Can anyone help ? Show that the pair of triangles is congruent. Then find the missing measures in the diagram.

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1 Answer
Geoff K.
Jan 31, 2018

A triangle is uniquely defined by any one of these conditions:

  • SSS (knowing all 3 side lengths)
  • SAS (knowing 2 sides and the angle between them)
  • ASA (knowing 2 angles and the length of the side between them)
  • AAS (equivalent to ASA, since if we know 2 angles, we can compute the 3rd)

To show two triangles are congruent, we need to show that they have enough equivalent corresponding sides/angles, so that one of the above options is satisfied.

For this pair of triangles, we see that #bar(LY) ~= bar(UG),# since they both have length 7.2.

Next, both triangles have a side of length 6, those being #bar(YF)# and #bar(GB),# meaning #bar(YF) ~= bar(GB).#

Finally, #mangle Y =56^@,# and so is #mangle G.# Thus, #angleY ~=angle G.#

Since all three of these congruent values appear in the same order in each triangle (i.e. in SAS order), this shows #triangle LYF ~= triangle UGB.# It is important to list the vertices in this order, because it's the order in which they match up (or correspond). That is:

#L# goes with #U#,
#Y# goes with #G#, and
#F# goes with #B#.

To determine the missing values, we look at the corresponding value in the other triangle. For instance, #x=BU,# and since #bar(BU)~=bar(FL)#, and #FL=6.3,# that gives us #x=6.3.#

Similarly,

#z^@=m angle B=mangle F=72^@,#

and

#y^@ = mangleU = mangleL = 52^@.#

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