Congruence: Congruence via Rigid Motions (CCSS.G-CO.6)
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Common Core High School Geometry › Congruence: Congruence via Rigid Motions (CCSS.G-CO.6)
Given: In △ABC and △DEF, AB = DE = 6, AC = DF = 8, and ∠A = ∠D = 40°. △ABC ≅ △DEF by which criterion?
SSS
ASA
SAS
AAS
Explanation
Two pairs of corresponding sides and the included angle (between those sides) are congruent, so SAS applies. ASA/AAS would require two angles; here the angle is included between the two known sides.
Given: ∠A = ∠D = 50°, ∠C = ∠F = 60°, and AC = DF = 9 in △ABC and △DEF. △ABC ≅ △DEF by which criterion?
ASA
AAS
SSA
SSS
Explanation
Two angles and the included side (AC between ∠A and ∠C, and DF between ∠D and ∠F) are congruent, so ASA. SSA does not guarantee congruence.
Given: In △ABC and △DEF, AB = DE = 5, BC = EF = 7, and AC = DF = 9. △ABC ≅ △DEF by which criterion?
SAS
AAS
ASA
SSS
Explanation
All three pairs of corresponding sides are congruent, so SSS. Angle-based criteria (ASA or AAS) are not applicable because no angles are given.
Given: ∠B = ∠E = 35°, ∠C = ∠F = 65°, and AB = DE = 10 in △ABC and △DEF. The side given is not between the two known angles. △ABC ≅ △DEF by which criterion?
ASA
AAS
SSA
HL
Explanation
Two angles and a non-included side are congruent, so AAS. HL would require right triangles, which are not specified.
Given: ∠A and ∠D are right angles, BC = EF (hypotenuse), and AC = DF (a leg) in △ABC and △DEF. △ABC ≅ △DEF by which criterion?
HL
SSS
SSA
SAS
Explanation
For right triangles, congruent hypotenuse and one corresponding leg establish congruence by HL. SSA is not a valid congruence test in general.
Given $AB=6$, $AC=8$, and $\angle A=40^\circ$; and $DE=6$, $DF=8$, and $\angle D=40^\circ$. $\triangle ABC \cong \triangle DEF$ by which criterion?
SSS
SAS
AAS
ASA
Explanation
Two pairs of corresponding sides and the included angle are congruent, so SAS. It is not ASA or AAS because only one angle is given.
Given $\angle A=50^\circ$, $\angle C=70^\circ$, and $AC=9$; and $\angle D=50^\circ$, $\angle F=70^\circ$, and $DF=9$. $\triangle ABC \cong \triangle DEF$ by which criterion?
ASA
AAS
SAS
SSA
Explanation
Two angles and the included side ($AC$ between $\angle A$ and $\angle C$) match, so ASA. A common error is to choose AAS, but here the given side lies between the two angles. SSA is not a valid congruence test.
Given $AB=7$, $BC=5$, $AC=9$; and $DE=7$, $EF=5$, $DF=9$. $\triangle ABC \cong \triangle DEF$ by which criterion?
SAS
AAS
ASA
SSS
Explanation
All three corresponding sides are congruent, so SSS. No angles are provided, so SAS/ASA/AAS do not apply.
Given $\angle B=35^\circ$, $\angle C=75^\circ$, and $AB=10$; and $\angle E=35^\circ$, $\angle F=75^\circ$, and $DE=10$. $\triangle ABC \cong \triangle DEF$ by which criterion?
SAS
SSA
AAS
ASA
Explanation
Two angles and a non-included side are given, so AAS. ASA would require the side between the two angles (which would be $BC$), and SSA is not a guaranteed congruence condition.
Given $\angle A=62^\circ$, $\angle B=48^\circ$, and $AB=11$; and $\angle D=62^\circ$, $\angle E=48^\circ$, and $DE=11$. $\triangle ABC \cong \triangle DEF$ by which criterion?
ASA
AAS
SSS
SAS
Explanation
Two angles and the included side ($AB$ between $\angle A$ and $\angle B$) match, so ASA. Although one could compute the third angle, that is unnecessary here.