Question

the figure below shows the angles formed by intersecting lines l and m. based on the figure, which of the following statements cannot be proven? a. (mangle1 - mangle4=mangle3 - mangle2) b. (mangle3 + mangle4 = 180^{circ}) c. (angle1congangle3) d. (mangle1 + mangle3=mangle2 + mangle4)

Explanation:

Step1: Recall angle - pair relationships

When two lines intersect, vertical angles are congruent and adjacent angles are supplementary. $\angle1$ and $\angle3$ are vertical angles, so $\angle1\cong\angle3$ (C is provable). Also, $\angle3$ and $\angle4$ are adjacent angles forming a linear - pair, so $m\angle3 + m\angle4=180^{\circ}$ (B is provable).

Step2: Analyze option A

Since $\angle1\cong\angle3$ and $\angle2\cong\angle4$ (vertical angles), we can rewrite $m\angle1 - m\angle4=m\angle3 - m\angle2$ as $m\angle1 - m\angle2=m\angle3 - m\angle4$ which is true because of the equality of vertical angles.

Step3: Analyze option D

There is no angle - pair relationship that would imply $m\angle1 + m\angle3=m\angle2 + m\angle4$. The sum of vertical angles does not have this property.

Answer:

D. $m\angle1 + m\angle3=m\angle2 + m\angle4$