The diagonals of an isosceles trapezoid have the same length.
This theorem also holds true in reverse, meaning, we can determine that a certain trapezoid is isosceles if we know that its diagonals are equal.
You can use this theorem in the exam as you see it and you will not need to prove it.
Given the isosceles trapezoid it holds that:
Angles of the Diagonals of an Isosceles Trapezoid
Observe- You will need to prove this property in the exam. We can prove it based on the congruence theorem .
The diagonals of an isosceles trapezoid create, along with the bases, equal angles.
We will mark them.
Observe- You will need to prove this property on the exam.
The diagonals of an isosceles trapezoid create, along with the bases, two isosceles triangles. Let E be the point where the diagonals intersect. We will notice that these are and .
We will mark in orange the triangles that are created from the diagonals and the bases:
Isosceles Triangles in a Trapezoid
Observe- You will need to prove this property on the exam.
Below is an isosceles trapezoid
If \( ∢D=50° \)
Determine the value of \( ∢B \)?
The diagonals of an isosceles trapezoid are very special and have main properties that will help us a lot when we want to solve problems and prove geometric properties on exams. The properties of the diagonals are very logical, and we are here to make them clear and easy to understand.
Shall we start?
The diagonals of an isosceles trapezoid have the same length.
This theorem also holds true in reverse, that is, we can determine that a certain trapezoid is isosceles if we know that its diagonals are equal.
You can use this theorem in the exam as you see it and you will not have to prove it.
However, understanding why this property is true helps you remember it. Let's look at the trapezoid :
Diagonals of an Isosceles Trapezoid
The only data we have are:
1) ABCD is a trapezoid with
2)
We have to prove:
Solution:
Given that , we can deduce that the trapezoid is isosceles.
In the exam, you will already be able to deduce that since the diagonals of an isosceles trapezoid have the same length.
But, to give an example, let's see how this theorem is proven.
We can prove it based on the congruence of two triangles formed by the diagonals and the sides of the trapezoid:
and
| Argument | Explanation |
| (Side) | Dato |
| (Angle) | ABCD is an isosceles trapezoid. In an isosceles trapezoid, the base angles have the same measure. |
| (Side) | A shared side is equal to itself |
| According to SAS | |
| Therefore- | Corresponding sides in congruent triangles are equal |
In fact, we have seen that we can superimpose the triangles that have been formed by the diagonals with the base and non-parallel sides, and in this way demonstrate the first property of the diagonals of an isosceles trapezoid.
Given: \( ∢A=120° \)
The isosceles trapezoid
Find a: \( ∢C \)
Given: \( ∢C=2x \)
\( ∢A=120° \)
isosceles trapezoid.
Find x.
True OR False:
In all isosceles trapezoids the base Angles are equal.
In the isosceles trapezoid the following is true:
Diagonals of an Isosceles Trapezoid
You may need to prove this property on exams. But, do not worry, we can do it easily and quickly by superimposing the triangles according to SSS.
Observe: It is true that we have proven the congruence of these triangles in the first property, but now we will do it based on the first property of the diagonals of an isosceles trapezoid.
Let's see the proof:
| Argument | Explanation |
| (Side) | Given - In an isosceles trapezoid, the legs have the same length. |
| (Side) | ABCD is an isosceles trapezoid. In an isosceles trapezoid, the diagonals are equal. |
| (Side) | A shared side is equal to itself |
| According to SSS |
The diagonals of an isosceles trapezoid create, along with the bases, equal angles.
You may need to prove this property on exams to use it We know that this property might seem a bit complicated at first glance, so we think it's very important that you start by observing in the illustration which angles we are talking about:
We will mark all the angles that are between a diagonal and the base, as follows:
Isosceles Triangles in a Trapezoid
Great! Now that we know which angles we are dealing with, we can demonstrate that they are equal based on the congruence of the triangles we made above. That is, congruence between
Observe- To demonstrate this property, you should know what the corresponding angles and alternate interior angles are.
After having demonstrated the congruence of the triangles, we can claim that:
| Argument | Explanation |
| Corresponding angles in congruent triangles are equivalent. | |
| Alternate interior angles formed by parallel lines (AB∥DC) and transversal AC | |
| Alternate interior angles formed by parallel lines (AB∥DC) and transversal BD | |
| Transitive Property of Equality |
Do isosceles trapezoids have two pairs of parallel sides?
Look at the polygon in the diagram.
What type of shape is it?
In an isosceles trapezoid ABCD
\( ∢B=3x \)
\( ∢D=x \)
Calculate the size of angle \( ∢B \).
The diagonals of an isosceles trapezoid create, along with the bases, two isosceles triangles. You may need to prove this property on exams to use it.
First, let's see which triangles we are talking about.
We will mark in orange the triangles that are created with the diagonals and the bases:
Note that these are and , where E is the intersection point of the diagonals.
In the previous property, we have proven that the green angles that arise from the diagonals of the isosceles trapezoid and the bases are equal .
We already know that the in a triangle, sides opposite equal angles have equal length Therefore, the demonstration will look as follows:
After we prove that the green angles are equal, we can claim that:
| Argument | Explanation |
| Corresponding angles in congruent triangles are equivalent. This is proven with the third property. | |
| Therefore | In a triangle, sides opposite equal angles are equal. |
| Therefore is isosceles | A triangle with two equal sides is isosceles |
| By the third property | |
| Therefore | In a triangle, sides opposite equal angles are equal |
| Por lo tanto isosceles | A triangle with two equal sides is isosceles |
Given:
isosceles trapezoid.
Find x.
Given that the trapezoid is isosceles and the angles on both sides are equal, it can be argued that:
We know that the sum of the angles of a quadrilateral is 360 degrees.
Therefore we can create the formula:
We replace according to the existing data:
We divide the two sections by 4:
30°
Below is an isosceles trapezoid
If
Determine the value of ?
Let's recall that in an isosceles trapezoid, the sum of the two angles on each of the trapezoid's legs equals 180 degrees.
In other words:
Since angle D is known to us, we can calculate:
130°
Do isosceles trapezoids have two pairs of parallel sides?
To solve this problem, we'll follow these steps:
Now, let's work through each step:
Step 1: A trapezoid is defined as a quadrilateral with at least one pair of parallel sides.
Step 2: An isosceles trapezoid is a special type of trapezoid where the non-parallel sides (legs) are of equal length. Its defining feature is having exactly one pair of parallel sides, which is the same characteristic as a general trapezoid.
Step 3: Since the definition of a trapezoid inherently allows for only one pair of parallel sides, an isosceles trapezoid, as a type of trapezoid, cannot have two pairs of parallel sides. A quadrilateral with two pairs of parallel sides is typically designated as a parallelogram, not a trapezoid.
Therefore, the solution to the problem is that isosceles trapezoids do not have two pairs of parallel sides. No.
No
True OR False:
In all isosceles trapezoids the base Angles are equal.
True: in every isosceles trapezoid the base angles are equal to each other.
True
In an isosceles trapezoid, will the sum of the opposite angles always be 180°?
Let's closely examine the properties of an isosceles trapezoid to determine the relationship between its opposite angles.
An isosceles trapezoid is a type of trapezoid where the non-parallel sides (legs) are of equal length. It also implies that the angles adjacent to each leg are equal, due to the symmetry of the trapezoid. However, the most crucial aspect for this problem involves understanding the sum of angles in a trapezoid and their specific relationships.
Consider an isosceles trapezoid, , with base parallel to base , and legs and being equal.
In addition, the total sum of the interior angles of any quadrilateral (and thus any trapezoid) is . Therefore, opposite angles and (first set) sum to , and similarly, opposite angles and (second set) also sum to . Therefore, the sum of the opposite angles in an isosceles trapezoid is always .
Thus, through application of the properties of parallel lines and angle sums, we find that the sum of opposite angles in an isosceles trapezoid is indeed always .
The correct conclusion of this problem is that the statement is True.
The sum of the opposite angles in an isosceles trapezoid is always .
True
Great! Now you know the important properties of the diagonals of an isosceles trapezoid.
You will be able to prove them without any problem and use them as arguments whenever you need to demonstrate something.
In an isosceles trapezoid, will the sum of the opposite angles always be 180°?
Are the diagonals of an isosceles trapezoid equal and do they intersect each other?
Do the diagonals of the trapezoid necessarily bisect each other?
