Answer:
[tex]x=30\°[/tex]
Step-by-step explanation:
Find the measures of interior angles in each triangle
Triangle BGC
[tex]m<BCG=90\°-60\°=30\°[/tex]
The measures of triangle BGC are [tex]90\°-60\°-30\°[/tex]
Triangle CGH
we know that
[tex]m<BCE+m<DEC=180\°[/tex] -----> by consecutive interior angles
we have that
[tex]m<DEC=90\°[/tex]
so
[tex]m<BCE=90\°[/tex]
[tex]m<GCH=90\°-m<BCG[/tex]
substitute
[tex]m<GCH=90\°-30\°=60\°[/tex]
we have
[tex]m<GCH=60\°[/tex]
[tex]m<CGH=2x\°[/tex]
[tex]m<CHG=180\°-4x[/tex]
remember that
[tex]m<GCH+m<CGH+m<CHG=180\°[/tex]
[tex]60\°+2x+180\°-4x=180\°[/tex]
[tex]60\°=2x[/tex]
[tex]x=30\°[/tex]
The measures of triangle CGH are [tex]60\°-60\°-60\°[/tex]
Triangle GHE
[tex]m< EGH=90\°-2x=90-2(30\°)=30\°[/tex]
[tex]m< GHE=4x=4(30\°)=120\°[/tex]
remember that
[tex]m<EGH+m<GHE+m<GEH=180\°[/tex]
substitute and solve for m<GEH
[tex]30\°+120\°+m<GEH=180\°[/tex]
[tex]150\°+m<GEH=180\°[/tex]
[tex]m<GEH=30\°[/tex]
The measures of triangle GHE are [tex]30\°-120\°-30\°[/tex]
