Triangles are fundamental shapes in geometry that have fascinated mathematicians for centuries. They are formed by connecting three non-collinear points, and their properties and relationships provide a solid foundation for understanding more complex geometric concepts. In this article, we will explore the key aspects of homework 1 on triangle relationships and delve into the solutions.

The first question in homework 1 asks us to determine the type of triangle based on its angles. The three types of triangles classified by angles are acute, obtuse, and right triangles.

An acute triangle is defined as a triangle where all three angles are less than 90 degrees. This means that each angle inside the triangle is sharp or smaller than a right angle. For example, if we have a triangle with angles measuring 30 degrees, 60 degrees, and 90 degrees respectively, it would be considered an acute triangle.

On the other hand, an obtuse triangle has one angle greater than 90 degrees. It has one blunt or wider angle inside the triangle. Let’s say we have a triangle with angles measuring 70 degrees, 80 degrees, and 120 degrees respectively; it would be categorized as an obtuse triangle due to the presence of the angle larger than 90 degrees.

Lastly, a right triangle has one right angle which measures exactly 90 degrees. The other two angles will be acute angles because they must always add up to less than 180 degrees (the sum of all angles in any given triangle). For instance, if we have a right-angled triangle with two angles measuring 45 degrees each and one right angle (90-degree), it would be termed as a right-angled triangle.

Moving on to question two in our homework assignment – determining whether triangles are congruent or not based on their side lengths – we need to understand congruence criteria for triangles: Side-Side-Side (SSS), Side-Angle-Side (SAS), Angle-Side-Angle (ASA), and Angle-Angle-Side (AAS).

The SSS criterion states that if the lengths of all three sides of one triangle are equivalent to the lengths of all three sides of another triangle, then the two triangles are congruent. For example, if we have a triangle with side lengths 5 cm, 6 cm, and 7 cm, and another triangle with side lengths 5 cm, 6 cm, and 7 cm as well, we can conclude that the triangles are congruent.

Similarly, the SAS criterion asserts that if two sides of one triangle are proportional to two sides of another triangle and the included angle between these two sides is equal in both triangles, then they are congruent. So if we have a triangle with side lengths 4 cm, 5 cm and an angle measuring 40 degrees sandwiched between them, and another triangle with side lengths 4 cm, 5 cm and also an angle measuring 40 degrees between them, we can conclude that these two triangles are congruent.

Moreover, the ASA criterion indicates that if two angles of one triangle are equal to their corresponding angles in another triangle while the included side is proportional in both triangles, then they are congruent. For instance, if we have a triangle with angles measuring 30 degrees and 60 degrees along with a side length of 8 cm connecting those angles; and another triangle with angles measuring exactly the same but with a corresponding side length also being equal to 8 cm; then these two triangles would be considered congruent.

Finally, the AAS criterion states that if two pairs of corresponding angles in one triangle are equal to their respective pairs in another but there is only one pair of corresponding sides proportional between them; then they can be classified as congruent triangles. For example, if we have a triangle with angles measuring 30 degrees and 50 degrees, along with a side length of 6 cm; and another triangle with angles measuring exactly the same, yet with only one corresponding side of 6 cm as well; these two triangles would still be congruent.

To summarize, in this article, we explored the key aspects and solutions of homework 1 regarding triangle relationships. We discussed the classification of triangles based on their angle measurements: acute triangles (all angles less than 90 degrees), obtuse triangles (one angle greater than 90 degrees), and right triangles (one right angle). Additionally, we learned about the criteria for determining congruence between triangles: SSS (Side-Side-Side), SAS (Side-Angle-Side), ASA (Angle-Side-Angle), and AAS (Angle-Angle-Side). Understanding these concepts is vital as they serve as building blocks for more complex geometric principles.