Bellringer Your mission: Construct a perfect square using the construction techniques you have learned from Unit 1. You may NOT measure any lengths with your ruler. You may NOT measure any angles All sides must be perfectly perpendicular (90 degree angle) and all side segments must be congruent (hint hint ;) You have 10 minutes.

Unit 2 Angle Pairs Unit 2: This unit introduces angles, types of angles, and angle pairs. It defines complimentary and supplementary angles ?

Standards SPI’s taught in Unit 2: SPI Give precise mathematical descriptions or definitions of geometric shapes in the plane and space. SPI Use definitions, basic postulates, and theorems about points, lines, angles, and planes to write/complete proofs and/or to solve problems. SPI Use algebra and coordinate geometry to analyze and solve problems about geometric figures (including circles). SPI Define, identify, describe, and/or model plane figures using appropriate mathematical symbols (including collinear and non-collinear points, lines, segments, rays, angles, triangles, quadrilaterals, and other polygons). CLE (Course Level Expectations) found in Unit 2: CLE Use mathematical language, symbols, definitions, proofs and counterexamples correctly and precisely in mathematical reasoning. CLE Develop the structures of geometry, such as lines, angles, planes, and planar figures, and explore their properties and relationships. CFU (Checks for Understanding) applied to Unit 2: Recognize the capabilities and the limitations of calculators and computers in solving problems Use vertical, adjacent, complementary, and supplementary angle pairs to solve problems and write proofs.

Review We have already addressed much of what is covered in the section on angles We classify angles in 4 ways: Less than 90 degrees: Acute Angle Equal to 90 degrees: Right angle Greater than 90, but less than 180: Obtuse angle Equal to 180 degrees: Straight angle

Review We define an angle bisector as: An angle bisector is a ray that divides an angle into two congruent coplanar angles. Its endpoint is the angle vertex. You can also say that a ray or segment bisects the angle.

Angle Pairs –Vertical Angles Vertical Angles: Two angles whose sides are opposite rays Which angle pairs are vertical angles? – Angle A and Angle C – Angle D and Angle B What letter in the alphabet always creates vertical angles? A B C D Vertical Angles are ALWAYS equal

Angle Pair –Complementary Angles Complementary Angles –Two angles whose measures have a sum of 90 degrees Each angle is called the complement of the other Angle 1 is the complement of angle 2 Angle B is the complement of Angle A. What conclusion can we draw? –A–Angle B is 30 degrees 60 B 1 2 A

Angle Pairs –Adjacent Angles Adjacent Angles – Two coplanar angles with one common side, one common vertex, and no common interior points A B 1 2 Common Side Common Vertex

Angle Pairs –Supplementary Angles Supplementary Angles –Two angles whose measures have a sum of 180 degrees Each angle is called the supplement of the other The angles do not have to be touching, or share a vertex, to be supplementary. They just have to sum 180 degrees. AB These are also known as “Linear Pairs” because they make a line

Example Identify the given angle pairs – Complementary Angles – Supplementary Angles – Vertical Angles – Adjacent Angles

Conclusions Given the type of diagram we have seen, you can conclude that angles are: – Adjacent Angles – Vertical Angles – Adjacent supplementary Angles Without congruency marks, you cannot conclude that: – Angles or segments are congruent – An angle is a right angle – Lines are parallel or perpendicular – Adjacent angles are complementary

Example What conclusions can we make about this diagram?

Vertical Angle Theorem Vertical Angles are Congruent If angle ABC = 120 degrees, what is the measure of angle EBD? What is the measure of angle CBD? What is the measure of angle ABE? 120 A B C D E

Example Solve for X Since they are equal in measure, we set them equal to each other: 4X = 3X + 35 Therefore X = 35 4X 3X+35

Congruent Supplements Theorem If two angles are supplements of the same angle (or of congruent angles) then the two angles are congruent Here, angle 1 is a supplement of angle 3, and angle 2 is a supplement of angle 3. We can therefore conclude that angle 1 and angle 2 are congruent 123

Supplements of Congruent Angles Remember, if two angles are supplements of congruent angles, then those two angles are congruent as well Here, angle 1 is a supplement of angle 3, and angle 2 is a supplement of angle 4. Because angle 3 and 4 are congruent, angle 1 and angle 2 are also congruent 123 4

Congruent Complements Theorem If two angles are complements of the same angle (or of congruent angles), then the two angles are congruent – This is exactly the same idea as supplementary congruent angles. Just with 90 degrees instead of 180 degrees ALL right angles are congruent – This is common sense. All right angles are 90 degrees If two angles are congruent and supplementary, then each is a right angle – The only way for two angles to add up to 180 degrees, and be equal to each other, is for each angle to be 90 degrees, which is by definition a right angle

Assignment Text, Page problems 7-30, (guided practice) Worksheet P 1-5 Worksheet 2-5 Angles and Segments Worksheet IF YOU DO NOT USE THE ANGLE SYMBOL, THEN I WILL MARK -3 ON YOUR PAPER. LABEL PROPERLY!

Unit 2 Bellringer (2 points each) In your own words –in other words, don’t copy your notes word for word- define: 1.Vertical Angles 2.Adjacent Angles 3.Supplementary Angles 4.Complementary Angles 5.Linear Angles