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Transcript

Chapter 5: Congruence based on Triangles Definitions: Line Segments Associated with Triangles Altitude of a A line segment drawn from any vertex of the triangle, Triangle perpendicular to and ending at the opposite side. Median of a A line segment that joins any vertex of the triangle to the Triangle midpoint of the opposite side. Angle Bisector A line segment that bisects any angle of the triangle and of a Triangle ends in the side opposite that angle. Perpendicular A line or segment that is perpendicular to another segment at Bisector its midpoint. Circumcenter The point where the three perpendicular bisectors of a triangle intersect. Triangles Isosceles A triangle with at least two sides congruent. Triangle Equilateral A triangle with three sides congruent. Triangle Postulates, Theorems and Corollaries Isosceles If two sides of a triangle are congruent, the angles opposite those Triangle sides are congruent. Theorem OR The base angles of an isosceles triangle are congruent. Corollary Corollary Corollary Theorem Theorem Theorem Theorem Theorem Theorem Theorem Theorem The median from the vertex angle of an isosceles triangle bisects the vertex angle. The median from the vertex angle of an isosceles triangle is perpendicular to the base. (altitude) Every Equilateral Triangle is Equiangular. ** An Equilateral Triangle has all the properties of an Isosceles Triangle. If two points are each equidistant from the endpoints of a line segment, then the points determine the perpendicular bisector of the line segment. If a point is equidistant from the endpoints of a line segment, then it is on the perpendicular bisector of the line segment. If a point is on the perpendicular bisector of a line segment, then it is equidistant from the endpoints of the line segment. 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. The perpendicular bisectors of the sides of a triangle are concurrent. The perpendicular bisectors of a triangle intersect at a point that is equidistant from the vertices of the triangle. Radii of congruent circles are congruent. All radii of the same circle are congruent TIPS for Overlapping Triangles and Using 2 pair of Congruent Triangles: 1. 2. 3. 4. Draw good diagrams – make sure they are large enough and labeled. Mark the given(s) on the diagram (it’s the law!) Be neat and orderly. For Overlapping Triangles: a. Be sure to identify the correct triangles needed to prove congruent. b. You may need to prove a 1st pair of triangles congruent in order to prove a 2nd pair of triangles congruent. c. Sometimes it may be easier to separate the triangles or use different colors (or solid & dotted lines) to differentiate each triangle. TIPS for proving intersecting lines or segments are perpendicular: The lines intersect to form right angles. (defn of perpendicular lines) The lines intersect to form congruent adjacent angles (theorem – if two lines intersect to form congruent adjacent angles, then they are perpendicular). One line contains 2 points that are equidistant from the endpoints of a segment of the other line. (theorem - If a point is equidistant from the endpoints of a line segment, then it is on the perpendicular bisector of the line segment)