Geometry - Concepts 9-12 Congruent Triangles and Special Segments Concept 9 Parallel Lines and Triangles (Section 3.5) ANGLE Classifications Acute: Obtuse: Right: SIDE Classifications Scalene: Isosceles: Equilateral: Equiangular: Theorem 3.11 Triangle Sum Theorem The sum of the measures of the angles of a triangle is. Theorem 3.12 Triangle Exterior Angle Theorem The measure of each exterior angle of a triangle equals the sum of the measures of its two. mð1= mð2+mð3 Concept 10 Congruent Figures (Section 4.1) Congruent Polygons: - Have corresponding that are congruent - Have corresponding that are congruent Congruence Statement: - a statement saying that two figures are congruent - corresponding angles must be lined up Example: List all pairs of congruent angles and all pairs of congruent sides using the statement below. Theorem 4.1 - Third Angle Theorem If two angles of one triangle are congruent to two angles of another triangle, then the third angles.
Concept 10 Congruent Triangles (Sections 4.2, 4.3, 4.6) Side-Side-Side Postulate (SSS) If the sides of one triangle are congruent to the sides of another triangle, then. Side-Angle-Side Postulate (SAS) If of a triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent. Angle-Side-Angle Postulate (ASA) If of a triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent. Angle-Angle-Side Theorem (AAS) If two angles and a side of a triangle are congruent to two angles and the non included side of another triangles, then the triangles are congruent. Hypotenuse-Leg Theorem (HL) If the of one right triangle is congruent to the hypotenuse and leg of another right triangle, then the triangles are congruent. BAD PROOFS these do not work
Concept 10 Using Corresponding Parts of Congruent Triangles (4.4) CPCTC: Example: Concept 10 Isosceles and Equilateral Triangles (Section 4.5) Parts of an Isosceles Triangle Legs: Base: Base Angles: Vertex Angle: Theorem 4.3 Isosceles Triangle Theorem If two sides of a triangle are congruent, then the two angles opposite those sides are. Theorem 4.4 Converse of the Isosceles Triangle Theorem If two angles of a triangle are congruent, then the two sides opposite those angles are. Corollary to Theorem 4.3 If a triangle is equilateral, then it is. Corollary to Theorem 4.4 If a triangle is equiangular, then it is. Examples: Find the value of each variable.
Concept 11 Midsegment Theorem (Section 5.1) Midsegment of a Triangle: Theorem 5.1 Triangle Midsegment Theorem If a segment joins the midpoints of two sides of a triangle, then the segment is to the third side and. Concept 11 Bisectors, Medians, and Altitudes (Section 5.2 and 5.4) Perpendicular Bisector: Theorem 5.2 - Perpendicular Bisector Theorem C If a point is on the perpendicular bisector of a segment, then it is from the endpoints of the segment. A D B Angle Bisector: Theorem 5.4 - Angle Bisector Theorem If a point is on the bisector of an angle, then the point is from the two sides of the angle. Median of a Triangle: Altitude:
Example: Name the special segment in each triangle. Concept 11 Points of Concurrency (Section 5.3 and 5.4) Concurrent: Point of Concurrency: Circumcenter: Properties of the Circumcenter 1) 2) Incenter: Properties of the Incenter 1) 2) Centroid: Properties of the Centroid 1) 2) Orthocenter:
Concept 13 - Inequalities in Triangles (Section 5.6) Corollary to the Triangle Exterior Angle Theorem The measure of an exterior angle of a triangle is the measure of each of its remote interior angles. Theorem 5.10 If two sides of a triangle are not congruent, then the larger angle lies opposite the. A Largest angle = Mid-sized angle = Smallest angle = C 14 20 17 B Theorem 5.11 If two angles of a triangle are not congruent, then the longer side lies opposite the. Longest side = Medium side = Shortest side = F D 112 o 32 o 36 o E Theorem 5.12 - Triangle Inequality Theorem The sum of the lengths of any two sides of a triangle is the length of the third side. Simplified Version: Example: If two sides of a triangle are 3 inches and 9 inches long, write a range of possible lengths for the third side of the triangle.