Question

100 POINTS WORTH PLEASE ANSWER I NEED BETTER THAN C 78%

Answer 1

Answer (with explanation):

1. Given

2. UY ≅ WY

3. Definition of Midpoint

4(statement). ∠UYX ≅ ∠VYW

4(reason). Side-Angle-Side SAS

5. Vertical Angle Theorem

in most proofs, (almost ALL the proofs) the first reason is always given.

you need given before you do anything.

on 2., the reason is Definition of Midpoint so you can tell it's UY is congruent to WY (uses the theorem)

on 3., it's the other way around and similar to 2. so the answer is definitely Definition of a Midpoint

on 4. statement and reason, you're side UY, angle Y, and side YX to prove that side VY, angle Y, and side YW are congruent. so, use the ∠UYX ≅ ∠VYW with the SAS

and on 5., you use vertical angles to say that ∠UYX reflects to ∠VYW so they are the same (aka congruent)

Answer 2

`\large\begin{array}l\cline{1-2}\sf Statement&\sf Reason\\\cline{1-2}1.\;\textsf{$Y$ is the midpoint of $UW$ and $VX$}&1.\;\sf Given\\\cline{1-2}2.\;UY \cong WY & 2.\; \sf De\:\!finition\;of\;a\;midpoint\\\cline{1-2}3.\;VY \cong XY&3.\; \sf De\:\!finition\;of\;a\;midpoint\\\cline{1-2}4.\;\angle UYX \cong \angle VYW&4.\; \sf Vertical\;Angle\;Theorem\\\cline{1-2}5.\; \triangle UYX \cong \triangle VYW &5.\;\sf Side-Angle-Side\;(SAS)\\\cline{1-2}\end{array}`

Explanation:

Midpoint

The point that is halfway between the endpoints of the line segment.

Side-Angle-Side (SAS) Congruence Theorem

If two sides and the included angle in one triangle are congruent to two sides and the included angle in another triangle, the triangles are congruent.

Vertical Angles Theorem

When two straight lines intersect, the opposite vertical angles are congruent.

`\large\begin{array}\cline{1-2}\sf Statement&\sf Reason\\\cline{1-2}1.\;\textsf{$Y$ is the midpoint of $UW$ and $VX$}&1.\;\sf Given\\\cline{1-2}2.\;UY \cong WY & 2.\; \sf De\:\!finition\;of\;a\;midpoint\\\cline{1-2}3.\;VY \cong XY&3.\; \sf De\:\!finition\;of\;a\;midpoint\\\cline{1-2}4.\;\angle UYX \cong \angle VYW&4.\; \sf Vertical\;Angle\;Theorem\\\cline{1-2}5.\; \triangle UYX \cong \triangle VYW &5.\;\sf Side-Angle-Side\;(SAS)\\\cline{1-2}\end{array}`