Types of Triangles: Identifying and defining elements

An acute-angled triangle is defined as a triangle where all three interior angles are less than $90^\circ$.

In examining the visual depiction of the triangle provided in the problem, we need to see if it appears to satisfy this property. The assessment relies on observing the triangle's structure shown in the drawing and noting any geometric indications suggesting angle types.

Given the information from the drawing, if all angles seem to satisfy the condition of being less than $90^\circ$, then by definition, the triangle is an acute-angled triangle.

Conclusively, the answer to whether the triangle is acute-angled based on provided visual assessment and inherent assumptions in its illustration is: Yes.

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.

As is known, a scalene triangle is a triangle in which each side has a different length.

According to the given information, this is indeed a triangle where each side has a different length.

Due to the presence of the 90 degree angle symbol we can determine that this is indeed a right-angled triangle.

In an isosceles triangle, there are three sides: two sides of equal length and one distinct side. Our task is to identify what the equal sides are called.

To address this, let's review the basic properties of an isosceles triangle:

• An isosceles triangle is defined as a triangle with at least two sides of equal length.
• The side that is different in length from the other two is usually called the "base" of the triangle.
• The two equal sides of an isosceles triangle are referred to as the "legs."

Therefore, each of the two equal sides in an isosceles triangle is called a "leg."

In our problem, we confirm that the correct terminology for these two equal sides is indeed "legs," distinguishing them from the "base," which is the unequal side. This aligns with both the typical definitions and properties of an isosceles triangle.

Thus, the equal sides in an isosceles triangle are known as legs.

In a right-angled triangle, there is one angle that equals 90 degrees, and the other two angles sum up to 180 degrees (sum of angles in a triangle)

Therefore, the sum of the two non-right angles is 90 degrees

$90+90=180$

In a right triangle, there are specific terms for the sides. The two sides that form the right angle are referred to as the legs of the triangle. To differentiate, the side opposite the right angle is called the hypotenuse, which is distinct due to being the longest side. Hence, in response to the problem, the sides forming the right angle are correctly identified as Legs.

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: $sides, main$.

It can be seen that all angles in the given triangle are less than 90 degrees.

In a right-angled triangle, there needs to be one angle that equals 90 degrees

Since this condition is not met, the triangle is not a right-angled triangle.

Let's analyze the problem to understand how the angles are defined in a right triangle.

A right triangle is defined as a triangle that has one angle equal to $90^\circ$. This is known as a right angle. Because the sum of all angles in any triangle must be $180^\circ$, the two remaining angles must add up to $90^\circ$ (i.e., $180^\circ - 90^\circ$).

In a right triangle, the right angle is always present, leaving the other two angles to be less than $90^\circ$ each. These angles are called acute angles. An acute angle is an angle that is less than $90^\circ$.

To summarize, the angle types in a right triangle are:

• One angle that is $90^\circ$ (a right angle).
• Two angles that are each less than $90^\circ$ (acute angles).

Given the choices, the description "Straight, sharp" correlates to the angle types in a right triangle, as "Straight" can be associated with the $90^\circ$ angle (though it's generally called a right angle) and "Sharp" correlates with acute angles.

Therefore, the correct aspect of the other two angles in a right triangle are straight (right) and sharp (acute), which matches the correct choice.

Therefore, the solution to the problem is Straight, sharp.

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.

The problem requires us to determine if the triangle with given side lengths is a scalene triangle, which means all sides must be different.

We start by verifying if these side lengths form a triangle using the triangle inequality theorem, which states for any triangle with sides $a$, $b$, and $c$:

• $a + b > c$
• $a + c > b$
• $b + c > a$

Denote the given side lengths as follows:
$a = 28$, $b = 28$, $c = 25$.

Check the triangle inequalities:
$28 + 28 = 56$ which is indeed greater than $25$.
$28 + 25 = 53$ which is greater than $28$.
$25 + 28 = 53$ which is again greater than $28$.

Since all inequalities hold, these sides indeed form a triangle.

Next, determine if it is a scalene triangle. A scalene triangle has all sides of different lengths.

In our case, $a = 28 = b$, and $c = 25$. The sides $a$ and $b$ are not distinct, hence the triangle is not scalene but isosceles.

Therefore, the triangle does not have all different sides.

Thus, the correct answer is: No.

The problem requires us to identify the side of a right triangle that is opposite to its right angle.
In right triangles, one of the most crucial elements to recognize is the presence of a right angle (90 degrees).
The side that is directly across or opposite the right angle is known as the hypotenuse. It is also the longest side of a right triangle.
Therefore, when asked for the side opposite the right angle in a right triangle, the correct term is the hypotenuse.

Selection from the given choices corroborates our analysis:

• Choice 1: Leg - In the context of right triangles, the "legs" are the two sides that form the right angle, not the side opposite to it.
• Choice 2: Hypotenuse - This is the correct identification for the side opposite the right angle.

Therefore, the correct answer is $\text{Hypotenuse}$.

The triangle with sides 4, 4, and 5 is not a triangle with different sides. Therefore, the answer is No.

To ascertain whether the triangle in the drawing is acute, we need to examine the orientation and notation within the visual representation. The drawing vividly illustrates a triangle featuring a small square at one of the angles, a universal sign indicating a right angle. A right angle measures $90^\circ$, rendering it impossible for the triangle to be classified as acute since an acute triangle requires all angles to be less than $90^\circ$.

Therefore, given the right angle in the drawing, the triangle cannot be an acute-angled triangle. Consequently, the correct choice is:

No (:

No

To determine if the given triangle is an equilateral triangle, we must compare the lengths of all three sides. An equilateral triangle requires all three sides to have the same length.

Given the side lengths:

• $AB = 7$
• $BC = 10$
• $CA = 9$

Step-by-step solution:

• Step 1: Compare $AB$ with $BC$. We see that $7 \neq 10$. Therefore, $AB \neq BC$.
• Step 2: Compare $AB$ with $CA$. We see that $7 \neq 9$. Therefore, $AB \neq CA$.
• Step 3: Since $AB \neq BC$ and $AB \neq CA$, the sides are not all equal.

Since the sides $AB$, $BC$, and $CA$ are not all equal, the triangle is not an equilateral triangle.

Thus, the solution to the problem is No.

To determine if the given triangle is a right triangle, we will analyze its drawing:

• The triangle is depicted with one side perfectly horizontal and another side perfectly vertical.
• These two sides form an angle with a vertex placed where the horizontal and vertical sides meet.
• The angle formed at this point must be $90^\circ$ since the horizontal and vertical lines are by definition perpendicular to each other.
• When a triangle has one angle equal to $90^\circ$, it confirms that the triangle is a right triangle.

Thus, based on the positioning of the triangle with respect to the horizontal and vertical axes, we can assert that the triangle is indeed a right triangle. This fulfills the condition required for the triangle to be classified as such.

Therefore, the solution to the problem is: Yes.

To determine if the given triangle is a right triangle, we will analyze its geometrical properties. In a right triangle, one of its angles must be $90^\circ$. The easiest method to identify a right triangle without specific numerical coordinates is to check if any of the angles form a right angle just by visual assessment or conceptual understanding; this method can use the Pythagorean theorem in reverse if sides are measure-known.

In this setting, instead of physical measurements or accessible labeled SVG points, only the geometrical visual approach is taken. If based on generalized drawing inspections, assuming there is no visually postulated straight 90-degree form visible without a numerical validation, it's assumed to not initially exhibit such requirements when no arithmetic sides are comparatively used.

The lack of specific side lengths that conform to the Pythagorean theorem implies that, without other noticed forms or vectors increment constructs delivering a forced angle view, the triangle doesn't conform to being considered right.

Therefore, the triangle in the drawing is not a right triangle based on this lack of definitional evidence when view or vertex distinctions are ensured.

The correct answer to the problem is No.

To determine if the given triangle is a scalene triangle, we examine the side lengths $10$, $10$, and $7$.

A triangle is classified as scalene if all three side lengths are different. Therefore, we need to check the equality between any pairs of the given side lengths:

• Check if $10 = 10$: Yes, they are equal.
• Check if $10 = 7$: No, they are not equal.
• Check if $7 = 10$: No, they are not equal.

Since the triangle has two sides of equal length ($10$ and $10$), it does not satisfy the condition for being a scalene triangle.

In conclusion, the triangle is not a scalene triangle because two of its sides are equal.

Therefore, the solution to the problem is No.

To determine if the given triangle is scalene, we must examine the sides: $8X$, $10X$, and $7X + 2X$.

• Step 1: Simplify the expression for the third side:
  $7X + 2X = 9X$.
  Thus, the sides of the triangle are $8X$, $10X$, and $9X$.
• Step 2: Compare the side lengths:
  - The first side is $8X$.
  - The second side is $10X$.
  - The third side is $9X$.
• Step 3: Check for distinctness of each side:
  Since $8X \neq 9X$, $9X \neq 10X$, and $8X \neq 10X$, all sides are indeed different.

Therefore, the triangle is scalene, as all sides have different lengths.

The solution to the problem is: Yes