Parts of a Triangle: Identifying and defining elements

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.

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.

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 geometry, the distance or length of a line segment between two points is the same, regardless of the direction in which it is measured. Consequently, the segments denoted by $BC$ and $CB$ refer to the same segment, both indicating the distance between points B and C.

Hence, the statement "BC = CB" is indeed True.

In order to determine if segment CB is a side of one of the triangles, let's start by identifying the triangles and their corresponding vertices from the given diagram:

• Triangle 1 has vertices labeled as A, B, C.
• Triangle 2 has vertices labeled as D, E, F.

Now, to decide if CB is a side, we need to check if a line segment exists between points C and B in any of these triangles.

Upon examining the points:

• Point C is present in triangle 1.
• Point B is also present in triangle 1.
• The line segment connecting B and C is visible, forming the base of triangle 1.

Therefore, segment CB is indeed a side of triangle ABC, confirming that the answer is Yes.

Thus, the solution to the problem is $\text{Yes}$.

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: $sides, main$.

In the given diagram, we need to determine the height of triangle $\triangle ABC$. The height of a triangle is defined as the perpendicular segment from a vertex to the line containing the opposite side.

Upon examining the diagram:

• Point $A$ is at the top of the triangle.
• The side $BC$ is horizontal, lying at the base.
• Line segment $AD$ is drawn from point $A$ perpendicularly down to the base $BC$ at point $D$. This forms a right angle at $D$ with line $BC$.

Therefore, line segment $AD$ is the perpendicular or the height of triangle $\triangle ABC$.

Consequently, the height of triangle $\triangle ABC$ is represented by the segment $AD$.

In the given problem, we have a triangle depicted with a specific line drawn inside it. The question asks if this line represents the height of the triangle. To resolve this question, we need to discern whether the line is perpendicular to one of the sides of the triangle when extended, as only a line that is perpendicular from a vertex to its opposite side can be considered the height.

The line in question is shown intersecting one of the sides within the triangle but does not form a perpendicular angle with any side shown or the ground (as is required for it to be the height of the triangle). A proper height would typically intersect perpendicularly either at or along the extended line of the opposite side from a vertex.

Therefore, based on the visual clues provided and the typical geometric definition of a height (or altitude) in a triangle, this specific line does not fit the criteria for being a height.

Thus, we conclude that the line depicted is not the height of the triangle. The correct answer is No.

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.

Let's solve the problem step-by-step.

• We first consider the two triangles given in the diagram. The vertices of the first triangle are labeled $A$, $B$, and $C$. The vertices of the second triangle are labeled $D$, $E$, and $F$.
• Identify the sides of the first triangle: Since the vertices are $A$, $B$, and $C$, the sides of the triangle are $AB$, $BC$, and $CA$.
• Identify the sides of the second triangle: With vertices $D$, $E$, and $F$, the sides are $DE$, $EF$, and $FD$.
• Now, we ascertain whether $BC$ is a side. Upon inspection, $BC$ is clearly the side connecting vertex $B$ and vertex $C$ in the first triangle.

Thus, we conclude that Yes, $BC$ is indeed a side of one of the triangles.

The solution to the problem is: Yes

Let us determine whether $DB$ is a side of triangle $ABC$. This verification depends greatly on understanding the arrangement and positioning of the points given within the triangle.

From the diagram, points $A$, $B$, and $C$ form a triangle because they create a closed figure with three lines connecting one another, typical of triangle sides. On examining point $D$ and the line $DB$, we notice that $D$ seems to be an internal point, potentially serving roles like that of a median or an angle bisector, but not forming an external side of triangle $ABC$.

Based on this understanding, line $DB$ fails to fulfil the requirement of connecting two of the vertices of the triangle directly. Hence, it's internal and is not counted as an external side of triangle $ABC$.

The conclusion based on the given options from the prompt: $DB$ is not a side of triangle $ABC$.

Since line segment DE does not correspond to a full side of any of the triangles present within the given geometry, we conclude that the statement “DE is a side in one of the triangles” is Not true.

The height of a triangle is defined as the perpendicular distance from a vertex to the line containing the opposite side (base). In this problem, we observe a vertical line segment drawn from a point on the base (horizontal line at the bottom of the triangle) to some level above the base. To determine if this line is a height, it must be perpendicular to the base and also reach to the opposite vertex of the triangle.

In the provided figure, the vertical line extends vertically from the base but does not connect to the opposite vertex of the triangle (at the top). Instead, it terminates at some intermediate point above the base. Since the line does not satisfy the full condition of being perpendicular and reaching an opposite vertex, it cannot be considered the height of this triangle.

Therefore, the given straight line is not the height of the triangle.

The correct and final answer is: No.

The problem asks us to confirm if AB is a side of triangle ADB.

Triangle ADB is defined by its vertices, A, D, and B. A triangle is formed when three vertices are connected by three sides.

• Identify vertices: The vertices of the triangle are A, D, and B.
• Identify sides: The triangle's sides should be AB, BD, and DA.
• Observe: From the provided diagram, AB connects vertices A and B.

Therefore, based on the definition of a triangle and observing the connection between components, side AB indeed is a part of triangle ADB.

This confirms that the statement is True.

The problem involves classifying the angle represented visually, which looks like a semicircle with a central axis drawn. This indicates an angle that spans half a complete circle.

A complete circle measures $360^\circ$, so half of it, represented by a semicircle, measures half of $360^\circ$, which is $180^\circ$.

The four primary classifications for angles are:

• Acute: Less than $90^\circ$
• Right: Exactly $90^\circ$
• Obtuse: Greater than $90^\circ$ but less than $180^\circ$
• Straight: Exactly $180^\circ$

Since the angle measures exactly $180^\circ$, it is classified as a straight angle.

Therefore, the type of angle given is Straight.

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.

The task is to determine if the segment $AD$ is a side of any of the given triangles. Based on the diagram, we have two distinct triangles:

• $\triangle ABC$: Formed by the points $A, B, C$.
• $\triangle DEF$: Formed by the points $D, E, F$.

For $\triangle ABC$, the sides are $AB, BC,$ and $CA$.

For $\triangle DEF$, the sides are $DE, EF,$ and $FD$.

In analyzing both triangles, we observe that:

• The side $AD$ is not listed as one of the sides of either triangle.

Thus, the conclusion is clear: AD is not a side of either triangle.

Therefore, the answer is No.

The task is to determine whether the line shown in the diagram serves as the height of the triangle. For a line to be considered the height (or altitude) of a triangle, it needs to be a perpendicular segment from a vertex to the line that contains the opposite side, often referred to as the base.

Let's analyze the diagram:

• The triangle is described by its vertices, forming a shape, and one side is the base. There's a line drawn from one vertex directed toward the opposite side.
• To be the height, this line must be perpendicular to the side it meets (the base).
• Though the figure does not explicitly show perpendicularity with a right angle mark, the line appears as a straight, direct connection from the vertex to the base. This is typically indicative of it being a height.
• Assuming typical geometric conventions and the common depiction of heights in diagrams, the line shows properties consistent with being perpendicular to the opposite side, thereby functioning as the height.

Based on the analysis, the line is indeed the height of the triangle. Thus, the answer is Yes.

Therefore, the solution to the problem is Yes.

The triangle's altitude is a line drawn from a vertex perpendicular to the opposite side. The vertical line in the diagram extends from the triangle's top vertex straight down to its base. By definition of altitude, this line is the height if it forms a right angle with the base.

To solve this problem, we'll verify that the line in question satisfies the altitude condition:

• Step 1: Identify the triangle's vertices and base. From the diagram, the base appears horizontal, and the vertex lies directly above it.
• Step 2: Check the nature of the line. The line is vertical when the base is horizontal, indicating perpendicularity.
• Conclusion: The vertical line forms right angles with the base, thus acting as the altitude or height.

Therefore, the straight line depicted is indeed the height of the triangle. The answer is Yes.

To determine if a triangle can have a right angle, consider the following explanation:

• Definition of a Right Angle: An angle is classified as a right angle if it measures exactly $90^\circ$.
• Definition of a Right Triangle: A right triangle is a type of triangle that contains exactly one right angle.
• According to the definition, a right triangle specifically includes a right angle. This is a well-established classification of triangles in geometry.

Thus, a triangle can indeed have a right angle and is referred to as a right triangle.

Therefore, the solution to the problem is Yes.