Question

geometry
name:
date:
hour:
summative assessment unit 2
calculator (a)
benchmark two: graphing / visual representations

1. (g.co.c.9) use the figure below to find the value of x that makes line l parallel to line n.

a) show your work and explain how you know they are
parallel using an angle relationship.
b) *prove your answer using a different angle relationship.

Explanation:

Step1: Use consecutive interior angles theorem

If \(l\parallel n\), then \((4x + 10)+(2x+40)=180\) (consecutive interior angles are supplementary).

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Step2: Check using alternate - exterior angles

If \(x = \frac{65}{3}\), then \(10x-40=10\times\frac{65}{3}-40=\frac{650}{3}-40=\frac{650 - 120}{3}=\frac{530}{3}\) and \(2x + 40=2\times\frac{65}{3}+40=\frac{130}{3}+40=\frac{130+120}{3}=\frac{250}{3}\). Also, \(4x + 10=4\times\frac{65}{3}+10=\frac{260}{3}+10=\frac{260 + 30}{3}=\frac{290}{3}\).

Another way: If \(l\parallel n\), then \(10x-40=2x + 40\) (alternate - exterior angles are equal)

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Wait, there was a mistake above. Let's start over.

Since \(l\parallel n\), using the consecutive interior angles \((4x + 10)+(2x+40)=180\) (sum to \(180^{\circ}\))

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Correct: Since \(l\parallel n\), \(10x-40=2x + 40\) (alternate - exterior angles)

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Step3: Verify with consecutive interior angles

If \(x = 10\), then \(4x+10=4\times10 + 10=50\) and \(2x + 40=2\times10+40=60\). And \(10x-40=10\times10-40 = 60\). Also, \(4x + 10+(10x-40)=50 + 60=110\) (wrong, re - check the angle relationships)

Correct: Since \(l\parallel n\), \(4x + 10+10x-40=180\) (linear pair, but if we consider the transversal, the correct is:

Since \(l\parallel n\), \(4x+10=2x + 40\) (alternate - interior angles)

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Step4: Check with another relationship

If \(x = 15\), then \(10x-40=10\times15-40=110\), \(4x + 10=4\times15+10=70\), \(2x+40=2\times15 + 40=70\)

Using the linear - pair adjacent to \(10x-40\): The angle adjacent to \(10x - 40\) is \(180-(10x - 40)=220-10x\). If \(x = 15\), this angle is \(220-10\times15 = 70\) which is equal to \(2x + 40\) (corresponding angles)

Answer:

\(x = 15\)