Answered

Explore Westonci.ca, the premier Q&A site that helps you find precise answers to your questions, no matter the topic. Discover detailed answers to your questions from a wide network of experts on our comprehensive Q&A platform. Our platform offers a seamless experience for finding reliable answers from a network of knowledgeable professionals.

If isosceles triangle [tex]\(ABC\)[/tex] has a [tex]\(130^{\circ}\)[/tex] angle at vertex [tex]\(B\)[/tex], which statement must be true?

A. [tex]\(m_{\angle} A = 15^{\circ}\)[/tex] and [tex]\(m_{\angle} C = 35^{\circ}\)[/tex]
B. [tex]\(m_{\angle} A + m_{\angle} B = 155^{\circ}\)[/tex]
C. [tex]\(m_{\angle} A + m_{\angle} C = 60^{\circ}\)[/tex]
D. [tex]\(m_{\angle} A = 20^{\circ}\)[/tex] and [tex]\(m_{\angle} C = 30^{\circ}\)[/tex]

Sagot :

GinnyAnswer
To solve this problem, let's go through the known facts and reasoning step-by-step.

### Step 1: Understanding the properties of the triangle
We have an isosceles triangle [tex]\(ABC\)[/tex] with [tex]\( \angle B = 130^\circ \)[/tex]. In an isosceles triangle, two sides are equal, and thus two angles are equal. Let's denote the measures of these equal angles at vertices [tex]\(A\)[/tex] and [tex]\(C\)[/tex] as [tex]\(m_{\angle} A = x\)[/tex] and [tex]\(m_{\angle} C = x\)[/tex].

### Step 2: Sum of angles in a triangle
We know that the sum of the interior angles of any triangle is always [tex]\(180^\circ\)[/tex]. Therefore, we can write the equation:
[tex]\[ m_{\angle} A + m_{\angle} B + m_{\angle} C = 180^\circ \][/tex]

### Step 3: Substitute known values
We substitute [tex]\( m_{\angle} A = x \)[/tex], [tex]\( m_{\angle} B = 130^\circ \)[/tex], and [tex]\( m_{\angle} C = x \)[/tex] into the equation:
[tex]\[ x + 130^\circ + x = 180^\circ \][/tex]
[tex]\[ 2x + 130^\circ = 180^\circ \][/tex]

### Step 4: Solve for [tex]\( x \)[/tex]
To find [tex]\( x \)[/tex], we solve the equation:
[tex]\[ 2x = 180^\circ - 130^\circ \][/tex]
[tex]\[ 2x = 50^\circ \][/tex]
[tex]\[ x = 25^\circ \][/tex]

So, [tex]\( m_{\angle} A = 25^\circ \)[/tex] and [tex]\( m_{\angle} C = 25^\circ \)[/tex].

### Step 5: Verify each statement
Now let’s verify each provided statement based on these angle measures:

1. [tex]\( m_{\angle} A = 15^\circ \)[/tex] and [tex]\( m_{\angle} C = 35^\circ \)[/tex]:
- This statement is false because we found [tex]\( m_{\angle} A = 25^\circ \)[/tex] and [tex]\( m_{\angle} C = 25^\circ \)[/tex].

2. [tex]\( m_{\angle} A + m_{\angle} B = 155^\circ \)[/tex]:
- [tex]\( m_{\angle} A + m_{\angle} B = 25^\circ + 130^\circ = 155^\circ \)[/tex]
- This statement is true.

3. [tex]\( m_{\angle} A + m_{\angle} C = 60^\circ \)[/tex]:
- [tex]\( m_{\angle} A + m_{\angle} C = 25^\circ + 25^\circ = 50^\circ \)[/tex]
- This statement is false.

4. [tex]\( m_{\angle} A = 20^\circ \)[/tex] and [tex]\( m_{\angle} C = 30^\circ \)[/tex]:
- This statement is false because we found [tex]\( m_{\angle} A = 25^\circ \)[/tex] and [tex]\( m_{\angle} C = 25^\circ \)[/tex].

### Conclusion
Among the given statements, the one that must be true is:
[tex]\[ m_{\angle} A + m_{\angle} B = 155^\circ \][/tex]
We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. Get the answers you need at Westonci.ca. Stay informed by returning for our latest expert advice.