An isosceles triangle is a triangle with at least two congruent sides. This means that two of the sides of the triangle have the same length, while the third side may be of a different length.
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In an isosceles triangle, the two congruent sides are called the 'legs' of the triangle, and the third side is called the 'base'.
The two angles opposite the congruent sides in an isosceles triangle are also congruent, meaning they are equal in measure.
The angle opposite the base of an isosceles triangle is called the 'vertex angle', and it is always greater than the other two angles.
Isosceles triangles have a line of symmetry that passes through the vertex angle and bisects the base at a right angle.
The Pythagorean Theorem can be used to solve for missing side lengths in isosceles triangles, as long as one side length is known.
Review Questions
How can you identify an isosceles triangle, and what are the key features that distinguish it from other types of triangles?
An isosceles triangle can be identified by the presence of at least two congruent sides. These congruent sides are called the 'legs' of the triangle, and the third side is called the 'base'. Additionally, the two angles opposite the congruent legs are also congruent, meaning they are equal in measure. This is a key distinguishing feature of isosceles triangles compared to other triangle types, such as scalene triangles, which have all three sides of different lengths, or equilateral triangles, which have all three sides and angles congruent.
Explain how the Pythagorean Theorem can be applied to solve for missing side lengths in isosceles triangles.
The Pythagorean Theorem, which states that in a right triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides, can be used to solve for missing side lengths in isosceles triangles. This is because an isosceles triangle can be divided into two right triangles by drawing a line from the vertex angle to the midpoint of the base. Once one side length is known, the Pythagorean Theorem can be applied to the resulting right triangles to solve for the missing side lengths.
Describe the relationship between the vertex angle and the base angles in an isosceles triangle, and explain how this relationship can be used to solve for unknown angle measures.
In an isosceles triangle, the two angles opposite the congruent sides (the 'legs') are also congruent, meaning they are equal in measure. This relationship means that the vertex angle, which is the angle opposite the base, is always greater than the other two angles in the triangle. This property can be used to solve for unknown angle measures in isosceles triangles. For example, if the measure of the vertex angle is known, the measure of the base angles can be determined by subtracting the vertex angle measure from 180 degrees (the sum of the interior angles of a triangle) and then dividing the result by 2, since the base angles are congruent.