Question

what information is needed to prove triangles are congruent by asa?
a. all sides are equal
b. three angles are equal
c. two sides are equal
d. two angles and the included side are congruent

given that △jkl ≅ △mno by asa, which of the following statements is true?
a. ∠j ≅ ∠o
b. kl ≅ mn
c. ∠k ≅ ∠n
d. jk ≅ mo

if △abc ≅ △def by asa, which of the following pairs of angles are congruent?
a. ∠a and ∠f
b. ∠a and ∠e
c. ∠c and ∠d
d. ∠b and ∠e

given that △pqr ≅ △xyz by asa, which of the following pairs of sides are congruent?
a. pr and yz
b. pq and xy
c. pq and yz
d. qr and xz

why is the included side important in asa triangle congruence?
a. it proves that the triangles are equilateral
b. it determines the area of the triangles
c. it ensures the angles form a unique triangle
d. it verifies the similarity of the triangles

Explanation:

• First question: The ASA (Angle - Side - Angle) congruence criterion requires two angles and the included side to be congruent. Option a is SSS (Side - Side - Side), option b is not a valid congruence criterion (AAA gives similarity), option c is not enough for ASA.
• Second question: If \(\triangle JKL\cong\triangle MNO\) by ASA, then corresponding angles are congruent. \(\angle K\) and \(\angle N\) are corresponding angles. \(\angle J\) corresponds to \(\angle M\), \(KL\) corresponds to \(NO\), \(JK\) corresponds to \(MN\).
• Third question: If \(\triangle ABC\cong\triangle DEF\) by ASA, then \(\angle B\) and \(\angle E\) are corresponding angles (as the side between the two angles in \(\triangle ABC\) and \(\triangle DEF\) will have the two angles adjacent to it, and \(\angle B\) and \(\angle E\) are in the correct corresponding positions for ASA). \(\angle A\) corresponds to \(\angle D\), \(\angle C\) corresponds to \(\angle F\).
• Fourth question: If \(\triangle PQR\cong\triangle XYZ\) by ASA, then \(PQ\) and \(XY\) are the sides between the two angles (included sides) and are congruent. \(PR\) corresponds to \(XZ\), \(QR\) corresponds to \(YZ\).
• Fifth question: The included side in ASA ensures that the two angles form a unique triangle. It has nothing to do with the triangle being equilateral (option a), area (option b), or similarity (option d, ASA is for congruence).

Answer:

1. d. Two angles and the included side are congruent
2. c. \(\angle K\cong\angle N\)
3. d. \(\angle B\) and \(\angle E\)
4. b. \(PQ\) and \(XY\)
5. c. It ensures the angles form a unique triangle