In ΔABC, m∠ABC = 40°, BL is the angle bisector of ∠B with point L∈ AC . Point M ∈ AB so that LM ⊥ AB and N ∈ BC so that LN ⊥ BC

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In ΔABC, m∠ABC = 40°, BL is the angle bisector of ∠B with point L∈ AC . Point M ∈ AB so that LM ⊥ AB and N ∈ BC so that LN ⊥ BC Find the angles of △MNL

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    2021-09-10T09:20:21+00:00

    It can help to draw a diagram.

    ∆LBM ≅ ∆LBN by the hypotenuse-angle (HA) theorem of congruence of right trianges. Then LM ≅ LN and BM ≅ BN by CPCTC. Quadrliateral BMLN is a kite,  so diagonal MN ⊥ BL.

    Of course, angles BLN and BLM are the complements of the halves of bisected angle B, so are both 70°. And angles LMN and LNM are the complements of those, so are both 20°.

    ∆MNL is an isosceles triangle with base angles of 20° and an apex angle of 140°.

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