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3. Suppose we wanted to prove that the diagonals of a thombus are perpendicular. Show that mangle IMJ=90^circ

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3. Suppose we wanted to prove that the diagonals of a thombus are perpendicular. Show that mangle IMJ=90^circ

3. Suppose we wanted to prove that the diagonals of a thombus are perpendicular. Show that mangle IMJ=90^circ

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SophieProfessional · Tutor for 6 years

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The measure of angle IMJ is \(\mathbf{90^{\circ}}\), proving that the diagonals of a rhombus are perpendicular.

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We start by acknowledging that we have a rhombus ABCD. By definition, a rhombus has all four sides of equal length. The diagonals of a rhombus bisect each other, which means that AM equals MC and BM equals MD. By the SSS congruence postulate, we can say that triangle AMB is congruent to triangle CMD because they have three pairs of equal sides. This implies that the angles AMB and CMD are equal because corresponding angles in congruent triangles are equal. Since AMB and CMD form a straight line, their measures add up to 180 degrees. Knowing that these angles are equal, we can deduce that each is 90 degrees. Therefore, angle IMJ, which is the same as angle AMB, measures 90 degrees.
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