Question

1. which postulate or theorem states that two triangles are congruent if two angles and the included side of one triangle are congruent to two angles and the included side of the other triangle? a. sss congruence postulate b. asa congruence postulate c. hl congruence theorem d. sas congruence postulate 2. which of the following side lengths could not form a triangle congruent to one with sides 3 cm, 4 cm, and 5 cm by sss? a. 3 cm, 4 cm, 6 cm b. 4 cm, 5 cm, 3 cm c. 5 cm, 4 cm, 3 cm d. 4 cm, 3 cm, 5 cm 3. if △abc≅△def by asa, and ab = 5 cm, ∠a = 60°, and ∠b = 50°, what must be true? a. side df = 6 cm b. ∠f = 50° c. side de = 4 cm d. ∠d = 60° and ∠e = 50°

Explanation:

1. The ASA (Angle - Side - Angle) Congruence Postulate states that two triangles are congruent if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle.
2. By the SSS (Side - Side - Side) congruence, for two triangles to be congruent, their corresponding sides must be equal. A triangle with side - lengths 3 cm, 4 cm, 6 cm is not congruent to a triangle with side - lengths 3 cm, 4 cm, 5 cm as one side length is different.
3. If \(\triangle ABC\cong\triangle DEF\) by ASA, corresponding angles are equal. So if \(\angle A = 60^{\circ}\) and \(\angle B=50^{\circ}\) in \(\triangle ABC\), then \(\angle D = 60^{\circ}\) and \(\angle E = 50^{\circ}\) in \(\triangle DEF\) since they are corresponding angles.

Answer:

1. b. ASA Congruence Postulate
2. a. 3 cm, 4 cm, 6 cm
3. d. \(\angle D = 60^{\circ}\) and \(\angle E = 50^{\circ}\)