Question 698526: Carl has a piece of cake in the shape of an isosceles triangle with angles 26 ° ,77 ° , and 77 °. He wanted to divide it into two equal parts, so he cut it through the middle of the 26 ° angle to the midpoint of the opposite side. He says that because he is dividing it at the midpoint of a side, the two pieces are congruent. Is this enough information? Explain. please explain step by step and give a summery for how you got solved the question.
Answer by neatmath(302)   (Show Source):
You can put this solution on YOUR website!
Carl is correct, he has cut the large piece into 2 equal smaller pieces.
However, his explanation is not complete.
What he drew was not only an angle bisector, but more importantly, a median!
When you draw a median of a triangle, it does indeed cut the other side into 2 equal segments.
Now, Carl has 2 small triangles.
But are they congruent? That is the question.
Let's label our large triangle ABC, with A being the vertex with the 26 degree angle.
Let's call the midpoint of the segment opposite vertex A, point D.
Now, the 2 triangles that are formed are triangle ADB, and triangle ADC.
Can we tell if these are congruent? We can, here's how:
AB is congruent to AC since it is an isoscoles triangle.
BD is congruent to CD since D is the midpoint of BC.
AD is congruent to itself by the reflexive property.
Thus, triangle ADB is congruent to triangle ADC by the Side-Side-Side Theorem (SSS).
I hope this helps! Keep practicing! :)
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