Geometry R/H 1.4 – Angle Measures 1.5 – Angle Relationships

Line Segments A line segment is part of a line containing two endpoints and all points between them. Unlike lines, which extend forever in both directions, line segments have a definite beginning and end A line segment is named with the endpoints A B AB or BA

Rays A ray is part of a line that consists of one endpoint and all points of the line extending in one direction Name a ray using the endpoint first and then another point on the ray – When naming, make sure the arrow points away from the endpoint. A B AB, not BA

Rays, continued Are these two rays the same? A B A B No Different endpoints Extend in different directions BA AB

Opposite Rays Are two collinear rays with the same endpoint Always form a line J K L KJ and KL are opposite rays

Parallel Lines Lines that – Never intersect – Extend in the same directions – Coplanar – Have the same slope

Skew lines Lines that: Never intersect Are noncoplanar Extend in different directions

Parallel Planes Parallel planes are planes that never intersect A line and plane that never intersect are also parallel

Learning Check and Summary Which of the following has no endpoints? – Ray, Line Segment, Line Which of the following has two endpoints? – Ray, Line Segment, Line Which of the following has one endpoint and extends in one direction? – Ray, Line Segment, Line Which of the following extend in the same direction? – Parallel lines, skew lines Which of the following extend in different directions? – Parallel lines, skew lines

Types of Angles

Vertex Side

Naming Angles Angles are measured in degrees. The measure of is written as. Angles with the same measure are congruent. 1

Angle Addition If point B is the interior of, then If is a straight angle, then A A B B O O C C

PROTRACTOR EXERCISE (on Word doc)

What it is? Angle addition is adding (or subtracting) two (or more) Angles If ∠ ABC is a right angle m∠ ABD is 50 o find m ∠ DBC.

Variables (not scary!) If ∠ ABC is a right angle ∠ ABD is (2x + 3) o m ∠ DBC is (x + 6) o solve for x

Example #1 m ∠ 1 = x Find x and find m 3 m ∠ 2 = 2x – 10 m ∠ 3 = 2x + 10

Solve for x A B C D O

Bisecting an Angle An angle bisector is a ray that divides an angle into two adjacent angles that are congruent. Ray FH Bisects Angle GFI because it divides the angle into two congruent angles. In the book, matching congruence arcs identify congruent angles in diagrams. F G H I

Which angles are adjacent? *Think: What does adjacent mean? <1 & <2, <2 & <3, <3 & <4, <4 & <1 Vertical Angles – 2 angles that share a common vertex & whose sides form 2 pairs of opposite rays. Vertical Angles are congruent. <1 & <3, <2 & <4 Then what do we call <1 & <3?

Linear Pair (of angles) 2 adjacent angles whose non-common sides are opposite rays. The sum of a linear pair = 180 degrees 1 2

Ex 1: Vertical angles? Adjacent angles? Linear pair? (2 adjacent angles=180 degrees) Adjacent angles not a linear pair?

Ex 2: If m<5=130 o, find m<3 m<6 m<

Ex 3: Find the values of x & y & m<ABE m<ABD m<DBC m<EBC 3x+5 o y+20 o x+15 o 4y-15 o A B C D E

Complementary Angles 2 angles whose sum is 90 o o A 55 o B <1 & <2 are complementary <A & <B are complementary

Supplementary Angles 2 angles whose sum is 180 o o 50 o X Y <1 & <2 are supplementary. <X & <Y are supplementary.

Ex 4: <A & <B are supplementary. m<A is 5 times m<B. Find m<A & m<B. m<A + m<B = 180 o m<A = 5(m<B) Now substitute! 5(m<B) + m<B = 180 o 6(m<B)=180 o m<B=30 o m<A=150 o