Congruence: Theorems about Lines and Angles (CCSS.G-CO.8)
Common Core High School Geometry · Learn by Concept
Help Questions
Common Core High School Geometry › Congruence: Theorems about Lines and Angles (CCSS.G-CO.8)
Given: In $\triangle ABC$, $AB=6$, $BC=8$, and $\angle B=52^\circ$. In $\triangle DEF$, $DE=6$, $EF=8$, and $\angle E=52^\circ$. Which congruence condition proves $\triangle ABC \cong \triangle DEF$?
SSA
SAS
AAA
AAS
Explanation
SAS: two pairs of corresponding sides and their included angle match, so a rigid motion can carry one triangle onto the other (rigid motions preserve distances and angles). SSA can be ambiguous (can form two different triangles), and AAA fixes only shape (similarity), not size, so neither guarantees congruence.
Given: In $\triangle ABC$, $\angle A=43^\circ$, $\angle B=71^\circ$, and $AC=10$. In $\triangle DEF$, $\angle D=43^\circ$, $\angle E=71^\circ$, and $DF=10$. Which congruence condition proves $\triangle ABC \cong \triangle DEF$?
SSA
AAA
ASA
AAS
Explanation
AAS: two angles and a non-included side determine a unique triangle up to rigid motion, so one triangle can be mapped to the other via a distance- and angle-preserving motion. AAA gives only similarity (scale not fixed), and SSA does not guarantee a unique triangle.
Given: In $\triangle ABC$, $AB=7$, $BC=9$, and $AC=12$. In $\triangle DEF$, $DE=7$, $EF=9$, and $DF=12$. Which congruence condition proves $\triangle ABC \cong \triangle DEF$?
SSS
SAS
ASA
SSA
Explanation
SSS: all three corresponding sides match, so there is a rigid motion taking one triangle to the other (rigid motions preserve lengths). SSA is not sufficient because two sides and a non-included angle can produce two different non-congruent triangles.
Given: In $\triangle ABC$, $\angle A=35^\circ$, $\angle C=68^\circ$, and $AC=9$. In $\triangle DEF$, $\angle D=35^\circ$, $\angle F=68^\circ$, and $DF=9$. Which congruence condition proves $\triangle ABC \cong \triangle DEF$?
AAA
AAS
ASA
SSA
Explanation
ASA: two angles and the included side are equal, fixing the triangle up to a rigid motion. AAA alone gives only similarity (scale not determined), and SSA can be ambiguous and does not ensure congruence.
Given: In $\triangle ABC$, $\angle B=50^\circ$, $\angle C=65^\circ$, and $AB=11$. In $\triangle DEF$, $\angle E=50^\circ$, $\angle F=65^\circ$, and $DE=11$. Which congruence condition proves $\triangle ABC \cong \triangle DEF$?
AAS
ASA
SSS
AAA
Explanation
AAS: two angles and a non-included side match, so a rigid motion exists mapping one triangle to the other because distances and angles are preserved. AAA would only imply similarity (not fixed side lengths), and SSS is not supported by the given data.
Two triangles $\triangle ABC$ and $\triangle DEF$ satisfy: $AB=8$, $BC=5$, $\angle ABC=52^\circ$; $DE=8$, $EF=5$, $\angle DEF=52^\circ$. The given angle in each triangle is included between the two given sides ($AB$ with $BC$, and $DE$ with $EF$).
Which congruence condition proves $\triangle ABC \cong \triangle DEF$?
SSA
AAA
SAS
AAS
Explanation
SAS: two pairs of corresponding sides and the included angle match, so a rigid motion can map one triangle to the other. SSA is not a valid congruence test (it can lead to two different triangles), and AAA guarantees only similarity. AAS would require two angles and a non-included side, which we do not have here.
In $\triangle ABC$: $\angle A=40^\circ$, $\angle B=70^\circ$, and $AC=9$. In $\triangle DEF$: $\angle D=40^\circ$, $\angle E=70^\circ$, and $DF=9$. The given side in each triangle is not between the two given angles.
Which congruence condition proves $\triangle ABC \cong \triangle DEF$?
SAS
SSA
AAA
AAS
Explanation
AAS: two angles and a non-included side determine a unique triangle up to rigid motion. AAA gives only similarity (scale can change), and SSA is ambiguous (it may produce two different triangles). SAS would require the included angle between two given sides, which is not the case here.
All three pairs of corresponding sides match: in $\triangle ABC$, $AB=6$, $BC=7$, $AC=5$; in $\triangle DEF$, $DE=6$, $EF=7$, $DF=5$.
Which congruence condition proves $\triangle ABC \cong \triangle DEF$?
SSS
SAS
SSA
AAA
Explanation
SSS: three pairs of equal corresponding side lengths fix the shape up to a rigid motion. AAA does not fix scale (only similarity), and SSA is not a valid congruence test. SAS would require an included angle, but we do not need angles when all three sides match.
In $\triangle ABC$: $\angle A=60^\circ$, $\angle C=50^\circ$, and the included side $AC=10$. In $\triangle DEF$: $\angle D=60^\circ$, $\angle F=50^\circ$, and the included side $DF=10$. The side given in each triangle lies between the two given angles.
Which congruence condition proves $\triangle ABC \cong \triangle DEF$?
AAA
ASA
SAS
SSA
Explanation
ASA: two angles and the included side determine a unique triangle up to rigid motion. AAA is only similarity. SAS would need two sides with their included angle (different data). SSA is ambiguous and does not ensure congruence.
Two sides and the included angle match: in $\triangle ABC$, $AB=6$, $AC=9$, and $\angle A=45^\circ$; in $\triangle DEF$, $DE=6$, $DF=9$, and $\angle D=45^\circ$. The given angle is between the given sides in each triangle.
Which congruence condition proves $\triangle ABC \cong \triangle DEF$?
SAS
AAA
SSA
SSS
Explanation
SAS: two corresponding sides and their included angle fix the triangle up to a rigid motion. AAA gives only similarity, and SSA is not a valid congruence condition. SSS would require the third side to be given; incorrectly adding $6+9$ to claim equal third sides is an arithmetic slip—only SAS is justified by the given data.