Question

• $overleftrightarrow{tv}$ is parallel to $overleftrightarrow{wx}$. • the distance between $overleftrightarrow{tv}$ and $overleftrightarrow{wx}$ is always 6 units. • $overline{vw}$ is 10 units long. • $mangle 1 = 36.87^circ$ li was asked to find the measure of $angle 2$ and explain his reasoning. image of geometric figure with lines, right angles, and angles labeled 1 and 2 fill in the blanks in li’s solution. • we were given the distance between the parallel lines, so both $overline{uw}$ and $overline{yv}$ have a length of $square$ units. • $\triangle vuw$ and $\triangle wyv$ share a side, and we were given that this side is 10 units long. • both $\triangle vuw$ and $\triangle wyv$ are right triangles, so we can use the pythagorean theorem to find the lengths of $overline{vu}$ and $overline{wy}$. • $overline{vu}$ and $overline{wy}$ both have a length of $square$ units. • $\triangle vuw cong \triangle wyv$ by the side - side - side congruency postulate. • $mangle 2=square^circ$ since $angle 1$ and $angle 2$ are corresponding parts of congruent triangles.

Explanation:

Step1: Determine length of UW and YV

The distance between parallel lines \( \overleftrightarrow{TV} \) and \( \overleftrightarrow{WX} \) is given as 6 units. Since \( \overline{UW} \) and \( \overline{YV} \) represent the perpendicular distance between these parallel lines, their length is 6 units.

Step2: Calculate length of VU and WY

We know \( \overline{VW} = 10 \) units (hypotenuse of right triangle) and \( \overline{UW} = 6 \) units (one leg). Using the Pythagorean theorem \( a^2 + b^2 = c^2 \), where \( c = 10 \) and \( b = 6 \), we solve for \( a \) (length of \( \overline{VU} \) or \( \overline{WY} \)):
\( a^2 + 6^2 = 10^2 \)
\( a^2 + 36 = 100 \)
\( a^2 = 100 - 36 = 64 \)
\( a = \sqrt{64} = 8 \) units.

Step3: Determine measure of \( \angle 2 \)

Since \( \triangle VUW \cong \triangle WYV \) (SSS congruence), corresponding angles \( \angle 1 \) and \( \angle 2 \) are equal. Given \( m\angle 1 = 36.87^\circ \), so \( m\angle 2 = 36.87^\circ \).

Answer:

• First blank: 6
• Second blank: 8
• Third blank: 36.87