Question

\overline{WU}\cong\overline{ZV}\ and\ \overline{WX} = \overline{YZ}. \angle U, \angle V are right angles, find congruence criteria used to prove that the triangles are congruent.

Congruence
ANSWERED
asked 2021-07-30
To find: congruent criteria used to prove that the triangles are congruent.
Given information: \(\displaystyle\overline{{{W}{U}}}\stackrel{\sim}{=}\overline{{{Z}{V}}}\ {\quad\text{and}\quad}\ \overline{{{W}{X}}}=\overline{{{Y}{Z}}}\). \(\displaystyle\angle{U},\angle{V}\) are right angles
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Expert Answers (1)

2021-07-31
Consider \(\displaystyle\triangle{W}{U}{Y},\triangle{Z}{V}{X}\)
\(\displaystyle\overline{{{W}{U}}}\stackrel{\sim}{=}\overline{{{V}{Z}}}\) (Given)
As \(\displaystyle\angle{U},\angle{V}\) are right angles,
\(\displaystyle\angle{U}\stackrel{\sim}{=}\angle{V}\) (right angles are congruent)
\(\displaystyle\overline{{{W}{X}}}\stackrel{\sim}{=}\overline{{{Y}{Z}}}\) (Given)
Add \(\displaystyle\overline{{{X}{Y}}}\) to both sides.
\(\displaystyle\overline{{{W}{X}}}+\overline{{{X}{Y}}}=\overline{{{Y}{Z}}}+\overline{{{X}{Y}}}\)
\(\displaystyle\Rightarrow\overline{{{W}{Y}}}=\overline{{{X}{Z}}}\)
So,
\(\displaystyle\overline{{{W}{Y}}}\stackrel{\sim}{=}\overline{{{X}{Z}}}\) (line segments of equal measures are congruent)
Therefore,
\(\displaystyle\triangle{W}{U}{Y},\triangle{Z}{V}{X}\) (RHS congruence criteria)
Also,
Congruent triangles basically overlap each other.
So, \(\displaystyle\triangle{W}{U}{Y},\triangle{Z}{V}{X}\) is a pair of overlapping triangles.
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