Observe the map. Consider the triangle formed by Pine St., Main Ave., and Summer Rd. AND the triangle formed by Summer Rd., Spruce St., and Main Ave. Provide one reason why these TWO triangles MUST be congruent.

Question

Observe the map. Consider the triangle formed by Pine St., Main Ave., and Summer Rd. AND the triangle formed by Summer Rd., Spruce St., and Main Ave. Provide one reason why these TWO triangles MUST be congruent.

Answer 1

These two triangles must be congruent because they both have the same side lengths and angles.

Answer 2

To determine why these two triangles must be congruent, we can analyze their corresponding sides and angles.

To get a better understanding, let's break down the steps to prove the congruence between these triangles:

1. Identify the overlapping sides: In this case, we have Main Ave. as the common side between the two triangles.
2. Identify the common angles: Here, the angle at the intersection of Main Ave. and Summer Rd. is the common angle for both triangles.
3. Prove that the corresponding sides and angles are congruent: To do this, we can use the Congruence Postulates or the side-angle-side (SAS) congruence criterion.

Now, let's apply these steps to explain the congruence between the two triangles formed:

1. Corresponding sides: The segment of Summer Rd. is shared as a side by both triangles, making it one pair of corresponding sides.

2. Common angle: The angle formed at the intersection of Main Ave. and Summer Rd. is shared by both triangles, making it the common angle between the two.

3. To prove that these triangles are congruent, we need to demonstrate that all corresponding sides and angles are equal. Since the common side and angle are already identified, we only need to prove the remaining sides and angles.

4. Additional information: To ensure that the remaining sides and angles are congruent, we might need more information, such as measures of certain angles or lengths of specific sides.

In conclusion, without additional information, we cannot definitively claim that the two triangles formed by Pine St., Main Ave., and Summer Rd. and by Summer Rd., Spruce St., and Main Ave. are congruent. However, if we had additional information that could prove the remaining sides and angles were equal, we could then establish their congruence.

Answer 3

One reason why these two triangles must be congruent is that they share the same three side lengths. In both triangles, the sides are formed by Pine St., Main Ave., and Summer Rd., and the sides are also formed by Summer Rd., Spruce St., and Main Ave. Two triangles are congruent if and only if their corresponding angles and side lengths are equal, so since these two triangles have the same side lengths, they must be congruent.