We will be using the properties of the isosceles triangle to prove the converse as discussed below. This is exactly the reverse of the theorem we discussed above. The converse of isosceles triangle theorem states that if two angles of a triangle are congruent, then the sides opposite to the congruent angles are equal. Hence, we have proved that if two sides of a triangle are congruent, then the angles opposite to the congruent sides are equal. Proof: We know, that the altitude of an isosceles triangle from the vertex is the perpendicular bisector of the third side. Given: ∆ABC is an isosceles triangle with AB = AC.Ĭonstruction: Altitude AD from vertex A to the side BC. Let's draw an isosceles triangle with two equal sides as shown in the figure below. To understand the isosceles triangle theorem, we will be using the properties of an isosceles triangle for the proof as discussed below. The perpendicular segment from a vertex to the line that contains the opposite side.Ī line (or segment or ray) that is perpendicular to the segment at its midpoint.Isosceles triangle theorem states that if two sides of a triangle are congruent, then the angles opposite to the congruent sides are also congruent. If a line and a plane intersect and the line is perpendicular to all lines in the plane that pass through the point of intersection.Ī segment from the vertex to the midpoint of the opposite side. If a point is equidistant from the sides of an angle, then the point lies on the bisector of the angle.ĭefinition of a line perpendicular to a plane If a point lies on the bisector of an angle, then the point is equidistant from the sides of the angle. If a point is equidistant from the endpoints of a segment, then the point lies on the perpendicular bisector of the segment. If a point lies on the perpendicular bisector of a segment, then the point is equidistant from the endpoints of the segment. If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and a leg of another right triangle, then the triangles are congruent. If two angles and a non-included side of one triangle are congruent to two angles and a non-included side of another triangle, then the triangles are congruent. If two angles of a triangle are congruent, then the sides opposite those angles are congruent.Ĭorollary to the Converse of the Isosceles Triangle Theorem (Don't call it this)Īn equiangular triangle is also equilateral. The bisector of the vertex angle of an isosceles triangle is perpendicular to the base at its midpoint.Ĭonverse of the Isosceles Triangle Theorem If two sides of a triangle are congruent, then the angles opposite those sides are congruent.Ĭorollary 1 to the Isosceles Triangle Theorem (Don't call it this!)Īn equilateral triangle is also equiangular.Ĭorollary 2 to the Isosceles Triangle Theorem (Don't call it this!)Īn equilateral triangle has three 60 degree angles.Ĭorollary 3 to the Isosceles Triangle Theorem (Don't call it this!) If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent. If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent. If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent.
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