Question

lesson 2.3.3 and 2.3.4 assignment
2 - 103. use the triangles at right to answer the following questions. are the triangles at right similar? homework help
a. how do you know if they are similar? show your reasoning in a flowchart.
b. examine your work from part (a). are the triangles also congruent? explain why or why not.
2 - 105. william thinks that the hypotenuse must be the longest side of a right triangle, but chad does not agree. who is correct? support your answer with an explanation or a counterexample, if possible. homework help

Explanation:

Step1: Recall similarity criterion

Two triangles are similar if their corresponding angles are equal. In \(\triangle ABC\) and \(\triangle DEF\), \(\angle A = 74^{\circ}\), \(\angle D=74^{\circ}\), \(\angle B = 9^{\circ}\), \(\angle F = 9^{\circ}\). By the AA (angle - angle) similarity criterion, since two pairs of corresponding angles are equal, the triangles are similar.

Step2: Recall congruence criterion

Two triangles are congruent if their corresponding angles are equal and corresponding sides are equal. In \(\triangle ABC\) and \(\triangle DEF\), we know \(\angle A=\angle D\), \(\angle B=\angle F\), \(\angle C=\angle E\) (because the sum of angles in a triangle is \(180^{\circ}\)), and \(AC = DF=4\). Since all corresponding angles are equal and one pair of corresponding sides are equal, by the ASA (angle - side - angle) congruence criterion, the triangles are congruent.

Answer:

a. The triangles \(\triangle ABC\) and \(\triangle DEF\) are similar by the AA (angle - angle) similarity criterion because \(\angle A=\angle D = 74^{\circ}\) and \(\angle B=\angle F = 9^{\circ}\).
b. The triangles \(\triangle ABC\) and \(\triangle DEF\) are congruent. We know that \(\angle A=\angle D\), \(\angle B=\angle F\), so \(\angle C=\angle E\) (since the sum of angles in a triangle is \(180^{\circ}\)). Also, \(AC = DF = 4\). By the ASA (angle - side - angle) congruence criterion, the two triangles are congruent.