20! Ticket No. 1: 1. Formulate a definition of a convex polygon (perimeter, diagonal). Formulate the theorem about

Ticket No. 1

1. Convex polygon definition (perimeter, diagonal):

A convex polygon is a polygon where all its interior angles are less than 180 degrees, and the line segment joining any two points inside the polygon lies entirely within the polygon.

Perimeter refers to the total length of the boundary of a polygon. For a convex polygon, the perimeter is the sum of the lengths of all its sides.

Diagonal refers to a line segment that connects any two non-adjacent vertices of a polygon. For a convex polygon, the number of diagonals can be calculated using the formula: D = (n * (n - 3))/2, where "n" is the number of sides of the polygon.

Sum of angles in a convex polygon theorem:

The sum of the interior angles in a convex polygon can be calculated using the formula: Sum = (n - 2) * 180 degrees, where "n" is the number of sides of the polygon.

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2. Similarity criteria of triangles and proof of one criterion:

Triangle similarity criteria are used to determine if two triangles are similar. One criterion is the angle-angle (AA) criterion, which states that if two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.

Proof of the AA criterion:

Given triangle ABC and triangle DEF, where angle A = angle D and angle B = angle E.

To prove that triangle ABC ~ DEF, we need to show that their corresponding sides are proportional.

Using the angle-angle criterion, we know that angle A = angle D, so we have angle ABD = angle DBE (vertical angles).

Since angle B = angle E, we have angle ABC = angle DBE (vertical angles).

By angle-angle similarity, we can conclude that triangle ABC ~ DEF.

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3. Finding triangle angles when triangle ABC is inscribed in a circle with AV as the diameter and arc VS is 134°:

Since AV is the diameter, angle VAB is a right angle (90°).

By the inscribed angle theorem, the measure of an inscribed angle is equal to half the measure of its intercepted arc. Therefore, we have angle VSB = 134°/2 = 67°.

Since triangle ABC is inscribed in a circle, the sum of its angles is 180°.

Let angle C = x, then angle B = 180° - 90° - x = 90° - x.

Using the angles of a triangle, we can write the equation: x + (90° - x) + 67° = 180°.

Simplifying the equation gives: 157° = x.

Therefore, angle A = 90°, angle B = 90° - 157° = -67° (or 293°), and angle C = 157°.

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Ticket No. 2

1. Definitions of sine, cosine, and tangent in a right triangle:

In a right triangle, the three trigonometric ratios are defined as follows:

- Sine (sin): The ratio of the length of the side opposite the angle to the length of the hypotenuse.
- Cosine (cos): The ratio of the length of the adjacent side to the length of the hypotenuse.
- Tangent (tan): The ratio of the length of the side opposite the angle to the length of the adjacent side.

2. Area of a rectangle (formulation and proof):

The area (A) of a rectangle is determined by multiplying its length (l) by its width (w). So, A = l * w.

For the proof, consider a rectangle with length l and width w. We can divide the rectangle into smaller squares of side length 1, as shown:

[Visualization of the rectangle divided into squares]

The total number of squares will be equal to the product of the number of rows (l) and the number of columns (w). Therefore, the area of the rectangle is equal to its length multiplied by its width, which proves the formula A = l * w.

3. The sum of two opposite sides of a circumscribed quadrilateral:

In a circumscribed quadrilateral (a quadrilateral with a circumcircle), the sum of the lengths of two opposite sides is equal to the sum of the lengths of the other two opposite sides.

This can be proved using the property of an inscribed angle: An inscribed angle intercepts an arc that is twice its measure.

Let ABCD be a circumscribed quadrilateral, with AB and CD as opposite sides, and BC and DA as the other opposite sides.

By the inscribed angle theorem, we know that angle BAC intercepts arc BC, and angle BDA intercepts arc DA. Similarly, angle CBD intercepts arc CD, and angle CAB intercepts arc AB.

Since the sum of the measures of the angles around a point is 360 degrees, we can write:

angle BAC + angle BDA + angle CBD + angle CAB = 360 degrees

Since the intercepted arcs are twice the measure of the corresponding angles, we have:

arc BC + arc DA + arc CD + arc AB = 360 degrees

But the sum of the measures of all the arcs of a circle is 360 degrees, so we can write:

arc BC + arc DA + arc CD + arc AB = arc ABCD

Therefore, the sum of the lengths of two opposite sides (AB and CD) is equal to the sum of the lengths of the other two opposite sides (BC and DA).