Question

complete each step to show that ( mangle3 + mangle2 = mangle4 ).
( mangle4 + mangle1 = 180^{circ} ) ( angle4 ) ( mangle1 ) ( mangle2 ) ( mangle3 ) ( mangle4 ) supplementary.
( mangle3 + mangle2 + mangle1 = 180^{circ} ) a triangles ( mangle1 ) ( mangle2 ) ( mangle3 ) ( mangle4 ) gles sum to ( 180^{circ} ).
( mangle3 + mangle2 + mangle1 = mangle4 + mangle1 ) sums equal ( 180^{circ} ).
( mangle3 + mangle2 = mangle4 ) subtract ( ? ) from both sides.

Explanation:

Step1: Use the property of supplementary angles

We know that \(m\angle4 + m\angle1=180^{\circ}\) because \(\angle4\) and \(\angle1\) are supplementary.

Step2: Use the triangle - angle sum property

For a triangle, \(m\angle3 + m\angle2 + m\angle1 = 180^{\circ}\) (sum of interior angles of a triangle is \(180^{\circ}\)).

Step3: Set the two equations equal

Since \(m\angle4 + m\angle1=180^{\circ}\) and \(m\angle3 + m\angle2 + m\angle1 = 180^{\circ}\), we can say \(m\angle3 + m\angle2 + m\angle1=m\angle4 + m\angle1\) (both sums equal \(180^{\circ}\)).

Step4: Subtract \(m\angle1\) from both sides

By the subtraction property of equality, if \(a + c=b + c\), then \(a=b\). Here, \(a = m\angle3 + m\angle2\), \(b=m\angle4\), and \(c = m\angle1\).

Answer:

\(m\angle1\)