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 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 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 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.

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.

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.

To determine if angles $X$ and $Y$ are alternate angles, let's analyze the configuration:

Step 1: Identify the Transversal:
The line labeled in orange cuts across the two parallel lines $AB$ and $CD$. This line acts as a transversal.

Step 2: Locate Angles $X$ and $Y$:
Angle $X$ is situated between lines $AB$ and the transversal. Angle $Y$ is between $CD$ and the transversal, but not in symmetric opposite with respect to the transversal line.

Step 3: Analyze Relative Positioning:
For $X$ and $Y$ to be alternate interior angles, they must lie between the parallel lines and on opposite sides of the transversal. Since both angles $X$ and $Y$ are not on alternate sides of the transversal line, they do not fit the definition of alternate angles.

Conclusion:
Since $X$ and $Y$ do not lie on opposite sides of the transversal and between the parallel lines, they are not alternate interior angles.

Therefore, the statement is False.

To determine if angles X and Y are corresponding angles, we need to consider the geometry involved.

Given that lines AB and CD are parallel, a transversal (a third line intersecting both AB and CD) creates multiple angles at the intersection points.

Corresponding angles are angles that are in the same relative position at each intersection where a straight line crosses two others. In other words, corresponding angles are matching angles that appear in similar locations relative to their parallel lines and the transversal.

In the problem's context, we look for angles X and Y, and analyze their relative positioning. By inspecting their placement:

• Identify the transversal which intersects both parallel lines AB and CD, creating angles at each intersection with these lines.
• Locate angle X created at the intersection of the transversal with line AB, and angle Y formed at the intersection of the transversal with line CD.
• By observation, angles X and Y are in the same relative position concerning the parallel lines and the transversal, hence they are corresponding angles.

By the Corresponding Angles Postulate, since AB || CD, angles X and Y must be equal, confirming they are indeed corresponding.

Thus, the statement that X and Y are corresponding angles is True.

To determine if $X$ and $Y$ are alternate angles, let's first identify the necessary components of the diagram:

• Lines $AB$ and $CD$ are parallel, stated by $AB \parallel CD$.
• There is a transversal intersecting both parallel lines $AB$ and $CD$.
• Angles $X$ and $Y$ are formed by this intersection.

According to the alternate interior angles theorem, when a transversal crosses two parallel lines, each pair of alternate interior angles is equal. Alternate angles appear on opposite sides of the transversal and between the two lines.

In the given diagram:
- Angle $X$ appears below point $B$ where the transversal intersects $AB$.
- Angle $Y$ appears above point $C$ where the transversal intersects $CD$.
These angles are formed on opposite sides of the transversal and between the lines $AB$ and $CD$, fulfilling the condition for alternate angles.

Therefore, $X$ and $Y$ are indeed alternate angles according to the given conditions.

The conclusion is that the statement "X and Y are alternate angles" is True.