Isosceles Triangle ABC With A 130-Degree Angle Vertex B

In the realm of geometry, isosceles triangles hold a special place due to their unique properties and symmetrical nature. An isosceles triangle is defined as a triangle that has two sides of equal length. This equality of sides leads to another crucial characteristic: the angles opposite these equal sides are also equal. When we delve into problems involving isosceles triangles, understanding these fundamental properties is essential for finding solutions. This article explores a specific scenario involving an isosceles triangle ABC with a $130^{\circ}$ angle at vertex B. We will analyze the implications of this given information and determine which statement about the triangle's angles must be true. This exploration will not only reinforce our understanding of isosceles triangles but also enhance our problem-solving skills in geometry.

Before diving into the specifics of the problem, let's recap the key properties of isosceles triangles. These properties are the foundation for solving any problem related to these triangles. As previously mentioned, an isosceles triangle has two sides of equal length. These equal sides are often referred to as the legs of the triangle. The angle formed by these two equal sides is called the vertex angle, while the other two angles, opposite the equal sides, are known as the base angles. One of the most important properties to remember is that the base angles in an isosceles triangle are congruent, meaning they have the same measure. This property stems directly from the Side-Angle-Side (SAS) congruence theorem, which states that if two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, then the triangles are congruent. In the context of an isosceles triangle, the two sides and the included angle (the vertex angle) are congruent to themselves, leading to the congruence of the two base angles. Another crucial property, applicable to all triangles, is the angle sum property. This property states that the sum of the interior angles in any triangle is always $180^{\circ}$. This is a fundamental theorem in Euclidean geometry and is vital for calculating unknown angles when some angles are already known. In the case of an isosceles triangle, if we know the measure of the vertex angle, we can easily find the measure of the base angles by subtracting the vertex angle from $180^{\circ}$ and then dividing the result by 2, since the two base angles are equal. Understanding these properties allows us to approach problems involving isosceles triangles with a clear strategy and logical framework.

The problem presents us with an isosceles triangle, labeled ABC, where the angle at vertex B measures $130^{\circ}$. This piece of information is crucial because it immediately tells us the measure of one of the triangle's angles. The problem also provides several statements about the angles of the triangle, and our task is to determine which statement must be true. To solve this, we need to apply our knowledge of the properties of isosceles triangles, particularly the relationship between the sides and angles, and the angle sum property. The fact that angle B is $130^{\circ}$ is significant because it helps us deduce the possible measures of the other two angles, angles A and C. Since the sum of angles in a triangle is $180^{\circ}$, the remaining two angles must add up to $180^{\circ} - 130^{\circ} = 50^{\circ}$. Furthermore, because the triangle is isosceles, two of its sides are equal, and consequently, the angles opposite those sides are also equal. This means that either angles A and C are equal, or one of them is equal to angle B. However, since angle B is $130^{\circ}$, which is greater than $90^{\circ}$, it cannot be equal to either angle A or angle C because the sum of all angles in a triangle is only $180^{\circ}$. Therefore, angles A and C must be the equal angles. This deduction is a critical step in solving the problem because it narrows down the possibilities and allows us to focus on the specific relationships between the angles. The statements provided as options likely involve different combinations and relationships between angles A, B, and C, and by logically analyzing the given information, we can identify the statement that holds true for this particular isosceles triangle.

To effectively solve this problem, we need to delve deeper into the implications of having a $130^{\circ}$ angle at vertex B in an isosceles triangle ABC. The initial deduction that angles A and C must be equal is crucial. Since the sum of all angles in a triangle is $180^{\circ}$, and angle B is $130^{\circ}$, the remaining $50^{\circ}$ must be divided between angles A and C. Given that angles A and C are equal, each of these angles must measure half of $50^{\circ}$, which is $25^{\circ}$. This calculation is a direct application of the angle sum property and the property of equal base angles in an isosceles triangle. Now that we know the measures of all three angles—angle A is $25^{\circ}$, angle B is $130^{\circ}$, and angle C is $25^{\circ}$—we can evaluate the given statements to determine which one must be true. The statements likely involve various relationships between these angles, such as their individual measures or sums. For example, a statement might claim that angle A is $15^{\circ}$ and angle C is $35^{\circ}$, which we now know is false. Another statement might propose a relationship between the sum of two angles, such as angle A plus angle B. By substituting the calculated values into these statements, we can systematically check their validity. The correct statement will be the one that holds true based on the established angle measures. This analytical process highlights the importance of accurately applying geometric principles and performing careful calculations. By combining the properties of isosceles triangles with the angle sum property, we can confidently determine the correct answer and reinforce our understanding of triangle geometry.

Now that we have determined the measures of angles A, B, and C, which are $25^{\circ}$, $130^{\circ}$, and $25^{\circ}$ respectively, we can systematically evaluate the given answer choices to identify the statement that must be true. Let's consider each option:

A. $m \angle A = 15^{\circ}$ and $m \angle C = 35^{\circ}$

This statement is incorrect because we have already established that both angles A and C measure $25^{\circ}$. The given measures do not align with our calculations, making this option false.

B. $m \angle A + m \angle B = 155^{\circ}$

To check this statement, we add the measures of angles A and B: $25^{\circ} + 130^{\circ} = 155^{\circ}$. This sum matches the value given in the statement, making this option a potential correct answer.

C. $m \angle A + m \angle C = 60^{\circ}$

We add the measures of angles A and C: $25^{\circ} + 25^{\circ} = 50^{\circ}$. This sum does not equal $60^{\circ}$, so this statement is false.

D. $m \angle C = 90^{\circ}$

This statement is also incorrect because we know that angle C measures $25^{\circ}$, not $90^{\circ}$. This eliminates option D as a possibility.

After evaluating all the options, only statement B, which states that $m \angle A + m \angle B = 155^{\circ}$, holds true based on our calculations. This systematic evaluation demonstrates the importance of verifying each answer choice against the known information to arrive at the correct solution.

In conclusion, by applying the properties of isosceles triangles and the angle sum property, we successfully determined the measures of angles A and C in the given triangle ABC. The problem highlighted the significance of understanding the relationship between equal sides and equal angles in isosceles triangles, as well as the fundamental principle that the sum of angles in any triangle is $180^{\circ}$. Through careful analysis and calculation, we found that angle A and angle C each measure $25^{\circ}$. By evaluating the given statements, we identified that the statement $m \angle A + m \angle B = 155^{\circ}$ must be true. This exercise reinforces the importance of a systematic approach to problem-solving in geometry, where each piece of information is carefully considered and applied. The combination of theoretical knowledge and practical application is key to mastering geometric concepts and solving related problems effectively. The ability to dissect a problem, identify relevant properties, and perform accurate calculations is a valuable skill that extends beyond the realm of mathematics and into various aspects of logical thinking and reasoning.