BellringerYour 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 anglesAll 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. 12345?

StandardsSPIs taught in Unit 2:SPI 3108.1.1 Give precise mathematical descriptions or definitions of geometric shapes in the plane and space. SPI 3108.1.4 Use definitions, basic postulates, and theorems about points, lines, angles, and planes to write/complete proofs and/or to solve problems. SPI 3108.3.1 Use algebra and coordinate geometry to analyze and solve problems about geometric figures (including circles). SPI 3108.4.2 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 3108.1.1 Use mathematical language, symbols, definitions, proofs and counterexamples correctly and precisely in mathematical reasoning. CLE 3108.4.1 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 2:3108.1.7 Recognize the capabilities and the limitations of calculators and computers in solving problems. 3108.4.5 Use vertical, adjacent, complementary, and supplementary angle pairs to solve problems and write proofs.

ReviewWe have already addressed much of what is covered in the section on anglesWe classify angles in 4 ways:Less than 90 degrees: Acute AngleEqual to 90 degrees: Right angleGreater than 90, but less than 180: Obtuse angleEqual to 180 degrees: Straight angle

ReviewWe 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 AnglesVertical Angles: Two angles whose sides are opposite rays

Which angle pairs are vertical angles?Angle A and Angle CAngle D and Angle BWhat letter in the alphabet always creates vertical angles?ABCDVertical Angles are ALWAYS equal

Angle Pair Complementary AnglesComplementary Angles Two angles whose measures have a sum of 90 degreesEach angle is called the complement of the other

Angle 1 is the complement of angle 2Angle B is the complement of Angle A. What conclusion can we draw?Angle B is 30 degrees60B12A

Angle Pairs Adjacent AnglesAdjacent Angles Two coplanar angles with one common side, one common vertex, and no common interior pointsAB12Common SideCommon Vertex

Angle Pairs Supplementary AnglesSupplementary Angles Two angles whose measures have a sum of 180 degreesEach 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.AB13545These are also known as Linear Pairs because they make a line

ExampleIdentify the given angle pairsComplementary AnglesSupplementary AnglesVertical AnglesAdjacent Angles12345

ConclusionsGiven the type of diagram we have seen, you can conclude that angles are:Adjacent AnglesVertical AnglesAdjacent supplementary AnglesWithout congruency marks, you cannot conclude that:Angles or segments are congruentAn angle is a right angleLines are parallel or perpendicularAdjacent angles are complementary

ExampleWhat conclusions can we make about this diagram?12345

Vertical Angle TheoremVertical 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?120ABCDE

ExampleSolve for X

Since they are equal in measure, we set them equal to each other: 4X = 3X + 35Therefore X = 354X3X+35

Congruent Supplements TheoremIf 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 congruent123

Supplements of Congruent AnglesRemember, 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 congruent1234

Congruent Complements TheoremIf two angles are complements of the same angle (or of congruent angles), then the two angles are congruentThis is exactly the same idea as supplementary congruent angles. Just with 90 degrees instead of 180 degreesALL right angles are congruentThis is common sense. All right angles are 90 degreesIf two angles are congruent and supplementary, then each is a right angleThe 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

AssignmentText, Page 38-39 problems 7-30, 33-36 (guided practice)Worksheet P 1-5Worksheet 2-5Angles and Segments WorksheetIF 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, dont copy your notes word for word- define:Vertical AnglesAdjacent AnglesSupplementary AnglesComplementary AnglesLinear Angles

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