Question

in the diagram below of $\triangle age$ and $\triangle old$, $\angle gae \cong \angle lod$, and $\overline{ae} \cong \overline{od}$.

to prove that $\triangle age$ and $\triangle old$ are congruent by sas, what other information is needed?
a. $\overline{ge} \cong \overline{ld}$
b. $\overline{ag} \cong \overline{ol}$
c. $\angle age \cong \angle old$
d. $\angle aeg \cong \angle odl$

determine if each pair of triangles is congruent. if so, write the postulate or theorem.
options: sas, asa, the triangles are not congruent, aas

Explanation:

Step1: Recall SAS Congruence

SAS (Side - Angle - Side) congruence states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent.

In $\triangle AGE$ and $\triangle OLD$, we know that $\angle GAE\cong\angle LOD$ (the included angle) and $\overline{AE}\cong\overline{OD}$ (one side). For SAS, we need the other pair of sides that form the included angle with the given side. The sides forming $\angle GAE$ in $\triangle AGE$ are $\overline{AG}$ and $\overline{AE}$, and the sides forming $\angle LOD$ in $\triangle OLD$ are $\overline{OL}$ and $\overline{OD}$. So we need $\overline{AG}\cong\overline{OL}$.

Let's analyze the options:

• Option a: $\overline{GE}\cong\overline{LD}$: These are not the sides forming the included angle, so this is not for SAS.
• Option b: $\overline{AG}\cong\overline{OL}$: This is the other side forming the included angle with the given side, so this satisfies SAS.
• Option c: $\angle AGE\cong\angle OLD$: This is an angle, not a side, so it doesn't satisfy SAS.
• Option d: $\angle AEG\cong\angle ODL$: This is an angle, not a side, so it doesn't satisfy SAS.

Step2: Conclusion

Based on the SAS congruence postulate, the additional information needed is $\overline{AG}\cong\overline{OL}$, which is option b.

Answer:

B. $\overline{AG}\cong\overline{OL}$