Acute Angles - Examples, Exercises and Solutions

To solve this problem, we'll follow these steps:

• Step 1: Identify the angle measure provided.
• Step 2: Match the angle measure to the corresponding type of angle.
• Step 3: Choose the correct type of angle from the multiple-choice options.

Let's break down the process:

Step 1: The problem specifies that $\angle \text{ABC} = 90^\circ$.

Step 2: Recall the definitions of angle types. A right angle is defined as an angle that measures exactly 90 degrees.

Step 3: Out of the provided choices, select the one that represents a right angle.
- Right angle: $\angle = 90^\circ$ (Correct Choice)
- Acute angle: $\angle < 90^\circ$
- Obtuse angle: $\angle > 90^\circ$ and $\angle < 180^\circ$
- Flat angle: $\angle = 180^\circ$

Thus, the conclusion is that $\angle \text{ABC}$ is a Right angle.

To determine the type of angle $\angle \text{ABC}$, we will consider the following:

• Step 1: Observe the positions of points A, B, and C, which are depicted in the diagram as lying on a straight line.
• Step 2: Recognize that when three points lie on a straight line and a central point (B here) is between two others (A and C), it forms a flat angle.
• Step 3: Recall that a flat angle is defined as an angle that measures $180^\circ$, which is the total rotation a line undergoes around a single point.

Visual inspection of the diagram confirms that points A, B, and C create a straight line. Hence, the angle $\angle \text{ABC}$ must be a flat angle.

Therefore, the angle $\angle \text{ABC}$ is a flat angle.

To determine what kind of angle $\angle \text{ABC}$ is, let's examine the given information: the angle is less than $90^\circ$.

• Step 1: Recall the types of angles based on their measurements:
• An acute angle measures less than $90^\circ$.
• A right angle measures exactly $90^\circ$.
• An obtuse angle measures greater than $90^\circ$ but less than $180^\circ$.
• A flat angle measures exactly $180^\circ$.
• Step 2: Match the given information with these definitions.

Since $\angle \text{ABC}$ measures less than $90^\circ$, it fits the definition of an acute angle.

Therefore, the angle $\angle \text{ABC}$ is an acute angle.

A right angle is equal to 90 degrees.

In diagrams (a) and (c), we can observe that the angle symbol is a symbol representing an angle that equals 90 degrees.

By definition, an obtuse angle is an angle that is greater than 90 degrees. We can observe that in one drawing there is an angle of 90 degrees and therefore it is not an obtuse angle, the other two angles are less than 90 degrees meaning they are also not obtuse, they are acute angles.

Therefore, none of the answers is correct.

To solve this problem, we need to recall the definition of a right angle.

A right angle is defined as an angle that measures exactly $90^\circ$. This is a standard definition found in geometry, describing an angle formed by the intersection of two perpendicular lines.

Given the multiple-choice options, we need to identify which one corresponds to a right angle:

• A right angle measures $90^\circ$.

Let's examine the provided choices:

1. Angle of 180°.
2. Minor angle of 90°.
3. Angle of 90°.
4. Angle greater than 90°.

Choice 3, "Angle of 90°," directly matches the definition of a right angle.

Therefore, the correct answer is Angle of 90°.

To solve this problem, let's begin by understanding what a flat angle is:

• A flat angle is an angle that measures exactly $180^\circ$.
• Visually, you can think of a flat angle as a straight line, where the two arms of the angle lie on opposite sides of the vertex, forming a straight angle.
• It is larger than a right angle, which measures $90^\circ$, and exactly twice the measure of a right angle.
• Flat angles are often used to understand the concept of supplementary angles, where two angles sum up to create a flat angle of $180^\circ$.
• None of the sides of a flat angle are considered to turn; they continue in a straight direction.

Among the given choices, the option that correctly defines a flat angle is the one that states, "Angle of 180°."

Therefore, the solution to the problem is that a flat angle is an angle of $180^\circ$.

The definition of an acute angle is an angle that is smaller than 90 degrees.

Since an angle that equals 90 degrees is a right angle, the statement is true.

One of the properties of a rectangle is that all its angles are right angles.

Therefore, it is not possible for an angle to be acute, that is, less than 90 degrees.

To solve this problem, we need to identify what type of triangle aligns with having both given angles, Angle A and Angle B, less than $90^\circ$.

• Step 1: Understand that for any triangle, the sum of the internal angles is always $180^\circ$.
• Step 2: Since both Angle A and Angle B are less than $90^\circ$, they are acute. A triangle with two acute angles implies that the third angle should also be acute because all angles should sum up to less than $180^\circ$.
• Step 3: We need to examine available options to determine if any comply with these properties of a triangle.

Now, let's analyze the given choices:

• Choice 1 shows a triangle with a right angle, which contradicts the condition that both Angle A and Angle B are less than $90^\circ$.
• Choice 2 explicitly indicates that none of the options are correct, suggesting no triangle fits the conditions given in the problem statement.
• Choice 3, similarly to Choice 1, can't have two angles being less than $90^\circ$ if one is a right angle.
• Choice 4 again has a right angle, contradicting the initial condition.

All given diagrammatic options have a right angle (based on the SVG descriptions or their right-angled appearance), which directly violates the condition of both Angle A and Angle B being acute.

Therefore, the most appropriate answer is: None of the options.

Let's note in which of the triangles angle B forms a right angle, meaning an angle of 90 degrees.

In answers C+D, we can see that angle B is smaller than 90 degrees.

In answer A, it is equal to 90 degrees.

In this problem, we need to determine which angle, $\angle C$ or $\angle A$, is larger in a scalene triangle with acute angles. However, we lack essential information such as specific side lengths or individual angle measurements, which would be necessary to apply triangle inequality or other angle-side relationships. The provided information is not sufficient to definitively compare the angles.

Given that no specific numeric information or other additional data to differentiate the angles is available, there is inherently no possible conclusion.

Therefore, the solution to this problem is: There is no way to know.

In an isosceles triangle, two sides are equal, meaning the angles opposite those sides are equal. Given that $∢A$ is the predominant (largest) angle, it follows that sides $AB$ and $AC$ are equal (assuming $∢A$ is opposite these sides, based on typical isosceles configuration). Therefore, the angles opposite these sides, $∢B$ and $∢C$, must be equal.

Applying the property of equal angles in an isosceles triangle:

• The sum of the angles in a triangle is always 180°.
• If $∢A$ is the largest angle, then $∢B + ∢C = 180° - ∢A$.
• Since $∢B = ∢C$ in an isosceles triangle, we can state $2∢B = 180° - ∢A$ leading to each angle $∢B = ∢C = \frac{180° - ∢A}{2}$.

Therefore, both angles $∢B$ and $∢C$ are equal.

The correct and final conclusion is: $∢C=∢B$.

To solve this problem, we need to compare $\angle B$ and $\angle A$ in an obtuse triangle ABC.

A triangle is classified as obtuse when one of its angles is greater than $90^\circ$. In such a triangle, the largest angle is the obtuse angle.

Without loss of generality, if we consider any angle of the triangle, say $\angle C$, to be the obtuse angle, it must be that $\angle C > 90^\circ$. This makes $\angle C$ the largest angle.

Given the angle sum property of triangles ($\angle A + \angle B + \angle C = 180^\circ$), the sum of the two non-obtuse angles ($\angle A$ and $\angle B$) must be less than $90^\circ$, hence ensuring $\angle C$ remains the largest.

Since $\angle B$ and $\angle A$ must both be less than $90^\circ$, and the problem requires determining which is larger without any specific constraints on $\angle C$, we observe:

• If $\angle C$ is indeed obtuse, then $\angle A$ and $\angle B$ must add up to less than $90^\circ$, leading to $\angle B$ generally being greater than $\angle A$ under typical conditions unless otherwise specified.
• This result denotes that $\angle B$ being comparably larger than $\angle A$ unless specified otherwise by additional conditions, which are absent here.

Therefore, $\angle B > \angle A$.

Hence, in the context of the problem's provided choices and lacking other conditions, the solution is $\angle B > \angle A$.

Thus, the larger angle is $\angle B > \angle A$.

In this problem, we need to determine which angle is larger between $∢B$ and $∢A$ in the equilateral triangle $\triangle ABC$.

Let's start by recalling what an equilateral triangle is. In an equilateral triangle, all three sides have equal length, and consequently, all three internal angles are of equal measure. This is a fundamental property of equilateral triangles.

Since $\triangle ABC$ is equilateral, we know that each angle, including $∢B$ and $∢A$, measures $60^\circ$. This is because the sum of internal angles in any triangle is $180^\circ$, and in an equilateral triangle, this total is divided equally among the three angles. Thus:
$\begin{aligned} ∢A &amp;= ∢B = ∢C = \frac{180^\circ}{3} = 60^\circ. \end{aligned}$

Since both $∢A$ and $∢B$ are $60^\circ$, neither angle is larger than the other; they are equal.

This means that the correct statement regarding their measures is that $∢A = ∢B$.

Thus, according to the choices provided, the correct answer is:

Choice 4: $∢A = ∢B$.