Question

given: ij = kj
prove: il ≠ lm (using the indirect method)

Explanation:

Step1: Use the Indirect Proof Assumption

Assume \(IL = LM\). If \(IL = LM\), then by the Isosceles Triangle Theorem (if two sides of a triangle are equal, then the angles opposite those sides are equal), \(\angle1=\angle2\). But we know from the given \(IJ = KJ\) and using the Hinge - Theorem (if two sides of one triangle are congruent to two sides of another triangle, but the included angle is larger, then the third side is larger).

Step2: Analyze the Angle - Side Relationship for Statement 6

Since the reason for statement 6 is "If Unequal Sides, then Unequal Angles", if we assume \(IL = LM\) (for the indirect proof), but we have \(m\angle K>m\angle2\) (from statement 5). If \(IL = LM\), then in \(\triangle ILM\), \(\angle1=\angle2\). Now, consider \(\triangle IKJ\) and other triangles in the figure. If we assume \(IL = LM\), and using the given \(IJ = KJ\), we know that in a triangle, the larger angle is opposite the larger side. So, if we consider the relationship between sides and angles, for statement 6, if we assume \(IL = LM\) (for the indirect proof), but we know that if we consider the sides related to \(\angle K\) and \(\angle2\), if \(IL = LM\), then we can get a contradiction. The statement for 6 should be \(IL eq LM\) (to start the indirect proof contradiction). But since we are building the proof step - by - step, and using the "If Unequal Sides, then Unequal Angles" (converse of the Isosceles Triangle Theorem in a non - equal sense). If we assume \(IL = LM\) (for contradiction), but from \(m\angle K>m\angle2\), and if \(IL = LM\) (so \(\angle1 = \angle2\)), then considering the sides opposite \(\angle K\) and \(\angle1\) (or \(\angle2\)). The correct statement for 6 is \(IL eq LM\) (to start the contradiction part of the indirect proof). But more precisely, if we assume \(IL = LM\) (for indirect proof), then in \(\triangle ILM\), \(\angle1=\angle2\). Now, using the Hinge - Theorem (if two sides of one triangle are equal to two sides of another triangle, but the included angle is different). Let's assume for the sake of building the proof:
Let's assume \(IL = LM\) (for indirect proof). Then, in \(\triangle ILM\), \(\angle1=\angle2\).
Since \(m\angle K>m\angle2\) (statement 5), and if \(\angle1=\angle2\), then \(m\angle K > m\angle1\).
By the "If Unequal Angles, then Unequal Sides" (converse of the Isosceles Triangle Theorem), in \(\triangle LKM\) (assuming some side - angle relationships from the figure), the side opposite \(\angle K\) is \(LM\) and the side opposite \(\angle1\) is \(LK\). But actually, if we consider the overall figure (assuming \(IJ = KJ\) and looking at triangles involving \(I\), \(J\), \(K\), \(L\), \(M\)).
The statement for 6: If we assume \(IL = LM\) (for indirect proof), then by the Isosceles Triangle Theorem \(\angle1=\angle2\). But since \(m\angle K>m\angle2\) (statement 5), then \(m\angle K > m\angle1\). By the "If Unequal Angles, then Unequal Sides" (in a triangle, the larger angle is opposite the larger side), if we consider the triangle where \(\angle K\) and \(\angle1\) are angles (say \(\triangle LKM\) or a related triangle), we get a contradiction. So the statement for 6 is \(IL eq LM\) (to start the contradiction in the indirect proof). But more accurately, since the reason is "If Unequal Sides, then Unequal Angles", we know that if \(IL = LM\) (assumption for indirect proof), then \(\angle1=\angle2\) (Isosceles Triangle Theorem). But \(m\angle K>m\angle2\) (statement 5), so \(m\angle K > m\angle1\). Then, by the converse (If Unequal Angles, then Unequal Sides), the side opposite \(\angle K\) and \(\angle1\) (say \(LM\) and \(LK\) or other sides) are unequal. But since we assumed \(IL = LM\), we get a contradiction. So the statement for 6 is \(IL eq LM\) (to start the contradiction).

Step3: Analyze Statement 7

Since the reason for statement 7 is "Transitive Property of Inequality". We know that \(m\angle K>m\angle2\) (statement 5) and if \(IL = LM\) (assumption for indirect proof, then \(\angle1=\angle2\)), so \(m\angle K>m\angle1\). The transitive property of inequality states that if \(a > b\) and \(b=c\), then \(a > c\). Here \(a=m\angle K\), \(b = m\angle2\), \(c=m\angle1\) (if \(IL = LM\)). So statement 7 is \(m\angle K>m\angle1\)

Step4: Analyze Statement 8

Since we have been using the indirect proof (reductio ad absurdum). The last step of an indirect proof is to state that the assumption (in this case \(IL = LM\)) is false. So statement 8 is "Assumption \(IL = LM\) is false", and the reason is "Contradiction"

Answer:

1. \(IL

eq LM\)

1. \(m\angle K>m\angle1\)
2. Assumption \(IL = LM\) is false; Contradiction