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Prove two triangles congruent by a side, a median drawn to that side, and the angle formed by the side and the median.

User Ashish M
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Let's call one triangle ABC, and M the midpoint of AB.

Let's call the other triangle DEF, and midpoint N of DE.

We have a congruent side, AB=DE, congruent medians CM=FN, and congruent angles, ∠AMC=∠DNF

AB ≅ DE Given

AM ≅ BM and DN ≅ EN Def median/midpoint

AM+BM=AB and DN+EN=DE Segment addition theorem

2AM=2BM=AB and 2DN=2EN=DE Substitution (eg BM for AM)

AM ≅ BM ≅ DN ≅ EN Transitivity, algebra

CM ≅ FN Given

∠AMC ≅ ∠DNF Given

ΔAMC ≅ ΔDNF Side Angle Side

AC ≅ DF CPCTC

∠BMC ≅ ∠ENF Supplements of congruent angles are congruent

ΔBMC ≅ ΔENF Side Angle Side

BC ≅ EF CPCTC

Δ ABC ≅ Δ DEF Side Side Side

Prove two triangles congruent by a side, a median drawn to that side, and the angle-example-1
User Mxro
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