Free Math Worksheets

- The hypothesis follows “if,” and the conclusion follows “then.” - Form the converse by switching the hypothesis and conclusion. - Relate congruent angles in a triangle to their opposite sides.

1. The hypothesis is that \(\triangle ABC\) is isosceles with base \(\overline{AB}\). The conclusion is that \(\alpha = \beta\). 2. The converse is: “If \(\alpha = \beta\) in \(\triangle ABC\), then \(\triangle ABC\) is isosceles with base \(\overline{AB}\).” 3. The converse is true. In a triangle, congruent angles have congruent opposite sides. Therefore, \(AC = BC\), so \(\triangle ABC\) is isosceles with base \(\overline{AB}\).

a) Hypothesis: \(\triangle ABC\) is isosceles with base \(\overline{AB}\). Conclusion: \(\alpha = \beta\). b) If \(\alpha = \beta\), then \(\triangle ABC\) is isosceles with base \(\overline{AB}\). c) The converse is true.

- Switch the hypothesis and conclusion. - Look for another quadrilateral with perpendicular diagonals. - A counterexample must satisfy the new hypothesis but not its conclusion.

1. The converse is: “If a quadrilateral has perpendicular diagonals, then it is a square.” 2. The converse is false. 3. A rhombus that is not a square has perpendicular diagonals but does not have four right angles. Therefore, perpendicular diagonals alone do not guarantee that a quadrilateral is a square.

a) If a quadrilateral has perpendicular diagonals, then it is a square. b) The converse is false. A nonsquare rhombus is a counterexample.