Types of Aangles (Right, Acute, Obtuse, Flat): Using angles in a triangle

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:
$\begin{aligned} ∢A &= ∢B = ∢C = \frac{180^\circ}{3} = 60^\circ. \end{aligned}$

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: $∢A = ∢B$.

In a right triangle $\triangle ABC$, angle $∢C$ is the right angle, which equals 90°. Therefore, the other two angles $∢A$ and $∢B$ must sum to 90°.

For any right triangle, if one acute angle is larger, the opposite side (leg) is longer. Conversely, if an angle is smaller, the opposing leg must be shorter.

Based on the triangle's diagram and fundamental right triangle properties, it appears that angle $∢A$ is larger than angle $∢B$, given that larger angles oppose longer sides.

This conclusion follows from practical geometry, where angle recognition and placement determine relative size.

Therefore, we can conclude that $∢A$ is greater than $∢B$, or $∢A > ∢B$.