Question

interior angles of triangles
using each picture or description of a triangle, write and solve an equation in order to find the
number of degrees in each angle.
1
2
3
4.

1. in triangle mno, \\( \angle n \\) is 5

times the measure of \\( \angle m \\), and
\\( \angle o \\) is 4 times the measure of
\\( \angle m \\).

Explanation:

Step1: Use the triangle angle - sum theorem

The sum of the interior angles of a triangle is \(180^{\circ}\).

For triangle \(ABC\):

\(4x+(7x + 2)+(5x-10)=180\)
\(4x+7x+2 + 5x-10=180\)
\(16x-8 = 180\)
\(16x=180 + 8\)
\(16x=188\)
\(x = 12\)

\(\angle A=4x=4\times12 = 48^{\circ}\)
\(\angle B=7x + 2=7\times12+2=86^{\circ}\)
\(\angle C=5x-10=5\times12-10 = 50^{\circ}\)

For triangle \(DEF\):

\(3x+(18x)+(5x - 2)=180\)
\(3x+18x+5x-2=180\)
\(26x-2=180\)
\(26x=180 + 2\)
\(26x=182\)
\(x = 7\)

\(\angle D=3x=3\times7=21^{\circ}\)
\(\angle E=18x=18\times7 = 126^{\circ}\)
\(\angle F=5x-2=5\times7-2=33^{\circ}\)

For triangle \(GHI\):

\(16x+16x+13x=180\)
\(45x=180\)
\(x = 4\)

\(\angle G=16x=16\times4=64^{\circ}\)
\(\angle H=16x=16\times4 = 64^{\circ}\)
\(\angle I=13x=13\times4=52^{\circ}\)

For triangle \(JKL\):

\((x + 8)+(2x-3)+(6x-5)=180\)
\(x+8+2x-3+6x-5=180\)
\(9x=180\)
\(x = 20\)

\(\angle J=x + 8=20+8=28^{\circ}\)
\(\angle K=6x-5=6\times20-5=115^{\circ}\)
\(\angle L=2x-3=2\times20-3=37^{\circ}\)

For triangle \(MNO\):

Let \(\angle M=x\), then \(\angle N = 5x\) and \(\angle O=4x\)
\(x+5x+4x=180\)
\(10x=180\)
\(x = 18\)

\(\angle M=18^{\circ}\)
\(\angle N=5x=5\times18=90^{\circ}\)
\(\angle O=4x=4\times18=72^{\circ}\)

Answer:

1. \(\angle A = 48^{\circ}\), \(\angle B=86^{\circ}\), \(\angle C = 50^{\circ}\)
2. \(\angle D=21^{\circ}\), \(\angle E = 126^{\circ}\), \(\angle F=33^{\circ}\)
3. \(\angle G=64^{\circ}\), \(\angle H=64^{\circ}\), \(\angle I=52^{\circ}\)
4. \(\angle J=28^{\circ}\), \(\angle K=115^{\circ}\), \(\angle L=37^{\circ}\)
5. \(\angle M=18^{\circ}\), \(\angle N=90^{\circ}\), \(\angle O=72^{\circ}\)