Question

Help me with this geometry question with proofs

Answer 1

m ∠ AMC = 75°

Explanation:

Given:

In Δ ABC, m ∠C=90°

m∠ B =30°

CM is angle bisector

We need to find m ∠AMC

In Δ ABC Sum of all angle is 180° so we get,

`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`

Now we know that CM is angle bisector of ∠C

∴
`m\angle ACM +m\angle BCM =90\\m\angle ACM +m\angle ACM =90\\2m\angle ACM =90\\m\angle ACM =(90)/(2)=45`

Now in Δ ACM we know that Sum of all angles is 180

`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`

Hence m ∠ AMC = 75°