Types of Triangles: Identifying and defining elements

The measure of angle C is 90°, therefore it is a right angle.

If one of the angles of the triangle is right, it is a right triangle.

As all the angles of a triangle are less than 90° and the sum of the angles of a triangle equals 180°:

$70+70+40=180$

The triangle is isosceles.

Given that in an obtuse triangle it is enough for one of the angles to be greater than 90°, and in the given triangle we have an angle C greater than 90°,

$C=107$

Furthermore, the sum of the angles of the given triangle is 180 degrees so it is indeed a triangle:

$107+34+39=180$

The triangle is obtuse.

Given that sides AB and AC are both equal to 9, which means that the legs of the triangle are equal and the base BC is equal to 5,

Therefore, the triangle is isosceles.

As we know that sides AB, BC, and CA are all equal to 6,

All are equal to each other and, therefore, the triangle is equilateral.

Since none of the sides have the same length, it is a scalene triangle.

Due to the presence of the 90 degree angle symbol we can determine that this is indeed a right-angled triangle.

In a right-angled triangle, there is one angle that equals 90 degrees, and the other two angles sum up to 180 degrees (sum of angles in a triangle)

Therefore, the sum of the two non-right angles is 90 degrees

$90+90=180$

As is known, a scalene triangle is a triangle in which each side has a different length.

According to the given information, this is indeed a triangle where each side has a different length.

It can be seen that all angles in the given triangle are less than 90 degrees.

In a right-angled triangle, there needs to be one angle that equals 90 degrees

Since this condition is not met, the triangle is not a right-angled triangle.

Let's note in which of the triangles angle B forms a right angle, meaning an angle of 90 degrees.

In answers C+D, we can see that angle B is smaller than 90 degrees.

In answer A, it is equal to 90 degrees.

In a right triangle, there are specific terms for the sides. The two sides that form the right angle are referred to as the legs of the triangle. To differentiate, the side opposite the right angle is called the hypotenuse, which is distinct due to being the longest side. Hence, in response to the problem, the sides forming the right angle are correctly identified as Legs.

The problem requires us to identify the side of a right triangle that is opposite to its right angle.
In right triangles, one of the most crucial elements to recognize is the presence of a right angle (90 degrees).
The side that is directly across or opposite the right angle is known as the hypotenuse. It is also the longest side of a right triangle.
Therefore, when asked for the side opposite the right angle in a right triangle, the correct term is the hypotenuse.

Selection from the given choices corroborates our analysis:

• Choice 1: Leg - In the context of right triangles, the "legs" are the two sides that form the right angle, not the side opposite to it.
• Choice 2: Hypotenuse - This is the correct identification for the side opposite the right angle.

Therefore, the correct answer is $\text{Hypotenuse}$.

In order to solve this problem, we need to understand the basic properties of an isosceles triangle.

An isosceles triangle has two sides that are equal in length, often referred to as the "legs" of the triangle. The angle formed between these two equal sides, which are sometimes referred to as the "sides", is called the "vertex angle" or sometimes more colloquially as the "main angle".

When considering the vocabulary of the given multiple-choice answers, choice 2: $sides, main$ accurately fills the blanks, as the angle formed between the two equal sides can indeed be referred to as the "main angle".

Therefore, the correct answer to the problem is: $sides, main$.

An isosceles triangle is one that has at least two sides of equal length. The angles opposite these two sides are known as the "base angles."
The side that is not equal to the other two is referred to as the "base" of the triangle. Thus, the "base angles" are the angles between each of the sides that are equal in length and the base.
Therefore, when we specify the angle in terms of its location or position, it is the angle between a "side" and the "base." This leads to the conclusion that the angle between the side and the base is the "base angle."

Therefore, the correct choice is Side, base.

To solve this problem, we need to understand what an isosceles triangle is and how its sides are labeled:

• In an isosceles triangle, there are two sides that have equal lengths. These are typically called the "legs" of the triangle.
• The third side, which is not necessarily of equal length to the other two sides, is known as the "base."

In terms of the problem, we want to determine the term used for the third side, which is the side that is not one of the two equal sides.

The correct term for the third side in an isosceles triangle is the "base." This is because the third side serves as a different function compared to the equal sides, which usually form the symmetrical parts of the triangle.

Among the given answer choices, choosing "Base" correctly identifies the third side of an isosceles triangle.

Therefore, the third side in an isosceles triangle is called the base.

Final Solution: Base

In an isosceles triangle, there are three sides: two sides of equal length and one distinct side. Our task is to identify what the equal sides are called.

To address this, let's review the basic properties of an isosceles triangle:

• An isosceles triangle is defined as a triangle with at least two sides of equal length.
• The side that is different in length from the other two is usually called the "base" of the triangle.
• The two equal sides of an isosceles triangle are referred to as the "legs."

Therefore, each of the two equal sides in an isosceles triangle is called a "leg."

In our problem, we confirm that the correct terminology for these two equal sides is indeed "legs," distinguishing them from the "base," which is the unequal side. This aligns with both the typical definitions and properties of an isosceles triangle.

Thus, the equal sides in an isosceles triangle are known as legs.

Since in the given triangle all angles are equal, all sides are also equal.

It is known that in an equilateral triangle the measure of the angles will always be equal to 60° since the sum of the angles in a triangle is 180 degrees:

$60+60+60=180$

Therefore, it is an equilateral triangle.

If we try to draw two obtuse angles and connect them to form a triangle (i.e: only 3 sides), we will see that it is not possible.

Therefore, the answer is no.

To demonstrate that triangle AED is isosceles, we must prove that its hypotenuses are equal or that the opposite angles to them are equal.

Given that angles ABC and ACB are equal (since they are equal opposite bisectors),

And since ED is parallel to BC, the angles ABC and ACB alternate and are equal to angles ADE and AED (alternate and equal angles between parallel lines)

Opposite angles ADE and AED are respectively sides AD and AE, and therefore are also equal (opposite equal angles, the legs of triangle AED are also equal)

Therefore, triangle ADE is isosceles.