Question

determine an expression for \\(x\\) in terms of \\(a\\) based on the following diagram:

\\(x = 5a - 3\\)
\\(x = -a + 5\\)
\\(x = -5a + 3\\)
\\(x = a - 5\\)

Explanation:

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<reasoning>

Identify the given angles

We are given a triangle with two interior angles:

• One interior angle is \(3a - 4\).
• Another interior angle is \(2a + 1\).
• The exterior angle opposite to these two interior angles is \(x\).

Apply the Exterior Angle Theorem

Using the Exterior Angle Theorem, the measure of an exterior angle of a triangle is equal to the sum of the measures of its two remote interior angles.
\[
x = (3a - 4) + (2a + 1)
\]

Simplify the expression

Combine the like terms to find \(x\) in terms of \(a\):
\[
x = 3a + 2a - 4 + 1
\]
\[
x = 5a - 3
\]
</reasoning>

<answer>
<mcq-correct>(A) \(x = 5a - 3\)</mcq-correct>
<mcq-option>(B) \(x = -a + 5\)</mcq-option>
<mcq-option>(C) \(x = -5a + 3\)</mcq-option>
<mcq-option>(D) \(x = a - 5\)</mcq-option>
</answer>

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Answer:

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<reasoning>

Identify the given angles

We are given a triangle with two interior angles:

• One interior angle is \(3a - 4\).
• Another interior angle is \(2a + 1\).
• The exterior angle opposite to these two interior angles is \(x\).

Apply the Exterior Angle Theorem

Using the Exterior Angle Theorem, the measure of an exterior angle of a triangle is equal to the sum of the measures of its two remote interior angles.
\[
x = (3a - 4) + (2a + 1)
\]

Simplify the expression

Combine the like terms to find \(x\) in terms of \(a\):
\[
x = 3a + 2a - 4 + 1
\]
\[
x = 5a - 3
\]
</reasoning>

<answer>
<mcq-correct>(A) \(x = 5a - 3\)</mcq-correct>
<mcq-option>(B) \(x = -a + 5\)</mcq-option>
<mcq-option>(C) \(x = -5a + 3\)</mcq-option>
<mcq-option>(D) \(x = a - 5\)</mcq-option>
</answer>

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