Exterior angles of a triangle - Examples, Exercises and Solutions

We must first add the three angles to see if they equal 180 degrees:

$30+60+90=180$

The sum of the angles equals 180, therefore they can form a triangle.

We add the three angles to see if they are equal to 180 degrees:

$56+89+17=162$

The sum of the given angles is not equal to 180, so they cannot form a triangle.

We add the three angles to see if they are equal to 180 degrees:

$90+115+35=240$
The sum of the given angles is not equal to 180, so they cannot form a triangle.

In an isosceles triangle, the median to the base is also the height to the base.

That is, side AD forms a 90° angle with side BC.

That is, two right triangles are created.

Therefore, angle ADC is equal to 90 degrees.

Remember that an acute angle is smaller than 90 degrees, an obtuse angle is larger than 90 degrees, and a straight angle equals 180 degrees.

Since the lines are perpendicular to each other, the marked angles are right angles each equal to 90 degrees.

An altitude in a triangle is the segment that connects the vertex and the opposite side, in such a way that the segment forms a 90-degree angle with the side.

If we look at the image it is clear that the above theorem is true for the line AE. AE not only connects the A vertex with the opposite side. It also crosses BC forming a 90-degree angle. Undoubtedly making AE the altitude.

Let's remember the definition of height of a triangle:

A height is a straight line that descends from the vertex of a triangle and forms a 90-degree angle with the opposite side.

The sides that form a 90-degree angle are sides AB and BC. Therefore, the height is AB.

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

$90+90=180$

The sum of angles in a triangle is 180 degrees. Since two angles of 90 degrees equal 180, a triangle can never have two right angles.

Every triangle has 3 sides. First let's go over the triangle on the left side:

Its sides are: AB, BC, and CA.

This means that in this triangle, side EC does not exist.

Let's then look at the triangle on the right side:

Its sides are: ED, EF, and FD.

This means that in this triangle, side EC also does not exist.

Therefore, EC is not a side in either of the triangles.

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

To determine if the statement that "the sum of the adjacent angles is 180" is true, follow these steps:

• Step 1: Define Adjacent Angles
  

  Adjacent angles are two angles that have a common vertex and a common side but do not overlap. In geometry, when these angles form a straight line, they are known as a linear pair.

• Step 2: Apply the Linear Pair Theorem
  

  The Linear Pair Theorem states that if two angles are adjacent and form a linear pair (i.e., the non-common sides form a straight line), then these angles are supplementary. This means that their sum is $180^\circ$.

• Step 3: Conclusion
  

  Therefore, when adjacent angles form a linear pair on a straight line, their sum is indeed $180^\circ$.

This validates the statement that "the sum of the adjacent angles is 180" for linear pairs, making the statement True.

This corresponds to the answer choice stating: True.

Remember that the sum of angles in a triangle is equal to 180.

In an equilateral triangle, all sides and all angles are equal to each other.

Therefore, we will calculate as follows:

$x+x+x=180$

$3x=180$

We divide both sides by 3:

$x=60$