Note:It is safe to assumethat the sun’s rays hit the groundat the sameangle.It is also properto assume
that the toweris vertical(perpendicularto the ground).
The diagramshowstwo similarright triangles.Theyare similarbecauseeachhas a right angle,and the
anglewherethe sun’s rays hit the groundis the samefor both objects.We can writea proportionwith only
one unknown, , the heightof the tower.
Thus,the toweris feet tall.Note:This is methodconsideredindirectmeasurementbecauseit wouldbe difficult todirectlymeasurethe
heightof tall tower. Imaginehow difficult it wouldbe to hold a tape measureup to a 51-foot-talltower.
LessonSummary
The mostbasicway—becauseit requiresthe leastinputof information—toassurethat trianglesare similar
is to showthat they havetwo pairsof congruentangles.The AA postulatestatesthis: If two triangleshave
two pairsof congruentangles,then the trianglesare similar.
Oncetrianglesare knownto be similar, we can writemanytrue proportionsinvolvingthe lengthsof their
sides.Theseproportionswerethe basisfor doingindirectmeasurement.
Pointsto Consider
Thinkaboutsomeright trianglesfor a minute.Supposetwo right trianglesboth havean acuteanglethat
measures. Thenthe ratio is the samein both triangles.In fact, this ratio,called
“the tangentof ” is the samein any right trianglewith a angle.As mentionedearlier, this is the
reasonfor trigonometricfunctionsof a givenanglebeingconstant,regardlessof the specifictriangleinvolved.
LessonExercises
Use the diagrambelowfor exercises1-5.
Giventhat :
- Nametwo similartriangles.