Question

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

in the diagram below, if ∠p ≅ ∠t, ∠q ≅ ∠u, and pq ≅ tu, which postulate can be used to prove that △pqr ≅ △tuv?
a. hl congruence theorem
b. sas congruence postulate
c. sss congruence postulate
d. asa congruence postulate

in asa, why must the side be included between the two congruent angles?
a. to make sure the triangle is unique and congruent
b. to confirm the triangles are isosceles
c. to show that all angles are equal
d. to ensure that the triangles are similar

which of the following is necessary for two triangles to be congruent by asa?
a. three pairs of congruent sides
b. one pair of congruent sides and any angle
c. two pairs of congruent angles and the included side
d. two pairs of congruent sides

given that △xyz ≅ △abc by asa, which pair of sides must be congruent?
a. yz and ab
b. xy and ab
c. xz and ac
d. xz and bc

Explanation:

• First Question: Congruent triangles (by any congruence rule like ASA) are identical in shape and size. So, if two triangles are congruent by ASA, they are congruent in both shape and size.
• Second Question: The ASA (Angle - Side - Angle) congruence postulate states that if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent. Here, we have two angles ($\angle P\cong\angle T$ and $\angle Q\cong\angle U$) and the included side ($PQ\cong TU$), so we use the ASA Congruence Postulate.
• Third Question: In the ASA (Angle - Side - Angle) criterion, the side must be included between the two congruent angles to ensure that the triangle is unique. If the side is not included, multiple non - congruent triangles can be formed with the given angle - side - angle information.
• Fourth Question: The ASA (Angle - Side - Angle) congruence rule requires two pairs of congruent angles and the included side. Three pairs of congruent sides is SSS, one pair of congruent sides and any angle is not a valid congruence rule (except in specific cases like AAS which is a derivation of ASA), and two pairs of congruent sides is not sufficient for ASA.
• Fifth Question: If $\triangle XYZ\cong\triangle ABC$ by ASA, the included sides must be congruent. If we assume the correspondence of angles (say $\angle X\cong\angle A$, $\angle Y\cong\angle B$), then the included side $XY$ (between $\angle X$ and $\angle Y$) must be congruent to $AB$ (between $\angle A$ and $\angle B$)

Answer:

1. a. They are congruent in both shape and size
2. d. ASA Congruence Postulate
3. a. To make sure the triangle is unique and congruent
4. c. Two pairs of congruent angles and the included side
5. b. $XY$ and $AB$