Parts of a Triangle: True / false

To determine if AD is the height of triangle ABC, we start by recalling the definition of a height in a triangle. A height is a perpendicular line segment from a vertex to the line containing the opposite side. In triangle geometry, for AD to be considered a height, it must be perpendicular to the line BC.

The given diagram shows that AD is indeed perpendicular to BC, as denoted by the perpendicular symbol (the small square at the intersection indicating a $90^\circ$ angle). This matches the definition of a height in a triangle, which is a line drawn perpendicular from a vertex (in this case, vertex A) to the line containing the opposite side (here, BC).

Since AD meets this criterion of being perpendicular to the opposite side, we can conclusively state that AD is indeed the height of the triangle ABC. Thus, the statement "AD is the height in the triangle ABC" is true.

Therefore, the solution to the problem is True.

To solve this problem, we will verify the information provided in the diagram and check if it suffices to calculate the measure of $\angle CAB$.

Firstly, consider the given data in the diagram:

• The only provided information is the measure of angle $\angle ABC = 76^\circ$.
• There are no side lengths, no additional angle measures, and no indication of the triangle being isosceles or equilateral.

Without further information, determining the measure of $\angle CAB$ is not possible:

• The sum of angles in any triangle is $180^\circ$. This requires knowing at least two angle measures to find the third.
• Only one angle measure is given, and no side lengths are known, making it impossible to calculate $\angle CAB$ directly or indirectly.

Therefore, the claim that the size of angle $\angle CAB$ can be calculated using the diagram is False.

To determine if AD is the median of triangle ABC, we recall that a median connects a vertex to the midpoint of the opposite side. In this scenario, we would need to verify if point D is indeed the midpoint of side BC.

As a median divides the opposite side into two equal halves, our task would be to confirm the equality of segments BD and DC with available information or measurements.

However, the problem does not provide specific measurements, coordinates, or other information necessary to confirm that D is the midpoint of BC. Without this critical information, it is impossible to ascertain whether AD is a median.

Therefore, the conclusion is that the statement about AD being a median cannot be determined with the given data.

Thus, the correct answer to the problem is: Impossible to determine.

To determine whether AD is the height of triangle ADB, we must inspect whether segment AD is perpendicular to the base BD. By definition, the height of a triangle from a vertex is a line segment perpendicular to the line containing the opposite side. The diagram describes that AD is a vertical line, indicating a perpendicular relationship to the horizontal line BC. Thus, within triangle ADB, AD perfectly aligns as the height from vertex A to the base BD.

Therefore, the statement is True.

Given that the angles alpha and beta are on the same straight line and given that they are adjacent angles. Together they are equal to 180 degrees and the statement is true.

In this problem, we are given that $BD$ is the median of triangle $\triangle ABC$ and that $BD$ is half the length of side $AC$. We want to determine if $\triangle ABC$ is a right triangle.

By the properties of triangles, if the median ($BD$) from the vertex to the hypotenuse ($AC$) of a triangle is half the length of the hypotenuse ($AC$), then the triangle is right-angled.

The key property here is that in a right triangle, the median to the hypotenuse is half the hypotenuse. This median property is a unique characteristic for right triangles.

Thus, since $BD = \frac{1}{2}AC$ and $BD$ is the median, we conclude that triangle $\triangle ABC$ is indeed a right triangle.

Therefore, the solution to the problem is: Yes, $\triangle ABC$ is a right triangle.