Based on the diagram above, the following relation holds true.
The Isosceles Triangle Theorem is also known as the Base Angles Theorem.
This theorem will be proven using congruent triangles. Consider a triangle with two congruent sides.
From the diagram, the following features of and can be observed.
|Definition of an angle bisector.|
|Reflexive Property of Congruence.|
Therefore, and have two pairs of corresponding congruent sides and one pair of included congruent angles. By the Side-Angle-Side Congruence Theorem, and are congruent triangles. Corresponding parts of congruent figures are congruent. Therefore, and are congruent.
It has been proven that if two sides in a triangle are congruent, the angles opposite them are congruent.
Classify the triangle by its sides and its angles.
We have found that two of the sides, and have the same length. Thus, it is an isosceles triangle. Consequently, the base angles, and are congruent and acute. Since the third angle, is also acute we can conclude that it is an acute triangle. Thus, the triangle can be classified as isosceles and acute.
The sum of the interior angles of is
To begin, draw a line, that passes through and is parallel to and create three angles and
Together, the three angles make a straight angle. Thus, the sum of their measures is Because, and is a transversal, and are alternate interior angles. Thus, according to the Alternate Interior Angles Theorem, By the same reasoning,
Two congruent angles have the same measure. This can be used to rewrite the sum of the three angles. Therefore, the sum of the interior angles of a triangle is This can be summarized in a flowchart proof.
Find the measures of the angles that are marked in the figure.