help me with this geometry question with proofs

Accepted Solution

Answer:m ∠ AMC = 75°Step-by-step explanation:Given:In Δ ABC, m ∠C=90°m∠ B =30°CM is angle bisector We need to find m ∠AMCIn Δ ABC Sum of all angle is 180° so we get,[tex]m\angle A+m\angle B+m\angle C =180\\m\angle A+90+30 =180\\m\angle A+120 =180\\m\angle A=180-120\\m\angle A=60[/tex]Now we know that CM is angle bisector of ∠C∴ [tex]m\angle ACM +m\angle BCM =90\\m\angle ACM +m\angle ACM =90\\2m\angle ACM =90\\m\angle ACM =\frac{90}{2}=45[/tex]Now in Δ ACM we know that Sum of all angles is 180 [tex]m\angle ACM + m\angle AMC + m\angle A=180\\45 + m\angle AMC + 60 =180\\105 + m\angle AMC =180\\m\angle AMC =180 -105 =75[/tex]Hence m ∠ AMC = 75°