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In /_\LMN, bar(MN)~= bar(LM) and m/_L=76^(@). Find m/_M.

In △LMN,MN‾≅LM‾ \triangle L M N, \overline{M N} \cong \overline{L M} and m∠L=76∘ \mathrm{m} \angle L=76^{\circ} . Find m∠M \mathrm{m} \angle M .

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Full solution

Q. In △LMN,MN‾≅LM‾ \triangle L M N, \overline{M N} \cong \overline{L M} and m∠L=76∘ \mathrm{m} \angle L=76^{\circ} . Find m∠M \mathrm{m} \angle M .
  1. Identify Triangle Type: Identify the type of triangle based on the given information.\newlineSince MN‾\overline{MN} is congruent to LM‾\overline{LM}, triangle LMNLMN is isosceles with two equal sides MNMN and LMLM.
  2. Properties of Isosceles Triangle: Determine the properties of an isosceles triangle.\newlineIn an isosceles triangle, the angles opposite the equal sides are also equal. Therefore, m/_M=m/_Nm/\_M = m/\_N.
  3. Calculate Third Angle: Calculate the measure of the third angle using the fact that the sum of angles in a triangle is 180180 degrees. \newlinem/_L+m/_M+m/_N=180∘m/\_L + m/\_M + m/\_N = 180^{\circ}\newline76∘+m/_M+m/_M=180∘76^{\circ} + m/\_M + m/\_M = 180^{\circ} (since m/_M=m/_Nm/\_M = m/\_N)\newline2×m/_M=180∘−76∘2 \times m/\_M = 180^{\circ} - 76^{\circ}\newline2×m/_M=104∘2 \times m/\_M = 104^{\circ}\newlinem/_M=104∘2m/\_M = \frac{104^{\circ}}{2}\newlinem/_M=52∘m/\_M = 52^{\circ}

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