If two sides of a triangle are congruent, then the angles opposite them are congruent.
Based on the diagram above, the following relation holds true.
NM≅NK ⇒ ∠M≅∠K
The Isosceles Triangle Theorem is also known as the Base Angles Theorem.
This theorem will be proven using congruent triangles. Consider △MNK, a triangle with two congruent sides.
In this triangle, let P be the point of intersection of the angle bisector of ∠N and MK.
From the diagram, the following features of △MNP and △KNP can be observed.
Feature | Reasoning |
---|---|
∠MNP ≅ ∠KNP | Definition of an angle bisector. |
MN ≅ KN | Given. |
NP ≅ NP | Reflexive Property of Congruence. |
Therefore, △MNP and △KNP have two pairs of corresponding congruent sides and one pair of included congruent angles. By the Side-Angle-Side Congruence Theorem, △MNK and △KNP are congruent triangles. △MNK≅△KNP Corresponding parts of congruent figures are congruent. Therefore, ∠M and ∠K are congruent.
∠M≅∠K
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.
Side | Distance formula | Length |
---|---|---|
AB | (3.6−0)2+(4.8−0)2 | 6 |
BC | (3.6−5)2+(4.8−0)2 | 5 |
AC | (5−0)2+(0−0)2 | 5 |
We have found that two of the sides, BC and AC, have the same length. Thus, it is an isosceles triangle. Consequently, the base angles, ∠A and ∠B, are congruent and acute. Since the third angle, ∠C, is also acute we can conclude that it is an acute triangle. Thus, the triangle can be classified as isosceles and acute.
Suppose one side of a triangle is extended. The angle created is called an exterior angle.
Because a triangle has three sides, three exterior angles exist. Each exterior angle and its corresponding interior angle are supplementary.
m∠A+m∠A′=180∘
At each vertex, the exterior angle can be defined in two ways.
Consider △ABC.
The sum of the interior angles of △ABC is 180∘.
m∠A+m∠B+m∠C=180∘
To begin, draw a line, PQ, that passes through B and is parallel to AB. PQ and △ABC create three angles ∠PBA, ∠ABC, and ∠CBQ.
Together, the three angles make a straight angle. Thus, the sum of their measures is 180∘. m∠PBA+m∠ABC+m∠CBQ=180∘ Because, PQ∥AC and AB is a transversal, ∠PBA and ∠A are alternate interior angles. Thus, according to the Alternate Interior Angles Theorem, ∠PBA≅∠A. By the same reasoning, ∠CBQ≅∠C.
Two congruent angles have the same measure. This can be used to rewrite the sum of the three angles. m∠A+m∠ABC+m∠C=180∘ Therefore, the sum of the interior angles of a triangle is 180∘. This can be summarized in a flowchart proof.
Find the measures of the angles that are marked in the figure.