Question

1. ∠abc and ∠def are supplementary angles.

______ = x
______ = m∠abc
______ = m∠def

Explanation:

Step1: Recall supplementary - angle property

Supplementary angles add up to 180°. So, \(m\angle ABC + m\angle DEF=180^{\circ}\).
Since \(m\angle ABC=(x + 10)^{\circ}\) and \(m\angle DEF=(3x + 50)^{\circ}\), we have the equation \((x + 10)+(3x + 50)=180\).

Step2: Simplify the left - hand side of the equation

Combine like terms: \(x+3x+10 + 50=180\), which simplifies to \(4x+60 = 180\).

Step3: Solve for \(x\)

Subtract 60 from both sides: \(4x=180 - 60\), so \(4x=120\). Then divide both sides by 4: \(x=\frac{120}{4}=30\).

Step4: Find \(m\angle ABC\)

Substitute \(x = 30\) into the expression for \(m\angle ABC\): \(m\angle ABC=x + 10=30+10 = 40^{\circ}\).

Step5: Find \(m\angle DEF\)

Substitute \(x = 30\) into the expression for \(m\angle DEF\): \(m\angle DEF=3x + 50=3\times30+50=90 + 50=140^{\circ}\).

Answer:

\(x = 30\), \(m\angle ABC=40^{\circ}\), \(m\angle DEF=140^{\circ}\)