Answer:
[tex]m\angle AMC=75^{\circ}[/tex]
Step-by-step explanation:
We have been given an image of a right triangle. We are asked to find the measure of angle AMC.
Let us find measure of angle A using angle sum property.
[tex]m\angle A+m\angle B+m\angle C=180^{\circ}[/tex]
[tex]m\angle A+30^{\circ}+90^{\circ}=180^{\circ}[/tex]
[tex]m\angle A+120^{\circ}=180^{\circ}[/tex]
[tex]m\angle A+120^{\circ}-120^{\circ}=180^{\circ}-120^{\circ}[/tex]
[tex]m\angle A=60^{\circ}[/tex]
Since CM is an angle bisector of our given triangle, so it will divide angle ACB into equal parts.
We can see that measure of angle ACB is 90 degrees, so measure of angle ACM will be [tex]\frac{90^{\circ}}{2}=45^{\circ}[/tex].
Now, we will angle sum property to find measure of angle AMC of traingle ACM.
[tex]45^{\circ}+60^{\circ}+m\angle AMC=180^{\circ}[/tex]
[tex]105^{\circ}+m\angle AMC=180^{\circ}[/tex]
[tex]105^{\circ}-105^{\circ}+m\angle AMC=180^{\circ}-105^{\circ}[/tex]
[tex]m\angle AMC=75^{\circ}[/tex]
Therefore, the measure of angle AMC is 75 degrees.
