Question

In ΔMNO, start overline, M, O, end overline
MO
is extended through point O to point P, m, angle, M, N, O, equals, left parenthesis, 2, x, minus, 3, right parenthesis, degreesm∠MNO=(2x−3)
∘
, m, angle, O, M, N, equals, left parenthesis, 3, x, plus, 14, right parenthesis, degreesm∠OMN=(3x+14)
∘
, and m, angle, N, O, P, equals, left parenthesis, 8, x, minus, 19, right parenthesis, degreesm∠NOP=(8x−19)
∘
. What is the value of x, question markx?

Answer 1

In triangle \( \Delta MNO \), the angles are related as follows:

1. \( m\angle MNO = (2x - 3)^\circ \)
2. \( m\angle OMN = (3x + 14)^\circ \)
3. \( m\angle NOP = (8x - 19)^\circ \)

Since \( OP \) is an extension from line segment \( MO \), the angles \( m\angle NOP \) and \( m\angle MNO \) are supplementary, meaning their sum should equal \( 180^\circ \):

\[
m\angle MNO + m\angle NOP = 180^\circ
\]

Substituting the expressions for \( m\angle MNO \) and \( m\angle NOP \):

\[
(2x - 3) + (8x - 19) = 180
\]

Now, simplify the equation:

\[
2x - 3 + 8x - 19 = 180
\]
\[
10x - 22 = 180
\]

Next, add \( 22 \) to both sides:

\[
10x = 202
\]

Now, divide by \( 10 \):

\[
x = 20.2
\]

Thus, the value of \( x \) is

\[
\boxed{20.2}
\]