Question

Carl had 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?

Answer 1

Carl correctly cut the isosceles triangle into two congruent parts by drawing a line from the 26-degree angle vertex to the midpoint of the opposite side.

Step-by-step explanation:

Carl's decision to cut an isosceles triangle from the 26-degree angle to the midpoint of the opposite side will indeed result in two congruent triangles. This is because when you draw a line from the vertex with the unequal angle (in this case, 26 degrees) to the midpoint of the opposite side in an isosceles triangle, it not only bisects that side but also bisects the angle at the vertex. Consequently, the line forms two equal angles at the base. This line is known as the median and it also acts as an altitude (height) of the triangle because it forms two right angles with the base. It can be proven that these two resulting triangles are congruent by the Side-Angle-Side (SAS) postulate or by recognizing that they are mirror images along the median.

Answer 2

It is not enough information to say that the two pieces of the cake are congruent just because the cut was made through the middle of the 26 degree angle to the midpoint of the opposite side. Congruence means that two figures have the same size and shape, and the angles and side lengths are all congruent. In order to determine congruence, we would need to know more information about the measures of the sides and angles of the original triangle. Without that information, we cannot say for certain whether the two pieces of the cake are congruent or not.

Step-by-step explanation: