Question

what is the included angle between sides ab and ac in △abc?
a. ∠bca
b. ∠bac
c. ∠abc
d. ∠acb

if ab = 7cm, ac = 5cm, and ∠bac = 50°, which additional triangle could be congruent to △abc using sas?
a. △def with de = 7cm, df = 6cm, ∠edf = 50°
b. △def with de = 7cm, df = 5cm, ∠edf = 50°
c. △def with de = 5cm, df = 7cm, ∠edf = 60°
d. △def with de = 8cm, df = 5cm, ∠edf = 50°

if two triangles satisfy the sas criterion, what can you conclude about them?
a. their corresponding angles are equal
b. they have the same area
c. they are congruent
d. they are similar but not congruent

Explanation:

Step1: Recall included - angle definition

The included angle between two sides of a triangle is the angle formed by those two sides. In $\triangle ABC$, the included angle between sides $AB$ and $AC$ is $\angle BAC$.

Step2: Recall SAS congruence criterion

The Side - Angle - Side (SAS) congruence criterion states that if two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, then the two triangles are congruent. Given $AB = 7$cm, $AC = 5$cm, and $\angle BAC=50^{\circ}$ in $\triangle ABC$, for $\triangle DEF$ to be congruent to $\triangle ABC$ by SAS, we need $DE = AB = 7$cm, $DF = AC = 5$cm, and $\angle EDF=\angle BAC = 50^{\circ}$.

Step3: Recall the meaning of SAS criterion

The SAS (Side - Angle - Side) criterion is a rule for triangle congruence. If two sides and the included - angle of one triangle are equal to two sides and the included - angle of another triangle, the two triangles are congruent.

Answer:

1. b. $\angle BAC$
2. b. $\triangle DEF$ with $DE = 7$cm, $DF = 5$cm, $\angle EDF = 50^{\circ}$
3. c. They are congruent