Geometry Review. IM3 Ms. Peralta
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1 Geometry Review IM3 Ms. Peralta
2 Ray: is a part of a line that consists of an endpoint, and all points on one side of the endpoint. P A PA Opposite Rays: are two rays of the same line with a common endpoint and no other points in common. PA is opposite to PB B P A
3 Angle: is the union of two rays having the same endpoint. side A vertex B side x C Naming angles: - Three capital letters, with vertex in the middle: - Single lowercase letter or number inside the angle: - Use the name of the vertex angle if it s the only angle at the vertex: ABC
4 1) Name the angle in four ways. ABC CBA B A 1 C 1 B 2) Identify the vertex and sides of this angle. vertex: sides: Point B BA and BC
5 1) Name all angles having W as their vertex. X W XWZ Y 2) What are other names for 1? XWY or YWX Z 3) Is there an angle that can be named? W No!
6 Once the measure of an angle is known, the angle can be classified as one of three types of angles. These types are defined in relation to a right angle. Types of Angles A A A obtuse angle 90 < m A < 180 right angle m A = 90 acute angle 0 < m A < 90
7 Classify each angle as acute, obtuse, or right Obtuse Right Acute Acute Obtuse Acute
8 Angles Right angle: Forms a square corner. Forms a 90 angle. Acute angle: Forms an angle that is less than Straight angle: Forms a straight line. The angle is 180. Obtuse angle: Forms an angle that is more than
9 Point: indicates a position or location in space.. P. A(2, 6) Y X Points are named using capital letters and/or coordinates.
10 Line: A line is an infinite set of adjacent points. Parallel Lines Lines do not intersect but are in the same plane Intersecting Lines Lines meet at one point Perpendicular Lines Lines form a right angle
11 Naming a Line: a) Two points on the line: A B C b) Single lowercase letter m
12 Plane: A plane is a set of points that forms a completely flat surface. Naming a Plane: a) Three points on the plane: Plane ABC A B C b) Single uppercase letter: Plane R R
13 Collinear Points A collinear set of points is a set of points all of which lie on the same straight line. E A B C D Points A, B, C and D are collinear. Points A, E and C are not collinear.
14 Line Segment A line segment is the set of two points on a line called endpoints, and all points on the line between the endpoints. A B Naming a Line Segment: Use the names of the endpoints. A B Line segment is part of Line
15 Measure of Line Segments The measure of a line segment is the length of the line segment. To indicate the measure of a line segment, use the name of the line segment without the line above the name. Ex: = measure of line segment Ex: The perimeter of ΔABC is 58 cm. If AB = x - 4, BC = 2x + 8, and CA= x 2, what is BC? Ans. BC = 36 cm
16 Congruent Line Segments Congruent line segments are segments that are equal in measure. Ex: A C B D If Then AB = CD Ex: If AB CD, AB = 9x 7, and CD = 4x + 13, what is CD? AB? Ans. CD = 29, AB = 29
17 Definition of Midpoint The midpoint of a line segment divides the line segment into two congruent segments. If M is the midpoint of AB A M B
18 Definition of Line Bisector The bisector of a line segment is a line that intersects the line segment at its midpoint. A P M Q B PQ intersects segment AB at its midpoint: M
19 Addition & Subtraction of Line Segments If several line segments belong to the same line, we can write addition and subtraction expressions using the names of these segments. P S R
20 When you split an angle, you create two angles. The two angles are called adjacent angles A adjacent = next to, joining. B 2 1 D <1 and <2 are examples of adjacent angles. They share a common ray. C Name the ray that <1 and <2 have in common.
21 Adjacent angles are angles that: A) share a common side B) have the same vertex, and C) have no interior points in common Definition of Adjacent Angles J R 2 1 M <1 and <2 are adjacent with the same vertex R and common side N
22 Determine whether <1 and <2 are adjacent angles. 1 2 No. They have a common vertex B, but no common side B 1 G 2 Yes. They have the same vertex G and a common side with no interior points in common. J 2 L 1 N No. They do not have a common vertex or a common side The side of <1 is The side of < 2 is
23 Determine whether <1 and <2 are adjacent angles. No. 1 2 Yes. 1 2 X D Z In this example, the noncommon sides of the adjacent angles form a. straight line These angles are called a linear pair
24 Two angles form a linear pair if and only if (iff): A) they are adjacent and B) their noncommon sides are opposite rays A B D Definition of Linear Pairs C 1 2 <1 and <2 are a linear pair.
25 In the figure, and are opposite rays. 1) Name the angle that forms a linear pair with <1. T H <ACE A E <ACE and <1 have a common side the same vertex C, and opposite rays M 1 C and 2) Do <3 and <TCM form a linear pair? Justify your answer. No. Their noncommon sides are not opposite rays.
26 Two angles are complementary if and only if (iff) The sum of their degree measure is 90. E A D 60 Definition of Complementary B 30 C F Angles m<abc + m<def = = 90
27 If two angles are complementary, each angle is a complement of the other. <ABC is the complement of <DEF and <DEF is the complement of <ABC. B A 30 C D 60 E F Complementary angles DO NOT need to have a common side or even the same vertex.
28 Some examples of complementary angles are shown below. H I m<h + m<i = 90 P H Q S m<phq + m<qhs = 90 T 60 U 30 V m<tzu + m<vzw = 90 Z W
29 If the sum of the measure of two angles is 180, they form a special pair of angles called supplementary angles. Two angles are supplementary if and only if (iff) the sum of their degree measure is 180. C D Definition of Supplementary Angles A 50 B E 130 F m<abc + m<def = = 180
30 Some examples of supplementary angles are shown below. 105 H 75 I m<h + m<i = 180 P Q H S U V Z T W m<phq + m<qhs = 180 m<tzu + m<uzv = 180 and m<tzu + m<vzw = 180
31 Recall that congruent segments have the same measure. Congruent angles also have the same measure.
32 Two angles are congruent iff, they have the same degree measure. Definition of Congruent Angles 50 B V 50 B V iff m B = m V
33 To show that <1 is congruent to <2, we use. arcs 1 2 To show that there is a second set of congruent angles, <X and <Z, we use double arcs. This arc notation states that: X Z m X = m Z X Z
34 When two lines intersect, four angles are formed. There are two pair of nonadjacent angles. These pairs are called. vertical angles
35 Two angles are vertical iff they are two nonadjacent angles formed by a pair of intersecting lines. Definition of Vertical angles: Vertical Angles <1 and <3 <2 and <4
36 Vertical angles are congruent. Theorem 3-1 Vertical Angle m n 1 3 Theorem 4 2 4
37 Find the value of x in the figure: 130 The angles are vertical angles. So, the value of x is 130. x
38 Find the value of x in the figure: (x 10) 125 The angles are vertical angles. (x 10) = 125. x 10 = 125. x = 135.
39 Suppose m<a = 52. Find the measure of an angle that is supplementary to <A. B 52 1 A <B + <A = 180 <B = 180 <B <B = <B = 128
40 G D 1) If m<1 = 2x + 3 and the m<3 = 3x + 2, then find the m<3 x = 17; <3 = A 4 B C 3 E H 2) If m<abd = 4x + 5 and the m<dbc = 2x + 1, then find the m<ebc x = 29; <EBC = 121 3) If m<1 = 4x - 13 and the m<3 = 2x + 19, then find the m<4 x = 16; <4 = 39 4) If m<ebg = 7x + 11 and the m<ebh = 2x + 7, then find the m<1 x = 18; <1 = 43
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