Question

3 - 95. for each pair of triangles below, determine if the triangles are congruent, similar but not congruent, or if their relationship cannot be determined from the information given. if they are congruent, organize your reasoning into a flowchart. remember that the triangles may not be drawn to scale. homework help a. b.

Explanation:

Part a

Step1: Calculate the third angle of each triangle

In \(\triangle ABC\), using the angle - sum property of a triangle (\(\angle A+\angle B+\angle C = 180^{\circ}\)), \(\angle A=180^{\circ}-\angle B - \angle C\). Given \(\angle B = 120^{\circ}\) and \(\angle C=35^{\circ}\), then \(\angle A=180^{\circ}-120^{\circ}-35^{\circ}=25^{\circ}\).
In \(\triangle DEF\), using the angle - sum property of a triangle (\(\angle D+\angle E+\angle F = 180^{\circ}\)), \(\angle D=180^{\circ}-\angle E - \angle F\). Given \(\angle E = 120^{\circ}\) and \(\angle F=35^{\circ}\), then \(\angle D=180^{\circ}-120^{\circ}-35^{\circ}=25^{\circ}\).

Step2: Check for congruence

We know the \(AAS\) (Angle - Angle - Side) congruence criterion. But we have no information about the side lengths. Just having three angles equal (\(\angle A=\angle D = 25^{\circ}\), \(\angle B=\angle E = 120^{\circ}\), \(\angle C=\angle F = 35^{\circ}\)) is not sufficient for congruence. However, by the \(AA\) (Angle - Angle) similarity criterion (if two angles of one triangle are equal to two angles of another triangle, the triangles are similar), since \(\angle B=\angle E\) and \(\angle C=\angle F\), \(\triangle ABC\sim\triangle DEF\).

Part b

Step1: Calculate the third angle of each triangle

In \(\triangle NOM\), using the angle - sum property of a triangle (\(\angle N+\angle O+\angle M=180^{\circ}\)), \(\angle N = 180^{\circ}-\angle O-\angle M\). Given \(\angle O = 80^{\circ}\) and \(\angle M = 15^{\circ}\), then \(\angle N=180^{\circ}-80^{\circ}-15^{\circ}=85^{\circ}\).
In \(\triangle PQR\), using the angle - sum property of a triangle (\(\angle P+\angle Q+\angle R = 180^{\circ}\)), \(\angle P=180^{\circ}-\angle Q-\angle R\). Given \(\angle Q = 80^{\circ}\) and \(\angle R\) (we know \(\angle M=\angle P = 15^{\circ}\), \(\angle O=\angle Q = 80^{\circ}\)), then \(\angle R=180^{\circ}-80^{\circ}-15^{\circ}=85^{\circ}\).

Step2: Check for congruence

We have \(\angle M=\angle P = 15^{\circ}\), \(\angle O=\angle Q = 80^{\circ}\), and \(NO = QR=2\). By the \(AAS\) (Angle - Angle - Side) congruence criterion (if two angles and a non - included side of one triangle are equal to two angles and the corresponding non - included side of another triangle, the triangles are congruent), \(\triangle NOM\cong\triangle RQP\).

Answer:

a. The triangles are similar but not congruent.
b. The triangles are congruent.