Congruence: Triangle Congruence from Rigid Motions (ASA, SAS, SSS) (CCSS.G-CO.7)
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Geometry › Congruence: Triangle Congruence from Rigid Motions (ASA, SAS, SSS) (CCSS.G-CO.7)
In △ABC, the side lengths are $AB=5$, $BC=12$, $AC=13$. In △DEF, the side lengths are $DE=5$, $EF=12$, $DF=13$.
Which congruence condition proves △ABC ≅ △DEF?
SSS
SAS
SSA
AAA
Explanation
All three pairs of corresponding sides are congruent, so by SSS the triangles are congruent under a rigid motion. SSA is not sufficient to fix a triangle, and AAA establishes only similarity. SAS would require an included angle measure, which is not given here.
In △ABC, $AB=7$, $AC=10$, and $m\angle A=58^\circ$. In △DEF, $DE=7$, $DF=10$, and $m\angle D=58^\circ$.
Which congruence condition proves △ABC ≅ △DEF?
SSA
SAS
AAA
ASA
Explanation
Two corresponding sides and the included angle are congruent, so SAS proves the triangles congruent via a rigid motion. SSA is ambiguous and does not guarantee congruence, AAA gives only similarity, and ASA would require two angles and the included side instead.
In △ABC, $m\angle A=45^\circ$, $m\angle B=65^\circ$, and $AB=9$. In △DEF, $m\angle D=45^\circ$, $m\angle E=65^\circ$, and $DE=9$.
Which congruence condition proves △ABC ≅ △DEF?
SSA
AAS
ASA
AAA
Explanation
Two angles and the included side between them are congruent, so ASA proves congruence by a rigid motion. AAS would use a side not included between the given angles, SSA is not sufficient, and AAA only ensures similarity.
In △ABC, $m\angle B=70^\circ$, $m\angle C=40^\circ$, and $AB=8$. In △DEF, $m\angle E=70^\circ$, $m\angle F=40^\circ$, and $DE=8$.
Which congruence condition proves △ABC ≅ △DEF?
ASA
SSA
AAA
AAS
Explanation
Two angles and a non-included side are congruent, which is the AAS condition; this guarantees congruence via a rigid motion. ASA would require the side between the two angles (here it would be $BC$/ $EF$), SSA is not sufficient, and AAA only establishes similarity.
In △ABC, $AB=6$, $BC=9$, and $m\angle B=92^\circ$. In △DEF, $DE=6$, $EF=9$, and $m\angle E=92^\circ$.
Which congruence condition proves △ABC ≅ △DEF?
AAA
SAS
AAS
SSA
Explanation
Two sides and their included angle are congruent in order, so SAS proves the triangles congruent by a rigid motion. AAA only gives similarity, AAS would require two angles and a non-included side, and SSA is ambiguous and not a valid congruence condition.
Triangles ABC and DEF are positioned so that A↔D, B↔E, C↔F. Given AB = 7, BC = 9, AC = 5 and DE = 7, EF = 9, DF = 5.
Which congruence condition proves △ABC ≅ △DEF?
SSS
SSA
AAA
SAS
Explanation
All three pairs of corresponding sides are congruent, so SSS guarantees a rigid motion maps △ABC to △DEF. SSA can be ambiguous and AAA gives only similarity, not congruence.
Triangles ABC and DEF are positioned so that A↔D, B↔E, C↔F. Given AB = 8, BC = 6, ∠B = ∠E = 40°, and DE = 8, EF = 6.
Which congruence condition proves △ABC ≅ △DEF?
AAA
SAS
SSA
SSS
Explanation
Two corresponding sides and their included angle are congruent, so SAS ensures a rigid motion matches the triangles. SSA can fail (ambiguous case), and AAA gives only similarity; SSS is not established because the third side is unknown.
Triangles ABC and DEF are positioned so that A↔D, B↔E, C↔F. Given ∠A = ∠D = 60°, ∠C = ∠F = 50°, and AC = DF = 7.
Which congruence condition proves △ABC ≅ △DEF?
SAS
AAA
ASA
SSA
Explanation
Two angles and the included side are congruent (ASA), which determines a unique triangle up to rigid motion. SAS is not met because two sides are not given; SSA is insufficient, and AAA is only similarity.
Triangles ABC and DEF are positioned so that A↔D, B↔E, C↔F. Given ∠A = ∠D = 35°, ∠B = ∠E = 65°, and AC = DF = 10 (note the given side is not between the two angles).
Which congruence condition proves △ABC ≅ △DEF?
AAA
SAS
SSA
AAS
Explanation
Two angles and a non-included side are congruent (AAS), so a rigid motion can map one triangle to the other. SSA can be ambiguous and AAA is only similarity; SAS is not satisfied because the given side is not between the two given angles.
Triangles ABC and DEF are positioned so that A↔D, B↔E, C↔F. Given ∠B = ∠E = 90°, AB = DE = 10, and BC = EF = 6.
Which congruence condition proves △ABC ≅ △DEF?
SAS
SSS
AAA
SSA
Explanation
Two sides with the included right angle are congruent, so SAS proves congruence via a rigid motion. SSS is not established (third side not given), SSA is ambiguous, and AAA gives only similarity.