Question

problem 2 writing a proof using asa
recreation members of a teen organization are building
a miniature golf course at your towns youth center. the
design plan calls for the first hole to have two congruent
triangular bumpers. prove that the bumpers on the first
hole, shown at the right, meet the conditions of the plan.
given: \\( \overline { a b } \cong \overline { d e } \\), \\( \angle a \cong \angle d \\), \\( \angle b \\) and \\( \angle e \\) are right angles
prove: \\( \triangle a b c \cong \triangle d e f \\)
statements | reasons

Explanation:

Step1: State the given information

• \( \overline{AB}\cong\overline{DE}\) (Given)
• \( \angle A\cong\angle D\) (Given)
• \( \angle B\) and \( \angle E\) are right angles, so \( \angle B\cong\angle E\) (All right angles are congruent)

Step2: Apply the ASA (Angle - Side - Angle) congruence criterion

The ASA congruence criterion states that if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent.
In \( \triangle ABC\) and \( \triangle DEF\):

• The included side between \( \angle A\) and \( \angle B\) in \( \triangle ABC\) is \( \overline{AB}\)
• The included side between \( \angle D\) and \( \angle E\) in \( \triangle DEF\) is \( \overline{DE}\)

Since \( \angle A\cong\angle D\), \( \overline{AB}\cong\overline{DE}\), and \( \angle B\cong\angle E\), by ASA, \( \triangle ABC\cong\triangle DEF\)

Answer:

The two triangles \( \triangle ABC\) and \( \triangle DEF\) are congruent by the ASA (Angle - Side - Angle) congruence criterion.