Here are the solutions for each problem on the worksheet. For each pair of triangles, we determine if they are congruent and identify the rule used (SSS, SAS, ASA, AAS, or HL).
1)
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Analysis: We see two pairs of sides marked with tick marks (one pair has 2 ticks, one pair has 3 ticks). The angles where these sides meet are vertical angles, which means they are equal.
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Rule: Side-Angle-Side (the angle is between the two known sides).
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Result: Congruent by
SAS.
2)
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Analysis: We have two pairs of angles marked equal. The side connecting these two angles is marked with a single tick mark in both triangles.
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Rule: Angle-Side-Angle (the side is between the two known angles).
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Result: Congruent by
ASA.
3)
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Analysis: There are two pairs of sides marked equal. They share a third side in the middle (Reflexive Property). This gives us three pairs of equal sides.
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Rule: Side-Side-Side.
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Result: Congruent by
SSS.
4)
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Analysis: We have two pairs of sides marked equal. However, the angle marked is *not* between these two sides (it is opposite the unmarked side). This is the "Side-Side-Angle" case, which does not prove congruence.
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Result: Not Congruent.
5)
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Analysis: These are right-angled triangles (indicated by the square symbol). They share a hypotenuse (the side opposite the right angle) in the middle. They also have a pair of legs marked equal.
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Rule: Hypotenuse-Leg (specific to right triangles).
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Result: Congruent by
HL.
6)
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Analysis: We have two pairs of sides marked equal. They share a third side in the middle. This creates three pairs of equal sides.
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Rule: Side-Side-Side.
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Result: Congruent by
SSS.
7)
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Analysis: We have two pairs of angles marked equal. We also have a pair of sides marked equal. Crucially, the side is *not* between the two angles; it is next to them.
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Rule: Angle-Angle-Side.
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Result: Congruent by
AAS.
8)
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Analysis: We have two pairs of sides marked equal and one pair of angles marked equal. However, the angle is not included between the sides (it's SSA), and there is no indication that these are right triangles (so we can't use HL).
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Result: Not Congruent.
9)
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Analysis: Looking at the markings, one triangle has an angle between two specific sides, while the other has an angle next to those sides. The corresponding parts do not match up correctly.
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Result: Not Congruent.
10)
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Analysis: We have two pairs of angles marked equal. The side marked with double ticks is located between these two angles in both triangles.
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Rule: Angle-Side-Angle.
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Result: Congruent by
ASA.
Final Answer:
1) SAS
2) ASA
3) SSS
4) Not Congruent
5) HL
6) SSS
7) AAS
8) Not Congruent
9) Not Congruent
10) ASA
Parent Tip: Review the logic above to help your child master the concept of sas sss asa aas worksheet.
