In △abc, m∠abc=40°, bl (l∈ ac ) is the angle bisector of ∠b. point m∈ ab so that lm ⊥ ab and n∈ bc so that ln ⊥ bc . find the measures of the angles of △abc if m∠cln=3m∠alm and prove that cn= 1/2 cl.

In △ABC, m∠ABC=40°, BL (L∈ AC ) is the angle bisector of ∠B.

Point M∈ AB so that LM ⊥ AB and N∈ BC so that LN ⊥ BC .

m∠CLN=3m∠ALM

Let m∠ALM = x so m∠CLN = 3x

According to the diagram,

m∠A = 90 - x and m∠c= 90 - 3x

As we know that sum of all the angles of a triangle is 180,

m∠A + m∠B + m∠C = 180

(90 - x) + 40 + (90 - 3x) = 180

220 - 4x = 180

-4x = -40

x = 10

So m∠A = 90 - x = 90 - 10 =80

m∠C = 90 - 3x = 90 - 30 = 60

In ΔCLN, ∠N= 90°, ∠C = 60° and ∠CLN = 30°

So by 30 -60-90 theorem

Side opposite to 90° = 1/2( side opposite to 30°)

CN = 1/2 (CL)

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step-by-step explanation:

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