AAS
Triangles A and B each have two matching angles and a corresponding side that is not between the two angles.
Are the triangles congruent by AAS?
Yes — matching two angles and a non-included side is enough for AAS congruence.
If two triangles each have two angles that match and the included side between those angles is the same, what can you conclude about the triangles?
They are congruent.
ASA states that two angles and the included side determine a unique triangle.
In triangle ABC, you know side AB, side AC, and the included angle ∠A. Can you determine triangle ABC uniquely using SAS?
Yes. SAS guarantees a unique triangle because the known angle is between the two known sides.
If triangle ABC has sides AB = BC = CA, what type of triangle is it?
It is an equilateral triangle.
Two right triangles each have a right angle. They also have congruent hypotenuses and one pair of congruent legs. Are the triangles congruent?
Yes. By the Hypotenuse–Leg (HL) Congruence Theorem, the triangles are congruent.
In two triangles, ∠X ≅ ∠P and ∠Y ≅ ∠Q.
Side XY ≅ side PQ.
Does this prove the triangles are congruent?
Yes — this directly fits the AAS pattern.
Triangle A and Triangle B have one side that is equal, and the angles at both ends of that side match. What congruence rule applies?
ASA congruence applies.
The side is between the two equal angles, which satisfies the ASA condition.
You are given two triangles, ΔABC and ΔDEF. If AB = DE, AC = DF, and ∠A = ∠D, what can you conclude about the two triangles?
The triangles are congruent by the SAS congruence theorem.
If triangle DEF has sides DE = EF but DF is different, what type of triangle is it?
It is an isosceles triangle.
Triangle A and Triangle B are right triangles. Their hypotenuses match in length, and their legs opposite the right angle on one side also match. Can you conclude Triangle A ≅ Triangle B?
Yes. Because both are right triangles and have a congruent hypotenuse and one congruent leg, they are congruent by HL.
Triangle RST has two angles congruent to two angles in triangle ABC.
A side in triangle RST opposite one of the known angles is congruent to the corresponding side in triangle ABC.
Does AAS apply?
Yes — the side doesn’t need to be between the angles. So the triangles are AAS congruent.
If two triangles have two corresponding angles equal and a side not between those angles equal, can ASA still be used?
Yes, ASA still works.
When two angles are known to be equal, the third angle is automatically equal, making the side effectively between two known angles.
Is SAS still valid if the known angle is not the included angle between the two known sides?
No. SAS only works when the angle is the included angle. If the angle is not included, the triangle may not be uniquely determined.
Two triangles, ABC and XYZ, have matching side lengths:
The triangles are congruent by SSS congruence.
You are told that Triangle RST and Triangle UVW are right triangles with congruent hypotenuses. You also know that RS ≅ UV. If ∠R and ∠U are the right angles, can HL be used to prove the triangles congruent?
Yes. Since ∠R and ∠U are right angles and they share a congruent hypotenuse and one corresponding leg, the triangles are congruent by HL.
Triangles MNO and DEF each have two congruent angles.
In addition, side MO ≅ side DF, and that side is not the one between the two angles.
Can you use AAS to show the triangles are congruent?
Yes — the non-included side still allows AAS.
Two triangles each have one angle equal, a side adjacent to that angle equal, and another angle on the other side of the same side equal. Does this guarantee the triangles are congruent?
Yes, the triangles are congruent by ASA.
The side lies between the two known equal angles.
If triangles ABC and XYZ satisfy the conditions AB = XY, BC = YZ, and ∠B = ∠Y, what corresponding parts of the triangles must also match because of SAS?
All corresponding parts match:
AC = XZ, ∠A = ∠X, and ∠C = ∠Z.
The triangles are fully congruent.
Triangle KLM has sides KL = LM, and triangle PQR has sides PQ = QR, with all three sides of triangle KLM equal to the three sides of triangle PQR (matching one-to-one). What additional conclusion can be made beyond congruence?
Since both are isosceles with all sides matching, the triangles are congruent and their matching angles are equal, including the angles opposite the equal sides.
Two right triangles share one leg in common. Their hypotenuses are also congruent. The shared leg corresponds to each other. Can these triangles be proven congruent using HL even though they share a side?
Yes. Shared sides are still congruent by the Reflexive Property, and with congruent hypotenuses, the triangles are congruent by HL.
A student argues that two triangles cannot be proven congruent because the side that matches is not between the two matching angles.
The triangles do have two pairs of congruent angles and a corresponding side that is outside the included angle pair.
Is the student correct?
No — the student is incorrect.
An included side is not required for AAS.
Two angles + any corresponding non-included side is enough to prove triangle congruence.
You know two triangles share one angle in common, and each triangle has another angle equal to each other. A side in one triangle lies between the known angles and matches the corresponding side in the other. Are the triangles congruent?
Yes, they must be congruent by ASA.
Two angles and the included side uniquely determine the triangle, even if one angle is shared.
If two triangles are shown to be congruent by SAS, what does this guarantee about the relationships between their corresponding angles and sides, and why does SAS alone ensure this?
SAS guarantees that all corresponding angles and sides are equal because once two sides and the included angle are fixed, only one possible triangle can be formed. This means the entire shape is determined, forcing all other parts to match.
In triangle STU, sides follow ST < TU < US. Another triangle ABC has sides AB < BC < CA, with each side corresponding in order to the inequalities of triangle STU (smallest to smallest, etc.). If all three pairs of sides match exactly, what can you conclude about the relationship between the two triangles?
The triangles are congruent by SSS, and their angles match in order:
Triangle ABC and Triangle DEF are right triangles. You know AB ≅ DE and AC ≅ DF. However, it is not stated which angle is the right angle in either triangle. Can you conclude the triangles are congruent using HL?
No. For HL to apply, you must know which angle is the right angle so you can identify the hypotenuse. Without knowing the right angle locations, you cannot use HL.