Question

sort the following triangles
congruent triangles
worksheet a
(attach triangle cards
below)
(attach triangle cards
below)
triangle
congruency
statement
sss
sas
asa
aas
hl
not
necessarily
congruent
n/a

Explanation:

Step1: Identify SSS (Side - Side - Side)

Triangles with three equal sides (marked by the same number of tick marks on each side) are congruent by SSS. For example, the triangle with vertices \(M, N, L\) (assuming the tick marks indicate equal sides) and the triangle with vertices \(P, Q, R\) (if they have three equal sides) would go here. We look for triangles where all three sides are marked as equal.

Step2: Identify SAS (Side - Angle - Side)

Triangles with two equal sides and the included angle equal. So we find triangles with two sides marked equal and the angle between them (the included angle) equal. For instance, a right - angled triangle with two legs marked equal and the right angle (included angle) would be SAS if another triangle has the same.

Step3: Identify ASA (Angle - Side - Angle)

Triangles with two equal angles and the included side equal. We search for triangles where two angles are marked equal and the side between them (included side) is equal.

Step4: Identify AAS (Angle - Angle - Side)

Triangles with two equal angles and a non - included side equal. So we look for triangles with two angles marked equal and a side that is not between them equal.

Step5: Identify HL (Hypotenuse - Leg)

For right - angled triangles, if the hypotenuse and one leg are equal, they are congruent by HL. So we find right - angled triangles with hypotenuse and one leg marked equal.

Step6: Identify Not necessarily congruent

Triangles that do not fit any of the above congruence criteria (SSS, SAS, ASA, AAS, HL) are placed here. For example, a triangle with only one side or one angle marked equal and no other corresponding equal parts to match the congruence criteria.

Let's take a more detailed look at each triangle:

SSS:

• Triangle \(MNL\) (assuming tick marks on all three sides) and triangle \(PQR\) (if all three sides have tick marks) and the triangle with vertices \(S, R, T\) (if all three sides are marked equal) would be in the SSS category.

SAS:

• Consider a right - angled triangle with legs of length \(a\) and \(b\) (marked equal) and the right angle (included angle). If another right - angled triangle has legs of the same length and right angle, it is SAS. For example, the triangle with vertices \(D, E, F\) (right - angled, two legs marked) and a similar - looking right - angled triangle.

ASA:

• A triangle with two angles (e.g., \(\angle A = \angle D\), \(\angle B=\angle E\)) and the included side \(AB = DE\) would be ASA. We look for triangles with two angles marked equal and the side between them equal.

AAS:

• A triangle with \(\angle A=\angle D\), \(\angle C = \angle F\) and side \(BC=EF\) (non - included side) would be AAS. We find triangles with two angles marked equal and a non - included side equal.

HL:

• Right - angled triangles with hypotenuse and one leg equal. For example, the right - angled triangle with vertices \(A, B, C\) (right - angled, hypotenuse \(AC\) and leg \(AB\) marked) and another right - angled triangle with the same hypotenuse and leg length.

Not necessarily congruent:

• Triangles with only one side or one angle marked equal and no other corresponding equal parts to satisfy SSS, SAS, ASA, AAS, or HL. For example, a triangle with only one side marked equal and the other sides and angles not matching any congruence criteria with another triangle.

Answer:

To sort the triangles, we use the congruence criteria:

• SSS: Triangles with three equal sides (all three sides marked with the same number of tick marks).
• SAS: Triangles with two equal sides and the included angle equal.
• ASA: Triangles with two equal angles and the included side equal.
• AAS: Triangles with two equal angles and a non - included side equal.
• HL: Right - angled triangles with hypotenuse and one leg equal.
• Not necessarily congruent: Triangles that do not satisfy any of the above congruence criteria.

For example, if we assume the triangle with vertices \(M, N, L\) (three equal sides), triangle \(P, Q, R\) (three equal sides) are SSS. A right - angled triangle with two legs equal and right angle (included angle) is SAS. A triangle with two angles and included side equal is ASA, etc. The final sorting would be done by visually inspecting the tick marks (for equal sides) and angle marks (for equal angles) on each triangle and matching them to the congruence criteria.

(Note: Since the actual tick - mark and angle - mark details are a bit unclear from the image, the general method is as above. If we had exact details, we could place each triangle in the correct category. For example, if a triangle has three sides with 3 tick marks, it goes to SSS; if two sides with 2 tick marks and the included angle marked equal, it goes to SAS, etc.)