Question

the diagram contains contradictory information. explain the contradiction.
a. since ∠deh and ∠ghj are consecutive interior angles, they are supplementary.
therefore, 90 - 3x + 3x - 90 = 180.
when you solve for x, you get 0 = 180. this equality is not valid.
b. since ∠deh and ∠ghj are consecutive interior angles, they are congruent. therefore,
90 - 3x = 3x - 90.
when you solve for x, you get x = 30.
this would make m∠deh and m∠ghj = 0°.
an angle cannot measure 0°, because it would not exist.
c. since ∠deh and ∠ghj are corresponding angles, they are supplementary. therefore,
90 - 3x + 3x - 90 = 180.
when you solve for x, you get 0 = 180.
this equality is not valid.
d. since ∠deh and ∠ghj are corresponding angles, they are congruent. therefore,
90 - 3x = 3x - 90.
when you solve for x, you get x = 30.
this would make m∠deh and m∠ghj = 0°.
an angle cannot measure 0°.

Explanation:

Step1: Analyze the property of corresponding angles

Corresponding angles are congruent. So, if \(\angle DEH\) and \(\angle GHJ\) are corresponding angles, then \(m\angle DEH=m\angle GHJ\). That is \(90 - 3x=3x - 90\).

Step2: Solve the equation \(90 - 3x=3x - 90\)

Add \(3x\) to both sides: \(90=6x - 90\).
Add \(90\) to both sides: \(180 = 6x\).
Divide both sides by \(6\): \(x = 30\).

Step3: Calculate the measure of the angles

Substitute \(x = 30\) into \(90-3x\) and \(3x - 90\).
For \(90-3x\): \(90-3\times30=90 - 90=0\).
For \(3x - 90\): \(3\times30-90=90 - 90=0\).
Since an angle with measure \(0^{\circ}\) does not exist in the geometric sense (an angle is formed by two non - collinear rays with a common vertex), this is a contradiction.

Answer:

D. Since \(\angle DEH\) and \(\angle GHJ\) are corresponding angles, they are congruent. Therefore, \(90 - 3x=3x - 90\). When you solve for \(x\), you get \(x = 30\). This would make \(m\angle DEH\) and \(m\angle GHJ=0^{\circ}\). An angle cannot measure \(0^{\circ}\).