Geometry Chapter 5 - Congruence based on Triangles
5.1 - Line Segments Associated with TrianglesVocabulary:
1. Altitude of a triangle - is a line segment drawn from any vertex of the triangle, perpendicular to and ending in the line that contains the opposite side.
2. Median of a triangle - is a line segment that joins any vertex of the triangle to the midpoint of the opposite side.
3. Angle bisector of a triangle - is a line segment that bisects any angle of the triangle and terminates in the side opposite that angle.
5.2 - Using Congruent triangles to prove line segments congruent and angles congruent.
1. SAS = SAS Congruence Postulate - if two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the two triangles are congruent.
Example: Given Triangle ABC and Triangle DEF,if AB = DE, BC = EF and angle B = angle E, then triangle ABC = triangle DEF.
2. SSS = SSS Congruence Postulate - if three sides of one triangle are congruent to three sides of a second triangle, then the two triangles are congruent.
Example: Given triangle ABC and triangle DEF,if AB = DE, BC = EF and AC = DF, then triangle ABC = triangle DEF.
3. ASA = ASA Congruence Postulate - if two angles and the included side of one triangle are congruent to two angles and the included side of a second triangle, then the two triangles are congruent.
Example: Given triangle ABC and triangle DEF,if angle A = angle D, AB = DE, and angle B = angle E, then triangle ABC = triangle DEF.
4. AAS = AAS Congruence Postulate - if two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of a second triangle, then the two triangles are congruent.
Example: Given triangle ABC and triangle DEF,if angle A = angle D, angle C = angle F, and BC = EF , then triangle ABC = triangle DEF.
5. HL = HL Congruence Postulate - if the leg and hypotenuse of one right triangle is congruent to the corresponding leg and hypotenuse of another right triangle, then the two triangles are congruent by hypotenuse - leg postulate.
Example: Given right triangle ABC and right triangle DEF, if angle B and angle E are both right angles and leg AB = leg DE and hypotenuse AC = hypotenuse DF, then triangle ABC = triangle DEF.
5.3 Isosceles and Equilateral Triangles:
Theorem: If two sides of a triangle are congruent, the angles opposite these sides are congruent.
Theorem: The median from the vertex angle of an isosceles triangle bisects the vertex angle.
Theorem: The median from the vertex angle of an isoscles triangle is perpendicular to the base.
Theorem: Every equilateral triangle is equiangular.
5.4 Using two pairs of Congruence Triangles:
5.5 Proving Overlapping Triangles Congruent:
5.6 Perpendicular Bisector of a line segment.
Definition: the perpendicular bisector of a line segment is any line or subset of a line that is perpendicular to the line segment at its midpoint.
Definition: Equidistant - equal distance from the endpoints.
Theorem: If two points are each equidistant from the endpoints of a line segment, then the points determine the perpendicular bisector of the line segment.
Theorem: If a point is equidistant from the endpoints of a line segment, then it is on the perpendicular bisector of the line segment.
Theorem: If a point is on the perpendicular bisector of a line segment, then it is equidistant from the endpoints of the line segment.
Theorem: A point is on the perpendicular bisector of a line segment if and only if it is equidistant from the endpoints of the line segment.
A. Methods of Proving lines or line segments perpendicular:Prove one of the following statements is true:1. The two lines form right angles at their point of intersection.2. The two lines form congruent adjacent angles at their point of intersection.3. Each of two points on one line is equidistant from the endpoints of a segment of the other.
B. Intersection of the Perpendicular Bisectors of the sides of a triangle.- the perpendicular bisectors of the sides of a triangle are concurrent (they intersect in one point).- The point where the three perpendicular bisectors of the sides of a triangle intersect is called the circumcenter.
