Question

if two triangles are congruent by asa, what does this mean for the two triangles?
a. their perimeters must be different
b. their angles are proportional
c. they are congruent in both shape and size
d. their areas must differ

if triangles △abc and △def are congruent by asa, which of the following statements must be true?
a. ∠b is congruent to ∠e
b. ∠c is larger than ∠d
c. ∠a is congruent to ∠f
d. side ab is equal to side df

in △abc and △def, if ∠a≅∠d, ∠c≅∠f, and ac≅df, which postulate proves that the triangles are congruent?
a. hl congruence theorem
b. sss congruence postulate
c. sas congruence postulate
d. asa congruence postulate

Explanation:

1. The ASA (Angle - Side - Angle) congruence postulate means that two triangles are identical in both shape and size when two angles and the included side of one triangle are congruent to two angles and the included side of another triangle. This implies that their perimeters and areas are equal, and angles are congruent (not just proportional).
2. For congruent triangles by ASA, corresponding angles are congruent. In \(\triangle ABC\) and \(\triangle DEF\) congruent by ASA, \(\angle B\) corresponds to \(\angle E\) and is congruent to it.
3. Given \(\angle A\cong\angle D\), \(\angle C\cong\angle F\), and \(AC\cong DF\), we have two angles and the included side of one triangle congruent to two angles and the included side of the other triangle, which is the ASA congruence postulate.

Answer:

1. c. They are congruent in both shape and size
2. a. \(\angle B\) is congruent to \(\angle E\)
3. d. ASA Congruence Postulate