PLS HELPP! In the diagram, PQ←→ ∥ RS←→
, and transversal t
intersects both lines.

Select all the true statements.

∡1 ≅ ∡4
because they are vertical angles

∡1 ≅ ∡8
because they are alternate exterior angles

∡1 ≅ ∡5
because they are corresponding angles

∡1 ≅ ∡3
because they are alternate interior angles

we have three similar triangles, because each has a right angle and shares an angle.   let's write the angles in order: opposite to short leg, long leg, hypotenuse.

cab similar to bad similar to cbd

or as ratios,

ca: ab: cb = ba: ad: bd = cb: bd: cd

we also know

ac = ad + cd

(ad+cd): ab: cb = ba: ad: bd = cb: bd: cd

(ad+cd)/cb=cb/cd

we have cb=6, ad=5 and seek x=cd.

we reject the negative root and conclude x=4

answer: 4