Isosceles Trapezoid

The isosceles trapezoid is, in fact, a trapezoid (that is, a four-sided polygon with two of them - the bases - being parallel), with two of its sides being equivalent and with its base angles of equal magnitude.

In the trapezoid, as is known, there are two bases and, each base has two adjacent base angles on both sides. In other words, in the isosceles trapezoid, there are two sets of equal base angles, as can be seen in the following illustration:

Isosceles Trapezoid

The properties detailed here are the unique characteristics of isosceles trapezoids among all other types of trapezoids. The following illustration describes the theorems in the best way:

Properties of the Isosceles Trapezoid

• The sides that are not parallel are congruent, that is, they have the same measure. That is, it is fulfilled: $NK=LM$
• In the isosceles trapezoid, there are two sets of equal angles for the larger base and for the smaller base.

That is, angles $L$ and $K$ are equivalent, just like angles $M$ and $N$ are also.

• The two diagonals of the isosceles trapezoid are equal. That is, it is fulfilled: $KM=LN$
• Any isosceles trapezoid can be inscribed in a circle

To demonstrate that a trapezoid is isosceles, we must make use of the properties specified earlier, in fact, these are reciprocal theorems. It is sufficient to demonstrate just one property.

That is, if we prove that:

• The two sides that are not parallel are congruent

or

• The angles at the base of the trapezoid are congruent

or

• The diagonals of the trapezoid are congruent

then, said trapezoid is an isosceles trapezoid.

The following properties refer to the diagonals of the isosceles trapezoid. To highlight these properties in the best way, we will use this illustration:

Diagonals of the Isosceles Trapezoid

• The two diagonals are equal. That is, it holds that: $PS=RT$
• The triangles $PTS$ and $RST$ are congruent according to the side - side - side congruence theorem
• The triangles $TPR$ and $SRP$ are congruent according to the side - side - side congruence theorem
• The triangles $PKR$ and $TKS$ are isosceles triangles with all that this implies
• The angles $P1$, $R1$, $S1$, and $T1$ are equivalent

The calculation of the area of an isosceles trapezoid is done exactly in the same way as the area of any other trapezoid is calculated.

That is, the lengths of the two bases are added, the total sum is multiplied by the height, and then, it is divided by $2$.

We will use this illustration to explain the steps of the calculation:

Finding the area of an Isosceles Trapezoid

The formula to calculate the area of the isosceles trapezoid (not exclusively) is:

$A=\frac{ ( AB+ DC) \times H}{2}$

Given the isosceles trapezoid described in the following scheme.

It is known that the sum of three angles is $240º$ degrees.

According to the data, we must calculate all the angles of this isosceles trapezoid.

Solution:

If the sum of the three angles of the given trapezoid is $240º$ and the total sum of the angles of a trapezoid (as with any quadrilateral) is $360º$, we can deduce that the fourth angle measures $120º$ degrees.

It is one of the adjacent angles to the smaller base, let's assume angle $A$. Since it is an isosceles trapezoid, the angles at the base are congruent, therefore, angle $B$ also measures $120º$.

Remember that it is a trapezoid and that the bases $AB$ and $DC$ are parallel, that is, angles $A$ and $D$ (just like $B$ and $C$) are collateral angles and, therefore, complement each other and together measure $180º$ degrees. Therefore, it will give us that angles $C$ and $D$ measure $60º$ degrees.

Answer:

The angles of the trapezoid are $120º, 120º, 60º, 60º$

Given the isosceles trapezoid described in the following scheme.

It is known that the sum of two of its sides is $120º$.

According to the data, we must calculate all the angles of this isosceles trapezoid.

Solution:

Let's go back to the rules related to the base and remember that it is a trapezoid and that the bases $AB$ and $DC$ are parallel, that is, the angles $A$ and $D$ (as well as $B$ and $C$) are collateral angles and, therefore, complement each other and together measure $180º$ degrees. Given that the amplitude we have is $120º$ degrees, we can deduce that these are not adjacent angles on the same side (i.e., unilateral), but angles that share the same base.

Being an isosceles trapezoid, the base angles are congruent, therefore, each of them measures $60º$ degrees.

The complementary angle (to reach $180º$degrees) of each of these angles measures $120º$ degrees.

Answer:

The angles of the trapezoid are $120º, 120º, 60º, 60º$