Question

1. solve for the values of x, showing all work. state the angle relationship used to find them.

a) b) c)
relationship: consecutive interior
relationship: alternate interior
relationship: alternate

Explanation:

Step1: Identify angle - relationship for part a

Since the angles are consecutive - interior angles, they are supplementary (sum to 180°). So, \(75+(5x - 10)=180\).

Step2: Simplify the equation for part a

Combine like - terms: \(75-10 + 5x=180\), which gives \(65 + 5x=180\).

Step3: Solve for x in part a

Subtract 65 from both sides: \(5x=180 - 65=115\). Then divide by 5: \(x=\frac{115}{5}=23\).

Step4: Identify angle - relationship for part b

The angles are alternate - interior angles, so they are equal. Thus, \(x + 15=80\).

Step5: Solve for x in part b

Subtract 15 from both sides: \(x=80 - 15=65\).

Step6: Identify angle - relationship for part c

The angles are alternate - exterior angles, so they are equal. So, \(x^{2}-12x = 45\).

Step7: Rearrange the equation for part c

Rewrite it as a quadratic equation: \(x^{2}-12x-45 = 0\).

Step8: Factor the quadratic equation for part c

\(x^{2}-12x - 45=(x - 15)(x + 3)=0\).

Step9: Solve for x in part c

Set each factor equal to zero: \(x-15 = 0\) gives \(x = 15\); \(x + 3=0\) gives \(x=-3\).

Answer:

a) \(x = 23\)
b) \(x = 65\)
c) \(x = 15\) or \(x=-3\)