Triangle Geometry Theorems and Classifications

A triangle is classified by its angles based on their measures. An acute triangle has all three angles measuring less than 90°. In this case, the angles of 35°, 75°, and 70° are all less than 90°, confirming that the triangle is acute. In contrast, a right triangle has one angle equal to 90°, and an obtuse triangle has one angle greater than 90°. Since none of the angles in this triangle meet those criteria, it is classified as an acute triangle.

The Triangle Inequality Theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. For the set of side lengths 5, 10, and 15, we can see that 5 + 10 equals 15, which does not satisfy the theorem since the sum is not greater than the third side. Therefore, this set of lengths cannot form a valid triangle, while the other sets do meet the requirements of the theorem.

In a triangle, the length of any side must be less than the sum of the other two sides and greater than the difference of the other two sides. For sides measuring 9 and 15, the third side (x) must satisfy two conditions: it must be less than 9 + 15 (24) and greater than |9 - 15| (6). Therefore, the third side must be greater than 6 and less than 24, resulting in the range 6 < x < 24.

In any triangle, the Side-Angle Relationship Theorem states that the side opposite the largest angle is the longest. Since the largest angle in this triangle measures 110°, the side that is directly opposite this angle must be the longest side. This relationship holds true because larger angles correspond to longer sides, making the side opposite the 110° angle the longest in this scenario.

An equiangular triangle is defined as having all three interior angles equal, each measuring 60 degrees. According to the properties of triangles, if all angles are equal, the sides opposite those angles must also be equal in length. This is a consequence of the triangle's congruence criteria, specifically the Angle-Side-Angle (ASA) theorem. Therefore, in an equiangular triangle, not only are the angles equal, but this equality necessitates that all three sides must also be congruent.

In triangle ABC, since sides AB and BC are congruent, the Isosceles Triangle Theorem states that the angles opposite these sides must also be congruent. Therefore, angle A, which is opposite side BC, is congruent to angle C, which is opposite side AB. This results in angles A and C being equal, while angle B can be different. Thus, the angles that must be congruent are angles A and C.

A triangle with no congruent sides is classified as a scalene triangle because all three sides are of different lengths. This lack of equal sides means that all three angles are also different, distinguishing it from isosceles triangles, which have at least two equal sides, and equilateral triangles, which have all sides equal. The scalene triangle's unique side lengths contribute to its distinct shape and properties in geometry.

In a triangle, the sum of all three interior angles is always 180°. Given two angles measuring 47° and 68°, we can find the third angle by subtracting the sum of the known angles from 180°. First, add 47° and 68° to get 115°. Then, subtract 115° from 180° to find the third angle: 180° - 115° = 65°. Thus, the measure of the third angle is 65°.

This triangle has two sides of equal length (5 and 5), which classifies it as an isosceles triangle. In isosceles triangles, the angles opposite the equal sides are congruent. Therefore, the base angles opposite the equal sides in this triangle are also congruent. The triangle does not have all sides equal, so it is not equilateral, and since it has two equal sides and one different side, it cannot be classified as scalene.