![]() 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. 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. The converse of the isosceles triangle theorem states that, if two angles of a triangle are congruent, then the sides opposite those angles are congruent. 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. ![]() since congruent angles are angles of equal measure.If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent.We have established that the rays forming these angles coincide under a reflection. The reflection of ∠ CAB will have the same measure as ∠ CBA since reflections preserve angle measure.The reflections of and the reflection of.The reflection of A is B since reflections preserve length and the segments share point C.The reflection of will have the same length as that of.Since these angles are equal in measure, the reflection of ray (side of the ∠) will coincide with its image.Since m∠ACD = m∠BCD and reflections preserve angle measure, the image of ∠ ACD will be the same measure as ∠ BCD.Under a reflection in, the reflection of C will be C, since C lies on the line of.m∠ACD = m∠BCD because an angle bisector forms two congruent angles which have equal measure.Label the intersection with the base as D. Construct an auxiliary line through point C bisecting ∠ C.
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