Question

if ∠3 + ∠8 = 180, and ∠4 + ∠8 = 180, then _______________*
your answer

if x is parallel to y, and z is parallel to y, then _______________.*
your answer

if ∠l ≅ ∠m, and ∠m is supplementary to ∠p, then _______________.*

Explanation:

First problem:

Step1: Use the property of supplementary angles

If \(\angle3+\angle8 = 180^{\circ}\) (so \(\angle3\) and \(\angle8\) are supplementary) and \(\angle4+\angle8=180^{\circ}\) (so \(\angle4\) and \(\angle8\) are supplementary), by the congruent - supplements theorem (if two angles are supplementary to the same angle, then the two angles are congruent), we can say \(\angle3\cong\angle4\)

Second problem:

Step1: Use the transitive property of parallel lines

If \(X\parallel Y\) and \(Z\parallel Y\), by the transitive property of parallel lines (if line \(a\) is parallel to line \(b\) and line \(c\) is parallel to line \(b\), then line \(a\) is parallel to line \(c\)), we can say \(X\parallel Z\)

Third problem:

Step1: Use the definition of congruent angles and supplementary angles

Since \(\angle L\cong\angle M\), then \(m\angle L=m\angle M\). If \(\angle M\) is supplementary to \(\angle P\) (i.e., \(m\angle M + m\angle P=180^{\circ}\)), substituting \(m\angle L\) for \(m\angle M\) (because of the congruence), we get \(m\angle L + m\angle P=180^{\circ}\), so \(\angle L\) is supplementary to \(\angle P\)

Answer:

1. \(\angle3\cong\angle4\)
2. \(X\parallel Z\)
3. \(\angle L\) is supplementary to \(\angle P\)