Triangle Angles Practice Problems - Sum Theorem Exercises

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.

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 the straight line is the height of the triangle, we'll analyze its role within the triangle:

• Step 1: Observe the triangle and the given line. The triangle seems to be made of three sides and a vertical line within it.
• Step 2: Recall that the height of a triangle, in geometry, is defined as a perpendicular dropped from a vertex to the opposite side.
• Step 3: Examine the positioning of the line: The vertical line starts at one vertex of the triangle and intersects the base, appearing to be perpendicular.
• Step 4: Verify perpendicularity: Given that the line is shown as a clear vertical (and a small perpendicular indicator suggests perpendicularity), we accept this line as the height.
• Step 5: Conclude with verification that the line is effectively meeting the definition of height for the triangle in the diagram.

Therefore, the vertical line in the figure is indeed the height of the triangle.

Yes

To determine whether a plane angle can be found in a triangle, we need to understand what a plane angle is and compare it to the angles within a triangle.

• A plane angle is an angle formed by two lines lying in the same plane.
• In the context of geometry, angles found within a triangle are the interior angles, which are the angles between the sides of the triangle.
• Although the angles in a triangle are indeed contained within a plane (since a triangle itself is a planar figure), when referencing "plane angles" in geometry, we usually consider angles related to different geometric configurations beyond those specifically internal to defined planar shapes like a triangle.
• The term "plane angle" typically refers to the measurement of an angle in radians or degrees within a plane, but this doesn't specifically pertain to angles of a triangle.

Therefore, based on the context and usual geometric conventions, the concept of a "plane angle" is not typically used to describe the angles found within a triangle. Thus, a plane angle as defined generally in geometry is not found specifically within a triangle.

Therefore, the correct answer to the question is $\text{No}$.

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.