Introduction to Angles and Triangles

Math is a language Line – extends indefinitely, no thickness or width Ray – part of a line, starts at a point, goes indefinitely Line segment – part of a line, begin and end point Angle - two lines, segments or rays from a common point Vertex - common point at which two lines or rays are joined

Degrees: Measuring Angles We measure the size of an angle using degrees. Example: Here are some examples of angles and their degree measurements.

Acute Angles An acute angle is an angle measuring between 0 and 90 degrees. Example:                                                                

Right Angles A right angle is an angle measuring 90 degrees. Example:                                                                 90°

Two angles are called complementary angles if the sum of their degree measurements equals 90 degrees. Example: These two angles are complementary.                                                                                 32° 58° Together they create a 90° angle

Obtuse Angles An obtuse angle is an angle measuring between 90 and 180 degrees. Example:                                                                  

Straight Angle A right angle is an angle measuring 180 degrees. Examples:                                       

Two angles are called supplementary angles if the sum of their degree measurements equals 180 degrees. Example: These two angles are supplementary.                                                                        139° 41° These two angles sum is 180° and together the form a straight line                                                 

Review State whether the following are acute, right, or obtuse. acute 3. 5. 1. right obtuse ? 4. 2. acute ? obtuse

Complementary and Supplementary Find the missing angle. 1. Two angles are complementary. One measures 65 degrees. 2. Two angles are supplementary. One measures 140 degrees. Answer : 25 Answer : 40

Complementary and Supplementary Find the missing angle. You do not have a protractor. Use the clues in the pictures. 2. 1. x x 55 165 X=35 X=15

1. x x = 90 y z y = z = x = 2. x 110 y z y = z =

Vertical Angles are angles on opposite sides of intersecting lines                     1. 90 x y z 90 90 and y are vertical angles x and z are vertical angles The vertical angles in this case are equal, will this always be true? 110 70 110 x y z 110 and y are vertical angles 2. x and z are vertical angles Vertical angles are always equal

Vertical Angles Find the missing angle. Use the clues in the pictures. x X=58 58

Can you find the missing angles? 20 90 D 70 G C 70 90 20 J H

Can you find these missing angles B C 68 A 52 G 60 D 60 52 68 F E

Parallel lines transversals and their angles

Parallel Lines What You'll Learn You will learn to identify the relationships among pairs of interior and exterior angles formed by two parallel lines and a transversal. In geometry, two lines in a plane that never intersect , have the same slope, are called ____________. parallel lines parallel lines are always the same distance apart

Parallel Lines and Transversals In geometry, a line, line segment, or ray that intersects two or more lines at different points is called a __________ transversal The lines cut by a transversal may or may not be parallel. l m 1 2 3 4 5 7 6 8 Parallel Lines t is a transversal for l and m. t 1 2 3 4 5 7 6 8 b c Nonparallel Lines r is a transversal for b and c. r

Parallel Lines and Transversals is an example of a transversal. It intercepts lines l and m. We will be most concerned with transversals that cut parallel lines. B A l m 1 2 4 3 When a transversal cuts parallel lines, special pairs of angles are formed that are sometimes congruent and sometimes supplementary. 5 6 8 7

Parallel Lines and Transversals Two lines divide the plane into three regions. The region between the lines is referred to as the interior. The two regions not between the lines is referred to as the exterior. Exterior Interior

Parallel Lines and Transversals When a transversal intersects two lines, _____ angles are formed. eight These angles are given special names. 1. Interior angles , 3,4,5,6 lie between the two parallel lines. l m 1 2 3 4 5 7 6 8 2. Exterior angles 1,2,7,8 lie outside the two lines. 3. Alternate Interior angles 4&6, 5&3 opposite sides of the transversal and lie between the parallel lines t 5. Consecutive Interior angles 4&5, 3&6 on the same side of the transversal and are between the parallel lines 4. Alternate Exterior angles 1&7, 2&8 Are on the opposite sides of the transversal and lie outside the two lines 6. Corresponding angles 1&5, 4&8, 2&6, 3&7 on the same side of the transversal one is exterior and the other is interior

Name the pairs of the following angles formed by a transversal. Line M B A Line N D E P Q G F Line L Line M B A Line N D E P Q G F Line L 500 1300 Line M B A Line N D E P Q G F Line L Alternate Interior angles Corresponding angles Consecutive Interior angles

Congruent: Same shape and size The symbol means that the shapes, lines or angles are congruent two shapes both have an area of 36 in2 , are they congruent? 6 in Area is 36 in2 9 in 4 in Area is 36 in2 Numbers, or expressions can have equal value….. In Geometry, we use “congruent” to describe two or more objects, lines or angles as being the same

Parallel Lines and Transversals Alternate interior angles are _________. congruent 1 2 4 3 5 6 8 7

Parallel Lines and Transversals Alternate exterior angles is _________. congruent 1 2 3 4 5 7 6 8 ?

Parallel Lines and Transversals consecutive interior angles is _____________. supplementary 1 2 3 4 5 7 6 8

Transversals and Corresponding Angles corresponding angles is _________. congruent

Transversals and Corresponding Angles Concept Summary Congruent Supplementary Types of angle pairs formed when a transversal cuts two parallel lines. alternate interior consecutive interior alternate exterior corresponding

Transversals and Corresponding Angles 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 s || t and c || d. Name all the angles that are congruent to 1. Give a reason for each answer. 3  1 corresponding angles 6  1 vertical angles 8  1 alternate exterior angles 9  1 corresponding angles 14  1 alternate exterior angles 11  9  1 corresponding angles 16  14  1 corresponding angles

Let’s Practice m<1=120° Find all the remaining angle measures. 1 4 6 5 7 8 3 m<1=120° Find all the remaining angle measures. 120° 60° 120° 60° 120° 60° 120° 60°

Another practice problem 40° 180-(40+60)= 80° Find all the missing angle measures, and name the postulate or theorem that gives us permission to make our statements. 60° 6 7 8 4 80° 5 60° 40° 80° 120° 100° 60° 11 9 2 1 10 12 3 80° 120° 60° 100°