Question

Based on the triangles shown below, which statements are true? Select All that apply.

Answer 1

All the options except the third choice are correct.

Step-by-step explanation:

In the given figure:

`\angle\text{GER}\cong\angle\text{TEA (Vertical Angles)}`

Since angles G and T are congruent:

• Triangles GER and TEA are similar triangles.

Therefore, the following holds:

`\begin{gathered} \triangle\text{GRE}\sim\triangle\text{TAE} \\ \triangle E\text{GR}\sim\triangle E\text{TA} \\ (GR)/(TA)=(RE)/(AE) \end{gathered}`

Similarly:

`\begin{gathered} (EG)/(ET)=(GR)/(TA) \\ ET=10,EG=5,TA=12,RG=\text{?} \\ (5)/(10)=(RG)/(12) \\ (1)/(2)=(RG)/(12) \\ 2RG=12 \\ RG=(12)/(2) \\ RG=6 \\ \text{Therefore if }ET=10,EG=5,and\; TA=12,then\; RG=6 \end{gathered}`

Finally, angles R and A are congruent.

`\begin{gathered} m\angle R=m\angle A \\ 80\degree=(x+20)\degree \\ x=80\degree-20\degree \\ x=60\degree \end{gathered}`

The correct choices are:

`\begin{gathered} \triangle\text{GRE}\sim\triangle\text{TAE} \\ \triangle E\text{GR}\sim\triangle E\text{TA} \\ (GR)/(TA)=(RE)/(AE) \\ I\text{f }ET=10,EG=5,and\; TA=12,then\; RG=6 \\ \text{If }m\angle R=80\degree\text{ and }m\angle A=(x+20)\degree,then\; x=60\text{ } \end{gathered}`

Only the third choice is Incorrect.