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Given: In triangle [tex]\(ABC\)[/tex], [tex]\(m \angle A = 6x - 10\)[/tex], [tex]\(m \angle B = 2x\)[/tex], and [tex]\(m \angle C = 8x - 2\)[/tex].

Prove: [tex]\(x = 12\)[/tex]

Statement:
1. [tex]\(m \angle A = 6x - 10\)[/tex]
2. [tex]\(m \angle B = 2x\)[/tex]
3. [tex]\(m \angle C = 8x - 2\)[/tex]
4. [tex]\(m \angle A + m \angle B + m \angle C = 180^\circ\)[/tex]
5. [tex]\(6x - 10 + 2x + 8x - 2 = 180\)[/tex]
6. [tex]\(16x - 12 = 180\)[/tex]
7. [tex]\(16x = 192\)[/tex]
8. [tex]\(x = 12\)[/tex]

Reason:
1. Given
2. Given
3. Given
4. Triangle Angle Sum Theorem
5. Substitution
6. Simplification
7. Addition Property of Equality
8. Division Property of Equality



Answer :

GinnyAnswer
To prove that [tex]\( x = 12 \)[/tex], we need to follow a series of logical steps, using given information about the angles in the triangle and algebraic techniques.

### Statement and Reason Analysis:

1. Statement: [tex]\( m \angle A = 6x - 10 \)[/tex]
Reason: Given

2. Statement: [tex]\( m \angle B = 2x \)[/tex]
Reason: Given

3. Statement: [tex]\( m \angle C = 8x - 2 \)[/tex]
Reason: Given

4. Statement: [tex]\( m \angle A + m \angle B + m \angle C = 180 \)[/tex]
Reason: Triangle Angle Sum Theorem (The sum of the angles in a triangle is always [tex]\( 180^\circ \)[/tex]).

5. Statement: [tex]\( 6x - 10 + 2x + 8x - 2 = 180 \)[/tex]
Reason: Substitution (Substitute the given expressions for [tex]\( m \angle A \)[/tex], [tex]\( m \angle B \)[/tex], and [tex]\( m \angle C \)[/tex] into the sum from step 4).

6. Statement: [tex]\( 16x - 12 = 180 \)[/tex]
Reason: Combine like terms (Combine all the [tex]\( x \)[/tex] terms and constant terms on the left side).

7. Statement: [tex]\( 16x = 192 \)[/tex]
Reason: Addition Property of Equality (Add 12 to both sides of the equation to isolate the [tex]\( x \)[/tex] term).

8. Statement: [tex]\( x = 12 \)[/tex]
Reason: Division Property of Equality (Divide both sides of the equation by 16 to solve for [tex]\( x \)[/tex]).

Therefore, the reason for the final statement [tex]\( x = 12 \)[/tex] is the Division Property of Equality. This refers to the algebraic property that allows us to divide both sides of an equation by the same nonzero number without changing the equality.

Let's write it out explicitly for clarity:

Step-by-Step Solution:
1. [tex]\( m \angle A = 6x - 10 \)[/tex] (Given)
2. [tex]\( m \angle B = 2x \)[/tex] (Given)
3. [tex]\( m \angle C = 8x - 2 \)[/tex] (Given)
4. The sum of the angles in a triangle is [tex]\( 180^\circ \)[/tex]:
[tex]\[ m \angle A + m \angle B + m \angle C = 180 \][/tex]
[tex]\[ (6x - 10) + (2x) + (8x - 2) = 180 \][/tex]
5. Combine like terms:
[tex]\[ 16x - 12 = 180 \][/tex]
6. Add 12 to both sides of the equation:
[tex]\[ 16x = 192 \][/tex]
7. Divide both sides by 16:
[tex]\[ x = 12 \][/tex]

Thus, we have proven step by step that [tex]\( x = 12 \)[/tex].

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