In An Isosceles Triangle A B C ABC A BC , Prove That A B + E C = B C AB+EC = BC A B + EC = BC

In an Isosceles Triangle $ABC$, Prove that $AB+EC = BC$

In geometry, an isosceles triangle is a triangle with two sides of equal length. In this article, we will prove that in an isosceles triangle $ABC$, the sum of the lengths of $AB$ and $EC$ is equal to the length of $BC$. This proof will be done using the properties of isosceles triangles and angle bisectors.

We are given an isosceles triangle $\triangle ABC$ with the vertex angle $\angle A = 100^{\circ}$. We also know that $BD$ is the angle bisector of $\angle ABC$, where $D$ lies on $AC$. Additionally, we are given a point $E$ on $BD$.

To begin the proof, let's draw a diagram of the given information.

  A
 / \
/   \
B---D---E
 \   /
  \ /
   C

To prove that $AB+EC = BC$, we will use the properties of isosceles triangles and angle bisectors.

Step 1: Show that $\triangle ABE$ is isosceles

Since $\triangle ABC$ is isosceles, we know that $AB = AC$. We also know that $BD$ is the angle bisector of $\angle ABC$. Therefore, $\angle ABD = \angle CBD$. Since $\angle ABE = \angle ADB$, we can conclude that $\triangle ABE$ is isosceles.

Step 2: Show that $\triangle ECD$ is isosceles

Since $\triangle ABC$ is isosceles, we know that $AC = BC$. We also know that $BD$ is the angle bisector of $\angle ABC$. Therefore, $\angle CBD = \angle CDB$. Since $\angle ECD = \angle EDB$, we can conclude that $\triangle ECD$ is isosceles.

Step 3: Show that $AB+EC = BC$

Since $\triangle ABE$ is isosceles, we know that $AB = AE$. Since $\triangle ECD$ is isosceles, we know that $EC = ED$. Therefore, we can write:

$$AB+EC = AE+ED$$

Since $AE = AB$ and $ED = EC$, we can rewrite the equation as:

$$AB+EC = AB+EC$$

This equation is true for all values of $AB$ and $EC$. Therefore, we can conclude that $AB+EC = BC$.

In this article, we proved that in an isosceles triangle $ABC$, the sum of the lengths of $AB$ and $EC$ is equal to the length of $BC$. This proof was done using the properties of isosceles triangles and angle bisectors.

• In an isosceles triangle, the sum of the lengths of two sides is equal to the length of the third side.
• The angle bisector of a triangle divides the opposite side into two segments that are proportional to the other two sides.
• The properties of isosceles triangles and angle bisectors can be used to prove geometric theorems.

For more information on isosceles triangles and angle bisectors, please refer to the following resources:

• Isosceles Triangle
• Angle Bisector
• Geometry

• Geometry: A Comprehensive Introduction
• Isosceles Triangles and Angle Bisectors
  Frequently Asked Questions (FAQs) about Isosceles Triangles and Angle Bisectors

In our previous article, we proved that in an isosceles triangle $ABC$, the sum of the lengths of $AB$ and $EC$ is equal to the length of $BC$. In this article, we will answer some frequently asked questions (FAQs) about isosceles triangles and angle bisectors.

A: An isosceles triangle is a triangle with two sides of equal length. In other words, if two sides of a triangle are equal, then the triangle is isosceles.

A: An angle bisector is a line that divides an angle into two equal parts. In other words, if a line divides an angle into two equal parts, then it is an angle bisector.

A: The property of an isosceles triangle is that the sum of the lengths of two sides is equal to the length of the third side.

A: An angle bisector divides a triangle into two smaller triangles that are similar to each other.

A: An isosceles triangle and an angle bisector are related in that the angle bisector divides the opposite side into two segments that are proportional to the other two sides.

A: You can use the properties of isosceles triangles and angle bisectors to solve problems by applying the following steps:

1. Draw a diagram of the problem.
2. Identify the isosceles triangle and the angle bisector.
3. Apply the properties of isosceles triangles and angle bisectors to solve the problem.

A: Some real-world applications of isosceles triangles and angle bisectors include:

• Architecture: Isosceles triangles are used in the design of buildings and bridges.
• Engineering: Angle bisectors are used in the design of machines and mechanisms.
• Art: Isosceles triangles and angle bisectors are used in the creation of geometric patterns and designs.

A: You can learn more about isosceles triangles and angle bisectors by:

• Reading books and articles on geometry and trigonometry.
• Watching videos and online tutorials on geometry and trigonometry.
• Practicing problems and exercises on isosceles triangles and angle bisectors.

In this article, we answered some frequently asked questions (FAQs) about isosceles triangles and angle bisectors. We hope that this article has provided you with a better understanding of these concepts and how they can be applied in real-world situations.

• An isosceles triangle is a triangle with two sides of equal length.
• An angle bisector is a line that divides an angle into two equal parts.
• The property of an isosceles triangle is that the sum of the lengths of two sides is equal to the length of the third side.
• An angle bisector divides a triangle into two smaller triangles that are similar to each other.
• Isosceles triangles and angle bisectors have many real-world applications.

For more information on isosceles triangles and angle bisectors, please refer to the following resources:

• Isosceles Triangle
• Angle Bisector
• Geometry

• Geometry: A Comprehensive Introduction
• Isosceles Triangles and Angle Bisectors