Question

4-7. use the triangles below to answer the following questions. homework help

a. are the triangles above similar? how do you know? show your reasoning in a flowchart.

b. examine your work from part (a). are the triangles also congruent? explain why or why not.

Explanation:

Step1: Analyze Triangle Angles

In triangle \(ABC\) and triangle \(EDF\), we have \(\angle A = 74^\circ\), \(\angle D = 74^\circ\), so \(\angle A=\angle D\). Also, \(\angle B\) and \(\angle E\) are equal (marked as \(9^\circ\) or same measure), so by AA (Angle - Angle) similarity criterion, the triangles are similar.

Step2: Check for Congruence

For congruence, corresponding sides must be equal. From the diagram, side \(AC = 4\) and side \(DF = 4\), side \(AB\) and \(ED\) (assuming the marked sides) - wait, actually, let's check: In similar triangles, if corresponding sides are equal, they are congruent. Let's see the sides: \(AC = DF = 4\), and the angles are equal, so let's check the other sides. Wait, the triangles have two angles equal (AA) and one side equal (the side between the angles? Wait, no, the sides: \(AC = 4\), \(DF = 4\), and the angles at \(A\) and \(D\) are \(74^\circ\), angles at \(B\) and \(E\) are equal. So by ASA (Angle - Side - Angle) congruence, if two angles and the included side are equal, triangles are congruent. Wait, but first, for part (a), similarity: AA similarity (two angles equal). For part (b), since they have equal angles and at least one pair of corresponding sides equal (AC = DF = 4, and the angles between them? Wait, \(\angle A\) and \(\angle D\) are equal, \(AC = DF\), and \(\angle C\) and \(\angle F\) (since sum of angles in triangle is \(180^\circ\), \(\angle C=180 - 74 - \angle B\), \(\angle F = 180 - 74 - \angle E\), and \(\angle B=\angle E\), so \(\angle C=\angle F\)). So by ASA, triangles are congruent. But let's structure part (a) first.

For part (a):

• Identify angles: \(\angle A = \angle D = 74^\circ\), \(\angle B=\angle E\) (given as equal measure, \(9^\circ\) or same). By AA (Angle - Angle) similarity, \(\triangle ABC \sim \triangle DEF\) (assuming notation).

For part (b):

• From part (a), triangles are similar. Now, check sides: \(AC = DF = 4\), and we have \(\angle A=\angle D\), \(\angle C=\angle F\) (since angles in triangle sum to \(180^\circ\), so third angles are equal). So by ASA congruence (Angle - Side - Angle: \(\angle A=\angle D\), \(AC = DF\), \(\angle C=\angle F\)), the triangles are congruent.

Answer:

a. The triangles are similar by the AA (Angle - Angle) similarity criterion. We know this because \(\angle A=\angle D = 74^\circ\) and \(\angle B=\angle E\) (equal marked angles), so two pairs of corresponding angles are equal.
b. The triangles are also congruent. Since they are similar (AA) and we have a pair of corresponding sides equal (\(AC = DF = 4\)) and the included angles (or corresponding angles) are equal, by ASA (Angle - Side - Angle) congruence criterion, the triangles are congruent.