Problem 64 If two triangles are exactly the... [FREE SOLUTION] (2024)

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First, let’s talk about triangle congruence. Congruent triangles are triangles that are identical in both shape and size. This means all their corresponding sides are equal in length, and all corresponding angles are equal. When two triangles are congruent, you can place one over the other, and they will match exactly.

There are several criteria to determine triangle congruence, including:

• Side-Side-Side (SSS) - where all three pairs of corresponding sides are equal.
• Side-Angle-Side (SAS) - where two pairs of corresponding sides and the angle between them are equal.
• Angle-Side-Angle (ASA) - where two pairs of corresponding angles and the side between them are equal.
• Angle-Angle-Side (AAS) - where two pairs of corresponding angles and a non-included side are equal.

So, when the problem states that two triangles are exactly the same shape and size, it means they are congruent by any of these criteria.

Next, let's explore the geometric properties of triangles. Triangles have various properties that differentiate them and help in identifying their type and comparability.

For example, the sum of the internal angles of any triangle is always 180 degrees. This is true whether the triangle is scalene, isosceles, or equilateral. Besides angles, triangles can be compared using their sides, like in the SSS or SAS criteria mentioned above.

Understanding these geometric properties helps to determine if two triangles are similar or congruent. Remember, similar triangles have the same shape but not necessarily the same size, whereas congruent triangles match exactly in both shape and size.

Now let's delve into corresponding angles. Corresponding angles are pairs of angles that occupy the same relative position at each intersection where a straight line crosses two others.

In the context of triangles, corresponding angles of two triangles are compared to see if they are equal. For two triangles to be similar, all the corresponding angles must be equal. Therefore, if two triangles have equal corresponding angles, it means each angle in one triangle is equal to the corresponding angle in the other triangle.

As the step-by-step solution mentions, if two triangles are the same shape and size, then their corresponding angles are naturally equal, aiding in their classification as congruent.

Lastly, let’s discuss proportional sides. This concept is crucial when identifying similar triangles. If two triangles are similar, the lengths of their corresponding sides are proportional. This means the ratio of one side in one triangle to the corresponding side in the other triangle remains constant across all three sides.

Mathematically, if triangle ABC is similar to triangle DEF, then:

• AB/DE = BC/EF = AC/DF

If the triangles are the same size and shape, not only are their corresponding angles equal, but their sides are also proportional with a ratio of 1:1. This proportionality confirms similarity, and when the ratio is 1:1, it establishes congruence as well.