All properties of the trapezoid with proof. What is a trapezoid: properties of a quadrilateral, theorems and formulas

\[(\Large(\text(Free trapezoid)))\]

Definitions

A trapezoid is a convex quadrilateral in which two sides are parallel and the other two sides are not parallel.

The parallel sides of a trapezoid are called its bases, and the other two sides are called its lateral sides.

The height of a trapezoid is the perpendicular drawn from any point of one base to another base.

Theorems: properties of a trapezoid

1) The sum of the angles at the side is \(180^\circ\) .

2) The diagonals divide the trapezoid into four triangles, two of which are similar, and the other two are equal in size.

Proof

1) Because \(AD\parallel BC\), then the angles \(\angle BAD\) and \(\angle ABC\) are one-sided for these lines and the transversal \(AB\), therefore, \(\angle BAD +\angle ABC=180^\circ\).

2) Because \(AD\parallel BC\) and \(BD\) are a secant, then \(\angle DBC=\angle BDA\) lie crosswise.
Also \(\angle BOC=\angle AOD\) as vertical.
Therefore, at two angles \(\triangle BOC \sim \triangle AOD\).

Let's prove that \(S_(\triangle AOB)=S_(\triangle COD)\). Let \(h\) be the height of the trapezoid. Then \(S_(\triangle ABD)=\frac12\cdot h\cdot AD=S_(\triangle ACD)\). Then: \

Definition

The midline of a trapezoid is a segment connecting the midpoints of the sides.

Theorem

The midline of the trapezoid is parallel to the bases and equal to their half-sum.

Proof*

1) Let's prove parallelism.

Let us draw through the point \(M\) the straight line \(MN"\parallel AD\) (\(N"\in CD\) ). Then, according to Thales’ theorem (since \(MN"\parallel AD\parallel BC, AM=MB\)) point \(N"\) is the middle of the segment \(CD\). This means that the points \(N\) and \(N"\) will coincide.

2) Let's prove the formula.

Let's do \(BB"\perp AD, CC"\perp AD\) . Let \(BB"\cap MN=M", CC"\cap MN=N"\).

Then, by Thales' theorem, \(M"\) and \(N"\) are the midpoints of the segments \(BB"\) and \(CC"\), respectively. This means that \(MM"\) is the middle line of \(\triangle ABB"\) , \(NN"\) is the middle line of \(\triangle DCC"\) . That's why: \

Because \(MN\parallel AD\parallel BC\) and \(BB", CC"\perp AD\), then \(B"M"N"C"\) and \(BM"N"C\) are rectangles. According to Thales' theorem, from \(MN\parallel AD\) and \(AM=MB\) it follows that \(B"M"=M"B\) . Hence, \(B"M"N"C"\) and \(BM"N"C\) are equal rectangles, therefore, \(M"N"=B"C"=BC\) .

Thus:

\ \[=\dfrac12 \left(AB"+B"C"+BC+C"D\right)=\dfrac12\left(AD+BC\right)\]

Theorem: property of an arbitrary trapezoid

The midpoints of the bases, the point of intersection of the diagonals of the trapezoid and the point of intersection of the extensions of the lateral sides lie on the same straight line.

Proof*
It is recommended that you familiarize yourself with the proof after studying the topic “Similarity of triangles”.

1) Let us prove that the points \(P\) , \(N\) and \(M\) lie on the same line.

Let's draw a straight line \(PN\) (\(P\) is the point of intersection of the extensions of the lateral sides, \(N\) is the middle of \(BC\)). Let it intersect the side \(AD\) at the point \(M\) . Let us prove that \(M\) is the midpoint of \(AD\) .

Consider \(\triangle BPN\) and \(\triangle APM\) . They are similar at two angles (\(\angle APM\) – general, \(\angle PAM=\angle PBN\) as corresponding at \(AD\parallel BC\) and \(AB\) secant). Means: \[\dfrac(BN)(AM)=\dfrac(PN)(PM)\]

Consider \(\triangle CPN\) and \(\triangle DPM\) . They are similar at two angles (\(\angle DPM\) – general, \(\angle PDM=\angle PCN\) as corresponding at \(AD\parallel BC\) and \(CD\) secant). Means: \[\dfrac(CN)(DM)=\dfrac(PN)(PM)\]

From here \(\dfrac(BN)(AM)=\dfrac(CN)(DM)\). But \(BN=NC\) therefore \(AM=DM\) .

2) Let us prove that the points \(N, O, M\) lie on the same line.

Let \(N\) be the midpoint of \(BC\) and \(O\) be the point of intersection of the diagonals. Let's draw a straight line \(NO\) , it will intersect the side \(AD\) at the point \(M\) . Let us prove that \(M\) is the midpoint of \(AD\) .

\(\triangle BNO\sim \triangle DMO\) along two angles (\(\angle OBN=\angle ODM\) lying crosswise at \(BC\parallel AD\) and \(BD\) secant; \(\angle BON=\angle DOM\) as vertical). Means: \[\dfrac(BN)(MD)=\dfrac(ON)(OM)\]

Likewise \(\triangle CON\sim \triangle AOM\). Means: \[\dfrac(CN)(MA)=\dfrac(ON)(OM)\]

From here \(\dfrac(BN)(MD)=\dfrac(CN)(MA)\). But \(BN=CN\) therefore \(AM=MD\) .

\[(\Large(\text(Isosceles trapezoid)))\]

Definitions

A trapezoid is called rectangular if one of its angles is right.

A trapezoid is called isosceles if its sides are equal.

Theorems: properties of an isosceles trapezoid

1) An isosceles trapezoid has equal base angles.

2) The diagonals of an isosceles trapezoid are equal.

3) Two triangles formed by diagonals and a base are isosceles.

Proof

1) Consider the isosceles trapezoid \(ABCD\) .

From the vertices \(B\) and \(C\), we drop the perpendiculars \(BM\) and \(CN\) to the side \(AD\), respectively. Since \(BM\perp AD\) and \(CN\perp AD\) , then \(BM\parallel CN\) ; \(AD\parallel BC\) , then \(MBCN\) is a parallelogram, therefore, \(BM = CN\) .

Consider the right triangles \(ABM\) and \(CDN\) . Since their hypotenuses are equal and the leg \(BM\) is equal to the leg \(CN\) , then these triangles are equal, therefore, \(\angle DAB = \angle CDA\) .

2)

Because \(AB=CD, \angle A=\angle D, AD\)- general, then according to the first sign. Therefore, \(AC=BD\) .

3) Because \(\triangle ABD=\triangle ACD\), then \(\angle BDA=\angle CAD\) . Therefore, the triangle \(\triangle AOD\) is isosceles. Similarly, it is proved that \(\triangle BOC\) is isosceles.

Theorems: signs of an isosceles trapezoid

1) If a trapezoid has equal base angles, then it is isosceles.

2) If a trapezoid has equal diagonals, then it is isosceles.

Proof

Consider the trapezoid \(ABCD\) such that \(\angle A = \angle D\) .

Let's complete the trapezoid to the triangle \(AED\) as shown in the figure. Since \(\angle 1 = \angle 2\) , then the triangle \(AED\) is isosceles and \(AE = ED\) . Angles \(1\) and \(3\) are equal as corresponding angles for parallel lines \(AD\) and \(BC\) and secant \(AB\). Similarly, angles \(2\) and \(4\) are equal, but \(\angle 1 = \angle 2\), then \(\angle 3 = \angle 1 = \angle 2 = \angle 4\), therefore, the triangle \(BEC\) is also isosceles and \(BE = EC\) .

Eventually \(AB = AE - BE = DE - CE = CD\), that is, \(AB = CD\), which is what needed to be proven.

2) Let \(AC=BD\) . Because \(\triangle AOD\sim \triangle BOC\), then we denote their similarity coefficient as \(k\) . Then if \(BO=x\) , then \(OD=kx\) . Similar to \(CO=y \Rightarrow AO=ky\) .

Because \(AC=BD\) , then \(x+kx=y+ky \Rightarrow x=y\) . This means \(\triangle AOD\) is isosceles and \(\angle OAD=\angle ODA\) .

Thus, according to the first sign \(\triangle ABD=\triangle ACD\) (\(AC=BD, \angle OAD=\angle ODA, AD\)– general). So, \(AB=CD\) , why.

- (Greek trapezion). 1) in geometry, a quadrilateral in which two sides are parallel and two are not. 2) a figure adapted for gymnastic exercises. Dictionary of foreign words included in the Russian language. Chudinov A.N., 1910. TRAPEZE... ... Dictionary of foreign words of the Russian language

Trapezoid- Trapezoid. TRAPEZE (from the Greek trapezion, literally table), a convex quadrilateral in which two sides are parallel (the bases of the trapezoid). The area of ​​a trapezoid is equal to the product of half the sum of the bases (midline) and the height. ... Illustrated Encyclopedic Dictionary

Quadrangle, projectile, crossbar Dictionary of Russian synonyms. trapezoid noun, number of synonyms: 3 crossbar (21) ... Synonym dictionary

- (from the Greek trapezion, literally table), a convex quadrangle in which two sides are parallel (the bases of a trapezoid). The area of ​​a trapezoid is equal to the product of half the sum of the bases (midline) and the height... Modern encyclopedia

- (from the Greek trapezion, lit. table), a quadrilateral in which two opposite sides, called the bases of the trapezoid, are parallel (in the figure AD and BC), and the other two are non-parallel. The distance between the bases is called the height of the trapezoid (at ... ... Big Encyclopedic Dictionary

TRAPEZOUS, a quadrangular flat figure in which two opposite sides are parallel. The area of ​​a trapezoid is equal to half the sum of the parallel sides multiplied by the length of the perpendicular between them... Scientific and technical encyclopedic dictionary

TRAPEZE, trapezoid, women's (from Greek trapeza table). 1. Quadrilateral with two parallel and two non-parallel sides (mat.). 2. A gymnastic apparatus consisting of a crossbar suspended on two ropes (sports). Acrobatic... ... Ushakov's Explanatory Dictionary

TRAPEZE, and, female. 1. A quadrilateral with two parallel and two non-parallel sides. The bases of the trapezoid (its parallel sides). 2. A circus or gymnastics apparatus is a crossbar suspended on two cables. Ozhegov's explanatory dictionary. WITH … Ozhegov's Explanatory Dictionary

Female, geom. a quadrilateral with unequal sides, two of which are parallel (parallel). Trapezoid, a similar quadrilateral in which all sides run apart. Trapezohedron, a body faceted by trapezoids. Dahl's Explanatory Dictionary. IN AND. Dahl. 1863 1866 … Dahl's Explanatory Dictionary

- (Trapeze), USA, 1956, 105 min. Melodrama. Aspiring acrobat Tino Orsini joins a circus troupe where Mike Ribble, a famous former trapeze artist, works. Mike once performed with Tino's father. Young Orsini wants Mike... Encyclopedia of Cinema

A quadrilateral in which two sides are parallel and the other two sides are not parallel. The distance between parallel sides is called. height T. If parallel sides and height contain a, b and h meters, then the area of ​​T contains square meters ... Encyclopedia of Brockhaus and Efron

Books

• Set of tables. Geometry. 8th grade. 15 tables + methodology, . The tables are printed on thick printed cardboard measuring 680 x 980 mm. The kit includes a brochure with teaching guidelines for teachers. Educational album of 15 sheets. Polygons...
• Set of tables. Mathematics. Polygons (7 tables), . Educational album of 7 sheets. Convex and non-convex polygons. Quadrilaterals. Parallelogram and trapezoid. Signs and properties of a parallelogram. Rectangle. Rhombus. Square. Square…

A polygon is a part of a plane bounded by a closed broken line. The angles of a polygon are indicated by the points of the vertices of the polygon. The vertices of the corners of a polygon and the vertices of a polygon are coincident points.

Definition. A parallelogram is a quadrilateral whose opposite sides are parallel.

Properties of a parallelogram

1. Opposite sides are equal.
In Fig. eleven AB = CD; B.C. = AD.

2. Opposite angles are equal (two acute and two obtuse angles).
In Fig. 11∠ A = ∠C; ∠B = ∠D.

3 Diagonals (line segments connecting two opposite vertices) intersect and are divided in half by the intersection point.

In Fig. 11 segments A.O. = O.C.; B.O. = O.D..

Definition. A trapezoid is a quadrilateral in which two opposite sides are parallel and the other two are not.

Parallel sides are called her reasons, and the other two sides are sides.

Types of trapezoids

1. Trapezoid, whose sides are not equal,
called versatile(Fig. 12).

2. A trapezoid whose sides are equal is called isosceles(Fig. 13).

3. A trapezoid in which one side makes a right angle with the bases is called rectangular(Fig. 14).

The segment connecting the midpoints of the lateral sides of the trapezoid (Fig. 15) is called the midline of the trapezoid ( MN). The midline of the trapezoid is parallel to the bases and equal to their half-sum.

A trapezoid can be called a truncated triangle (Fig. 17), therefore the names of trapezoids are similar to the names of triangles (triangles are scalene, isosceles, rectangular).

Area of ​​parallelogram and trapezoid

Rule. Area of ​​a parallelogram is equal to the product of its side and the height drawn to this side.

Related definitions

Trapezoid elements

• Parallel sides are called reasons trapezoids.
• The other two sides are called sides.
• The segment connecting the midpoints of the sides is called the midline of the trapezoid.
• The distance between the bases is called the height of the trapezoid.

Types of trapezoids

Rectangular trapezoid

Isosceles trapezoid

• A trapezoid whose sides are equal is called isosceles or isosceles.
• A trapezoid having right angles on its sides is called rectangular.

General properties

• The midline of the trapezoid is parallel to the bases and equal to their half-sum.
• The segment connecting the midpoints of the diagonals is equal to half the difference of the bases.
• Parallel lines intersecting the sides of an angle cut off proportional segments from the sides of the angle.
• A circle can be inscribed in a trapezoid if the sum of the bases of the trapezoid is equal to the sum of its sides.

Properties and signs of an isosceles trapezoid

• The straight line passing through the midpoints of the bases is perpendicular to the bases and is the axis of symmetry of the trapezoid.
• The height lowered from the top to the larger base divides it into two segments, one of which is equal to half the sum of the bases, the other - half the difference of the bases.
• In an isosceles trapezoid, the angles at any base are equal.
• In an isosceles trapezoid, the lengths of the diagonals are equal.
• If a trapezoid can be inscribed in a circle, then it is isosceles.
• A circle can be described around an isosceles trapezoid.
• If the diagonals in an isosceles trapezoid are perpendicular, then the height is equal to half the sum of the bases.

Inscribed and circumscribed circle

Square

These formulas are the same, since half the sum of the bases is equal to the midline of the trapezoid.

Let's consider several directions for solving problems in which a trapezoid is inscribed in a circle.

When can a trapezoid be inscribed in a circle? A quadrilateral can be inscribed in a circle if and only if the sum of its opposite angles is 180º. It follows that You can only fit an isosceles trapezoid into a circle.

The radius of a circle circumscribed by a trapezoid can be found as the radius of a circle circumscribed by one of the two triangles into which the trapezoid is divided by its diagonal.

Where is the center of the circle circumscribed by the trapezoid? It depends on the angle between the diagonal of the trapezoid and its side.

If the diagonal of a trapezoid is perpendicular to its side, then the center of the circle described around the trapezoid lies in the middle of its larger base. The radius of the circle circumscribed about the trapezoid in this case is equal to half of its larger base:

If the diagonal of a trapezoid forms an acute angle with its side, the center of the circle described around the trapezoid lies inside the trapezoid.

If the diagonal of a trapezoid forms an obtuse angle with its side, the center of the circle circumscribed about the trapezoid lies outside the trapezoid, behind the large base.

The radius of a circle circumscribed about a trapezoid can be found by a corollary of the theorem of sines. From triangle ACD

From triangle ABC

Another option to find the radius of the circumscribed circle is

The sines of angle D and angle CAD can be found, for example, from right triangles CFD and ACF:

When solving problems involving a trapezoid inscribed in a circle, you can also use the fact that the inscribed angle is equal to half of its corresponding central angle. For example,

By the way, you can also use the angles COD and CAD to find the area of ​​a trapezoid. Using the formula for finding the area of ​​a quadrilateral using its diagonals