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Warm-Up • Find the domain and range: Geometry Vocabulary & Notation Point • Name: Use only the capital letter, without any symbol. Line • Name: Use any two points on the line with a line symbol above. AB Line Segment • Name: Use the endpoints of the line segment with a line segment symbol above. AB Ray • Name: Use the endpoint and one other point on the ray with a ray symbol above. AB Plane • Name: Use the word Plane followed by any three non-collinear points. PlaneACB Angles • Name: Use three letters with the vertex in the middle preceded by the angle symbol. B A 1 C BAC A 1 Acute Angle • An angle that measures less than 90 degrees. Obtuse Angle • An angle that measures greater than 90 degrees but less than 180 degrees. Right Angle • An angle that measures 90 degrees Naming Geometric Figures • An angle can be named using three points. The middle point must be the vertex. How many different angles can you name from this diagram? AEB AEC AED BEC BED CED Measure or Length • Measure of an angle: 𝑚∠𝐴𝐵𝐶 • Length of a segment: 𝐴𝐵 Congruent Segments In geometry, two segments with the same length are called ________ congruent _________ segments Definition of Congruent Segments Two segments are congruent if and only if ________________________ they have the same length Congruent Segments In the figures at the right, AB is congruent to BC, and PQ is B A congruent to RS. The symbol is used to represent congruence. AB BC, and PQ RS. R C Congruent Segments Since congruence is related to the equality of segment measures, there are properties of congruence that are similar to the corresponding properties of equality. theorems These statements are called ________. Theorems are statements that can be justified by using logical reasoning. 2–1 Congruence of segments is reflexive. AB AB 2–2 Congruence of segments is symmetric. If AB CD, then CD AB 2–3 Congruence of segments is transitive. If AB CD, and CD EF then AB EF Congruent Segments A point M is the midpoint of a segment between S and T and SM = MT Definition of Midpoint S M ST if and only if M is T SM = MT The midpoint of a segment separates the segment into two segments of equal _____. length _____ congruent So, by the definition of congruent segments, the two segments are _________. §3.3 The Angle Addition Postulate The bisector of an angle is the ray with its endpoint at the vertex of the angle, extending into the interior of the angle. The bisector separates the angle into two angles of equal measure. Definition of an Angle Bisector P QA 1 Q 2 is the bisector of PQR. A m1 = m2 R Types of Triangles • Classifying by Sides – Equilateral • All sides are congruent – Isosceles • Two sides congruent – Scalene • No sides congruent Types of Triangles • Classifying by Angles – Equiangular • All angles congruent – Acute • All angles less than 90 – Right • One angle 90 – Obtuse • One angle greater than 90 Isosceles Triangles • Isosceles Triangle – A triangle with two congruent sides • Legs – Two congruent sides of an isosceles triangle • Base – Non-congruent side of an isosceles triangle • Vertex angle – Angle across from the base • Base angles – Two congruent angles, across from the legs. Isosceles Triangle Theorem • If two sides of a triangle are congruent, then the angles opposite those sides are congruent. • Example: If 𝐴𝐶 ≅ 𝐶𝐵, then ∠𝐴 ≅ ∠𝐵 C A B Converse of Isosceles Triangle Theorem • If two angles of a triangle are congruent, then the sides opposite those angles are congruent. • Example: If ∠𝐴 ≅ ∠𝐵, then 𝐴𝐶 ≅ 𝐶𝐵 C A B A triangle is equilateral if and only if it is equiangular. Triangle Angle-Sum Theorem • The sum of the measures of the angles of a triangle is 180 degrees. 2 1 3 3 3 2 1 1 2 1 2 3 Exterior Angles of a Polygon • An exterior angle is formed by a side and an extension of an adjacent side. 3 exterior angle 1 2 Two remote interior angles • Remote interior angles: for each exterior angle if a triangle, there are two nonadjacent interior angles. Triangle Exterior Angle Theorem • The measure of each exterior angle of a triangle equals the sum of the measures of its two remote interior angles. 𝑚∠1 = 𝑚∠2 + 𝑚∠3 2 1 3