Lesson Problems on congruence tests for triangles-2

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<H2>More problems on congruence tests for triangles</H2>

It is better to read this lesson after the lesson <A HREF = http://www.algebra.com/algebra/homework/Triangles/Congruence-tests-for-triangles.lesson> Congruence tests for triangles</A>, which is under the topic Triangles in the section Geometry in this site.
Here you can see the technique to prove geometry facts.

Problem 1

If two sides in one triangle are congruent to the two sides in another triangle and the median drawn to one of them in one triangle is congruent to the median 
drawn to the congruent side in the second triangle, then the triangles are congruent.
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Solution

The input data is shown in Figure 1.
We have two triangles ABC and EFG with the congruent sides AC = EG&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
and BC = FG.
The median AD of the triangle ABC drawn to the side BC is congruent 
to the median EH of the triangle EFG drawn to the side FG. 
We need to prove congruency of the triangles ABC and EFG.

The proof is very simple.
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{{{drawing( 400, 150, 0, 10, 0, 3, 
 line( 0.5, 0.5, 4.5, 0.5), 
 line( 0.5, 0.5, 3.0, 2.5),
 line( 3.0, 2.5, 4.5, 0.5),
 green(line( 0.5, 0.5, 3.75, 1.5)),

locate(0.5, 0.5, A),
            locate(4.5, 0.5, B),
            locate(3.0, 2.9, C),
            locate(3.8, 1.8, D),

line( 1.65,  1.5, 1.85,  1.5),

line( 3.35,  1.8, 3.60,  1.8),
            line( 3.35,  1.9, 3.60,  1.9),

line( 2.7,  1.3, 2.7,  1.1),
            line( 2.8,  1.3, 2.8,  1.1),
            line( 2.9,  1.3, 2.9,  1.1),

line( 5.5, 0.5, 9.5, 0.5), 
            line( 5.5, 0.5, 8.0, 2.5),
            line( 8.0, 2.5, 9.5, 0.5),
      green(line( 5.5, 0.5, 8.75, 1.5)),

locate(5.5, 0.5, E),
            locate(9.5, 0.5, F),
            locate(8.0, 2.9, G),
            locate(8.8, 1.8, H),

line( 6.65,  1.5, 6.85,  1.5),

line( 8.35,  1.8, 8.60,  1.8),
            line( 8.35,  1.9, 8.60,  1.9),

line( 7.7, 1.3, 7.7, 1.1),
 line( 7.8, 1.3, 7.8, 1.1),
 line( 7.9, 1.3, 7.9, 1.1)
)}}}
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Figure 1. To the Problem 1
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 </TR>
</TABLE>Since the sides BC and FG are congruent, their halves CD and GH are congruent, too. 
Hence, triangles ACD and EGH have three congruent sides. 
In accordance to the postulate P3 (SSS) of the lesson <A HREF = http://www.algebra.com/algebra/homework/Triangles/Congruence-tests-for-triangles.lesson> Congruence tests for triangles</A> (the topic Triangles in the section Geometry in this site), 
the triangles ACD and EGH are congruent. 
Therefore angles ACB and EGF are congruent.
Thus triangles ABC and EFG have two congruent sides AC = EG and BC = FG, as well as congruent angles ACB and EGF included by the pairs of these sides.
In accordance to the postulate P1 (SAS) of the lesson <A HREF = http://www.algebra.com/algebra/homework/Triangles/Congruence-tests-for-triangles.lesson> Congruence tests for triangles</A> (the topic Triangles in the section Geometry in this site), 
the triangles ABC and EGF are congruent. 
This is that to be demonstrated.

Other examples of problems on congruence tests for triangles are in the lesson <A HREF = http://www.algebra.com/algebra/homework/Triangles/Problems-on-congruence-tests-for-triangles.lesson> Problems on congruence tests for triangles</A> under the topic Triangles 
in the section Geometry in this site.

For navigation over the lessons on Properties of Triangles use this file/link &nbsp;<A HREF=https://www.algebra.com/algebra/homework/Triangles/Compendium-of-properties-of-triangles.lesson>Properties of Trianles</A>.

To navigate over all topics/lessons of the Online Geometry Textbook use this file/link &nbsp;<A HREF=https://www.algebra.com/algebra/homework/Triangles/GEOMETRY-your-online-textbook.lesson>GEOMETRY - YOUR ONLINE TEXTBOOK</A>.