Question

What are the angle measures of a 5-12-13 right triangle?

Answer 1

`m(/_A)~=67^0 22'37'', m(/_B)~=22^0 37'53'', and m(/_C)=90` degrees

Explanation:

Let the vertices of the triangle be `A`, `B`, and `C`.

Theorem:
`color(white)xxa^2+b^2=c^2<=>m(/_C)=90` degrees

`color(white)xxa^2+b^2=5^2+12^2`
`color(white)xxcolor(white)xxcolor(white)xxcolor(white)x=25+144`
`color(white)xxcolor(white)xxcolor(white)xxcolor(white)x=169`
`color(white)xxcolor(white)xxcolor(white)xxcolor(white)x=13^2`
`color(white)xxcolor(white)xxcolor(white)xxcolor(white)x=c^2`

`=>m(/_C)=90` degrees

`color(white)xxsin/_A=12/13`
`=>m(/_A)=arcsin(12/13)`
`color(white)xxcolor(white)xxcolor(white)xxcolor(white)x~=67^0 22'37''`

The sum of the measures of the interior angles of a triangle is 180 degrees:
`color(white)xxm(/_A)+m(/_B)+m(/_C)=180` degrees
`=>67^0 22'37''+m(/_B)+90~=180` degrees
`=>67^0 22'37''+m(/_B)+90-67^0 22'37''-90~=180-67^0 22'37''-90`
`=>m(/_B)~=90-67^0 22'37''`
`color(white)xxcolor(white)xxcolor(white)xxcolor(white)x=22^0 37'53''`