Question

1. what does the “included angle” mean in triangle congruence?

a. the angle opposite the given sides
b. the angle formed by the two given sides
c. the largest angle in the triangle
d. the smallest angle in the triangle

1. if ab = 7 cm, ac = 6 cm, and ∠bac = 50°, which additional triangle could be congruent to △abc using sas?

a. △def with de = 5 cm, df = 7 cm, ∠edf = 80°
b. △def with de = 8 cm, df = 5 cm, ∠edf = 50°
c. △def with de = 7 cm, df = 6 cm, ∠edf = 50°
d. △def with de = 7 cm, df = 5 cm, ∠edf = 50°

1. why is the sas method common in bridge design?

a. it allows for flexibility in side lengths.
b. it only uses angles for construction.
c. it minimizes the number of measurements needed.
d. it ensures stability by relying on fixed angles and sides.

Explanation:

1. In triangle congruence, the included - angle is the angle formed by two given sides. For example, in $\triangle ABC$, if we consider sides $AB$ and $AC$, $\angle BAC$ is the included - angle between them.
2. 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 $DE = 7$ cm, $DF = 5$ cm, and $\angle EDF = 50^{\circ}$.
3. In bridge design, the SAS (Side - Angle - Side) method is common because it ensures stability by relying on fixed angles and sides. Triangles with fixed side - lengths and included angles have a rigid structure, which is crucial for the stability of bridges.

Answer:

1. B. The angle formed by the two given sides
2. D. $\triangle DEF$ with $DE = 7$ cm, $DF = 5$ cm, $\angle EDF = 50^{\circ}$
3. D. It ensures stability by relying on fixed angles and sides.