# Isosceles Triangles with Congruent Lateral Sides

• Born
In summary: Again, since the vertex and lateral sides are congruent, you only have a short argument for both cases to say that neither cases can fit inside the original. Only the congruent triangle would fit on the original. 5) decrease leg length, 6) increase leg length -- This would only work if the base of the triangle was the same size as the height. However, if the base was smaller then the height then the triangle would not fit. 7) have both angles the same, 8) have one angle greater than the other. -- This would create a right triangle with one vertex on the original isosceles triangle and the other two on the new triangle.

## Homework Statement

Problem 99 from "Kiselev's Geometry Book I - Planimetry":

Two isosceles triangles with a common vertex and congruent lateral sides cannot fit one inside the other.

## The Attempt at a Solution

The statement is obviously true. If we visualize each isosceles triangle as made from two right triangles then we see that in order for one to fit in the other both the base and the height of one should be smaller than the bade and height of the other, however this is not possible since the sides of the smaller right triangle would not be able to connect to the endpoints of the hypotenuse by the Pythagorean theorem.

Sadly, the Pythagorean theorem comes later in the book. All I have are some more simple theorems, such as "the greater angle opposes the greater side" and "the greatest slant drawn from a point to a line is the one with the greatest distance from the foot of the perpendicular of the line to the point".

Any help is welcome

#### Attachments

• Planimetry Problem 99.jpg
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So I take i this is a true or false question?

If so, I'm not sure what the question means by "fit inside." If it means one has to be smaller than the other than I think you're right. If "fit inside" means their vertices and sides line up, I think you have an issue with your answer since the triangles are also congruent.

I just uploaded a picture to make it clearer.

Right. If you decrease the vertex angle but keep the leg length the same, you should be able to use the "greater angle opposes the greater side" property you mentioned to show the new triangle would pop out the bottom of the original.

Yes, that makes sense. However, what I want to pove is that if the base and height of a right triangle are inside another then the hypotenuses cannot be congruent. If I can prove that then the isosceles triangles can be argumented to be case of two pairs of juxtaposed right triangles

@Born, it sounds like you are still trying to use the logic based on the Pythagorean theorem. You are given the fact that your hypotenuses are congruent. Based on the theorems you listed in post 1, your best argument would be based upon angle measure.
Using the juxtaposed right triangle line of thinking, you should be able to list out the possible cases: 1) increase top angle, 2) decrease top angle -- Since the vertex and lateral sides are congruent, you only have a short argument for both cases to say that neither cases can fit inside the original. Only the congruent triangle would fit on the original.

## 1. What is an isosceles triangle with congruent lateral sides?

An isosceles triangle is a triangle with two equal sides and two equal angles. When both of the equal sides are also congruent, it means that they have the same length.

## 2. How can you identify an isosceles triangle with congruent lateral sides?

An isosceles triangle with congruent lateral sides can be identified by its two equal sides and two equal angles. It can also be identified by its symmetry, as the triangle will have two identical halves when folded along the line of symmetry.

## 3. What is the relationship between the congruent lateral sides and the base of an isosceles triangle?

The congruent lateral sides of an isosceles triangle are always equal in length. The base of an isosceles triangle is the third side, and its length may be different from the lateral sides.

## 4. Can an isosceles triangle have congruent lateral sides but unequal angles?

Yes, an isosceles triangle can have congruent lateral sides but unequal angles. As long as two sides are equal in length, the triangle is still considered an isosceles triangle.

## 5. How is the perimeter of an isosceles triangle with congruent lateral sides calculated?

The perimeter of an isosceles triangle with congruent lateral sides can be calculated by adding the length of all three sides together. As the lateral sides are congruent, you can simply multiply the length of one side by 2 and add it to the length of the base to find the perimeter.