Question

g.gsr.3.4 (mc)
triangle abc is an isosceles triangle where . also, . what else are we able to prove in order to prove these triangles are congruent using sas?
image of triangle abc with ab and ac marked congruent, bd and dc marked congruent, and ad as a segment from a to d on bc
options:

• first option text missing because they are corresponding sides.
• $\angle bad \cong \angle cad$ because they are corresponding angles.
• $\angle abd \cong \angle acd$ because the base angles of an isosceles triangle are congruent.
• fourth option text missing using the reflexive property.

Explanation:

Step1: Recall SAS Congruence

SAS (Side - Angle - Side) congruence criterion 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 $\triangle ABC$, $AB = AC$ (given as isosceles with markings), $BD=DC$ (markings on $BC$). We need the included angle between these sides. The common side is $AD$, but for the angle, we need the angle between $AB$ and $AD$ and between $AC$ and $AD$.

Step2: Analyze Each Option

• Option 1: Corresponding sides - but we need an angle for SAS, so this is incorrect.
• Option 2: $\angle BAD\cong\angle CAD$: In $\triangle ABD$ and $\triangle ACD$, $AB = AC$, $BD = DC$, and if $\angle BAD=\angle CAD$, then by SAS (since $AB$ and $AD$ with included $\angle BAD$ in $\triangle ABD$ and $AC$ and $AD$ with included $\angle CAD$ in $\triangle ACD$), the triangles are congruent. This angle is the included angle between the two sides.
• Option 3: $\angle ABD\cong\angle ACD$: These are base angles, but they are not the included angles for the sides $AB, BD$ and $AC, DC$ in the SAS criterion, so this is incorrect.
• Option 4: Reflexive property for a side - but we need an angle for SAS, so this is incorrect.

Answer:

$\angle BAD \cong \angle CAD$ because they are corresponding angles.