Question

1. use the relationships in the diagrams below to solve for x, if possible. if it is not possible, state how you know. if it is possible, justify your solution by stating which geometric relationships you use.

a.
2x + 3°
3x - 2°
x + 8°
b.
4x + 18°
6x - 28°
c.
56°
56°
56°
180 - 2(56)=
178(56)=

Explanation:

Step1: Use angle - sum property of a triangle

The sum of interior angles of a triangle is 180°. So for the first triangle, we have the equation \((2x + 3)+(3x-2)+(x + 8)=180\).
Combining like - terms: \(2x+3x+x+3 - 2+8 = 180\), which simplifies to \(6x+9 = 180\).
Subtract 9 from both sides: \(6x=180 - 9=171\).
Divide both sides by 6: \(x=\frac{171}{6}=28.5\).

Step2: Use vertical - angle property

Vertical angles are equal. In the second diagram, \(4x + 18=6x-28\).
Subtract \(4x\) from both sides: \(18=6x-4x - 28\), which simplifies to \(18 = 2x-28\).
Add 28 to both sides: \(18 + 28=2x\), so \(46 = 2x\).
Divide both sides by 2: \(x = 23\).

Step3: Analyze the third diagram

In the third diagram, we have an isosceles triangle. The base angles of an isosceles triangle are equal. But we don't have enough information related to \(x\) to form an equation to solve for \(x\). So it is not possible to solve for \(x\).

Answer:

a. \(x = 28.5\)
b. \(x = 23\)
c. Not possible to solve for \(x\)