Step 1

Taking \(\displaystyle\triangle{X}{Y}{Z}\)

\(\displaystyle\angle{X}={45}^{{\circ}},\angle{Y}={85}^{{\circ}}\)

Now,

Since sum of interior angle of triangle is \(\displaystyle{180}^{{\circ}}\)

\(\displaystyle\Rightarrow\angle{X}+\angle{Y}+\angle{Z}={180}^{{\circ}}\)

\(\displaystyle{45}^{{\circ}}+{85}^{{\circ}}+\angle{Z}={180}^{{\circ}}\)

\(\displaystyle\angle{Z}={180}^{{\circ}}-{130}^{{\circ}}\)

\(\displaystyle\angle{Z}={50}^{{\circ}}\)

Step 2

Taking \(\displaystyle\triangle{W}{Y}{U}\)

\(\displaystyle\angle{W}={50}^{{\circ}},\angle{Y}={85}^{{\circ}}\)

Now,

Since sum of interior angle of triangle is \(\displaystyle{180}^{{\circ}}\)

\(\displaystyle\Rightarrow\angle{W}+\angle{Y}+\angle{U}={180}^{{\circ}}\)

\(\displaystyle{50}^{{\circ}}+{85}^{{\circ}}+\angle{U}={180}^{{\circ}}\)

\(\displaystyle\angle{U}={180}^{{\circ}}-{135}^{{\circ}}\)

\(\displaystyle\angle{U}={45}^{{\circ}}\)

Step 3

In \(\displaystyle\triangle{X}{Y}{Z}\ {\quad\text{and}\quad}\ \triangle{U}{Y}{W}\)

\(\displaystyle\angle{X}=\angle{U}={50}^{{\circ}}\)

\(\displaystyle\angle{Y}=\angle{Y}={85}^{{\circ}}\)

\(\displaystyle\angle{Z}=\angle{W}={45}^{{\circ}}\)

Therefore

\(\displaystyle\triangle{X}{Y}{Z}\sim\triangle{U}{Y}{W}\) By: AA similarity

Taking \(\displaystyle\triangle{X}{Y}{Z}\)

\(\displaystyle\angle{X}={45}^{{\circ}},\angle{Y}={85}^{{\circ}}\)

Now,

Since sum of interior angle of triangle is \(\displaystyle{180}^{{\circ}}\)

\(\displaystyle\Rightarrow\angle{X}+\angle{Y}+\angle{Z}={180}^{{\circ}}\)

\(\displaystyle{45}^{{\circ}}+{85}^{{\circ}}+\angle{Z}={180}^{{\circ}}\)

\(\displaystyle\angle{Z}={180}^{{\circ}}-{130}^{{\circ}}\)

\(\displaystyle\angle{Z}={50}^{{\circ}}\)

Step 2

Taking \(\displaystyle\triangle{W}{Y}{U}\)

\(\displaystyle\angle{W}={50}^{{\circ}},\angle{Y}={85}^{{\circ}}\)

Now,

Since sum of interior angle of triangle is \(\displaystyle{180}^{{\circ}}\)

\(\displaystyle\Rightarrow\angle{W}+\angle{Y}+\angle{U}={180}^{{\circ}}\)

\(\displaystyle{50}^{{\circ}}+{85}^{{\circ}}+\angle{U}={180}^{{\circ}}\)

\(\displaystyle\angle{U}={180}^{{\circ}}-{135}^{{\circ}}\)

\(\displaystyle\angle{U}={45}^{{\circ}}\)

Step 3

In \(\displaystyle\triangle{X}{Y}{Z}\ {\quad\text{and}\quad}\ \triangle{U}{Y}{W}\)

\(\displaystyle\angle{X}=\angle{U}={50}^{{\circ}}\)

\(\displaystyle\angle{Y}=\angle{Y}={85}^{{\circ}}\)

\(\displaystyle\angle{Z}=\angle{W}={45}^{{\circ}}\)

Therefore

\(\displaystyle\triangle{X}{Y}{Z}\sim\triangle{U}{Y}{W}\) By: AA similarity