Warm-Up. Find the domain and range:

Transcription

1 Warm-Up Find the domain and range:

2 Geometry Vocabulary & Notation

3 Point Name: Use only the capital letter, without any symbol.

4 Line Name: Use any two points on the line with a line symbol above. AB

5 Line Segment Name: Use the endpoints of the line segment with a line segment symbol above. AB

6 Ray Name: Use the endpoint and one other point on the ray with a ray symbol above. AB

7 Plane Name: Use the word Plane followed by any three non-collinear points. PlaneACB

8 Angles Name: Use three letters with the vertex in the middle preceded by the angle symbol. B 1 A BAC A C 1

9 Acute Angle An angle that measures less than 90 degrees.

10 Obtuse Angle An angle that measures greater than 90 degrees but less than 180 degrees.

11 Right Angle An angle that measures 90 degrees

12 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

13 Measure or Length Measure of an angle: m ABC Length of a segment: AB

14 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

15 In the figures at the right, AB is Congruent Segments congruent to BC, and PQ is congruent to RS. A B C The symbol is used to represent congruence. AB BC, and PQ RS. R

16 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. These statements are called. theorems 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

17 Congruent Segments A point M is the midpoint of a segment between S and T and SM = MT ST if and only if M is Definition of Midpoint S M T SM = MT The midpoint of a segment separates the segment into two segments of equal. length So, by the definition of congruent segments, the two segments are. congruent

18 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 Q 1 2 A QA is the bisector of PQR. m 1 = m 2 R

19 Types of Triangles Classifying by Sides Equilateral All sides are congruent Isosceles Two sides congruent Scalene No sides congruent

20 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

21 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.

22 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

23 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

24 A triangle is equilateral if and only if it is equiangular.

25 Triangle Angle-Sum Theorem The sum of the measures of the angles of a triangle is 180 degrees

26 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.

27 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