

3. In an isosceles triangle, the center line (or height) of the base bisects the vertex angle.

4. The isosceles angle, internal dislocation angle or diagonal of the parallelogram of two parallel lines are equal.

5. The complementary angle (or complementary angle) of the same angle (or equal angle) is equal.

6. In the same circle (or circle), the central angles of a pair of equal chords (or arcs) are equal, the peripheral angles are equal, and the tangent angle is equal to the peripheral angles of a pair of arcs it clamps.

7. A point outside the circle leads to two tangents of the circle, and the connecting line between the center of the circle and this point bisects the included angle of the two tangents.

8. The corresponding angles of similar triangles are equal.

9. The outer angle of the inscribed quadrilateral of a circle is equal to the inner diagonal.

10. Two angles equal to the same angle.

3. Prove that two straight lines are perpendicular to each other.

1. The bisector of the top angle or the median line of the bottom edge of an isosceles triangle is perpendicular to the bottom edge.

2. If the median line of one side of a triangle is equal to half of this side, the angle subtended by this side is a right angle.

In a triangle, if two angles are complementary, the third angle is a right angle.

4. The bisectors of adjacent complementary angles are perpendicular to each other.

If a straight line is perpendicular to one of the parallel lines, it must be perpendicular to the other.

6. When two straight lines intersect at right angles, they are vertical.

7. Use points with the same distance from both ends of the line segment to be located on the middle vertical line of the line segment.

8. Use the inverse theorem of Pythagorean theorem.

9. Use diagonal lines of diamonds to be perpendicular to each other.

10. The diameter of a chord (or arc) bisected by a circle is perpendicular to the chord.

1 1. Use right angles on the semicircle.

4. Prove that two straight lines are parallel

1. Lines perpendicular to the same line are parallel.

2. The congruent angles are equal, and the internal angle is equal to or parallel to two lines with complementary internal angles.

3. The opposite sides of a parallelogram are parallel.

The center line of triangle is parallel to the third side.

5. The center line of the trapezoid is parallel to the two bottom sides.

6. Two lines parallel to the same line are parallel.

7. If the line segments obtained by cutting two sides (or extension lines) of a triangle are proportional, the line is parallel to the third side.

5. Prove the sum and difference multiples of line segments.

1. Do the sum of two line segments and prove that it is equal to the third line segment.

2. Intercept a segment equal to the first segment on the third segment, and prove that the rest is equal to the second segment.

3. Extend the short segment twice, and then prove that it is equal to the long segment.

4. Take the midpoint of the long line segment and prove that half of it is equal to the short line segment.

5. Using some theorems (midline of triangle, right triangle of 30 degrees, midline on hypotenuse of right triangle, center of gravity of triangle, properties of similar triangles, etc.). ).

6. Prove the sum and difference of angles.

1. and prove that the sum, difference, multiplication and division of line segments are the same.

2. Using the definition of angular bisector.

3. The outer angle of a triangle is equal to the sum of two non-adjacent inner angles.

7. Prove that the line segments are not equal

1. The same triangle, the big angle is opposite to the big side.

2. The vertical segment is the shortest.

3. The sum of two sides of a triangle is greater than the third side, and the difference between the two sides is less than the third side.

4. In two triangles, two sides are equal but the included angle is not equal, so the third side with larger included angle is larger.

5. In the same circle or equal circle, the arc is big and the chord is big, and the distance between the chord centers is small.

6. The total amount is greater than any part of it.

8. Prove the inequality of two angles.

1. In the same triangle, the big side faces the big corner.

2. The outer angle of a triangle is larger than any inner angle that is not adjacent to it.

3. In two triangles, two sides are equal, the third side is unequal, the third side is large, and the included angle between the two sides is also large.

*4. In the same circle or equal circle, the arc is large, and the fillet and central angle are large.

5. The total amount is greater than any part of it.

9. Prove the proportional formula or equal product formula

1. Line segments corresponding to similar triangle proportions.

2. Using the bisector theorem of inner angle and outer angle.

3. Parallel lines are in proportion.

4. The proportional mean value theorem in right triangle is a projective theorem.

5. Proportional theorems related to circles-intersecting chord theorem, line cutting theorem and their inference.

6. Use Billy formula or equal product formula.

10, prove four * * * cycles.

1. Vertex * * circle of diagonally complementary quadrilateral.

2. A quadrilateral with an outer angle equal to the inner diagonal is inscribed in a circle.

3. The top circle of a triangle with the same base and the same vertex (the vertex is on the same side of the base).

4. The vertex * * * circle of a right triangle with the same hypotenuse.

5. A circle with equal distance from each point to the vertex.