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OTCQ RGT is adjacent to angle TGL. If m ◦ RGL = 100 and What is m RGT? m ◦ TGL = 20 , OTCQ RGT is adjacent to angle TGL. If m ◦ RGL = 100 and What is m RGT? G 80 ◦ L ◦ TGL = 20 , m R 20 ◦ T Aims 1-6 & 1-7 How do we define angles and triangles? NY GG 31, GG 32, GG 33, GG 35, GG 28 Read 1-6 and 1-7 do problems 18 -25 on page 22. 1-6 & 1-7 objectives 1. 2. 3. 4. SWBAT name angles and angle parts. SWBAT define: congruent angles, complimentary angles, supplementary angles and adjacent angles Bisector of an angle Perpendicular line Distance from a point to a line SWBAT add angles. SWBAT define a triangle and its parts. • An angle consists of two different rays that have the same initial point. • The rays are the sides of the angle. • The initial point A is the vertex of the angle. • The angle that has rays AB and AC as sides may be named BAC, CAB, or A. B sides Vertex A C The set of all points between the sides of the angle is the interior of an angle. The exterior of an angle is the set of all points outside the angle. Angle Name R, SRT, TRS, or 1 If the point is the vertex of more than one angle, you must use all three points to name the angle. The middle point is always the vertex. The measure of A is denoted by mA. The measure of an angle can be approximated using a protractor, using units called degrees(°). BAC has a measure of 50°, which can be written as mBAC = 50°. B C A Angle Relationships Congruent Angles: Angles with equal measures in degrees. MSU RWE EWR or M USM W E R S U Angle Relationships • Complementary Angles: Two angles are called complementary angles if the sum of their degree measurements equals 90° degrees. • Supplementary Angles: Two angles are called supplementary angles if the sum of their degree measurements equals 180° degrees. Angle Relationships • Adjacent Angles: Share a vertex and a common side but no interior points. • Bisector of an angle: a ray that divides the angle into two congruent angles. OY is the angle bisector. What are perpendicular lines? • Two lines that ◦ intersect at a 90 angle (right angle) are Perpendicular lines. • Boxed vertex ◦ means 90 The shortest distance from a point to a line will always be the distance of the perpendicular line from the point to the line. Boxed vertex means 90 ◦ Adding Angles When you want to add angles, use the notation m1, meaning the measure of 1. If you add m1 + m2, what is your result? m1 + m2 = 58. A B 36° m1 + m2 = mADC also. 22° 1 Therefore, mADC = 58. D 2 C Congruent Angles that measure the same in degrees are congruent. PET TEJ J LEP E L T JEL P Symbol for congruent Congruent Explain why the angles are congruent. J PET LEP E L T TEJ JEL P Symbol for congruent Congruent They are all 90◦ PET J LEP E L T TEJ JEL P Symbol for congruent Congruent Segments that measure the same in inches, feet, miles, milimeters, centimeters, meters, or kilometers are called congruent. A T P C ___ AT ___ PC Symbol for congruent 1-6 & 1-7 objectives CHECK UP 1. 2. 3. SWBAT name angles and angle parts. SWBAT define: congruent angles, complimentary angles, supplementary angles and adjacent angles Bisector of an angle Perpendicular line Distance from a point to a line SWBAT add angles. 4. NEXT:SWBAT define a triangle and its parts. Polygon: A closed plane figure formed by three or more line segments that meet only at their endpoints. Triangle: a polygon with exactly 3 sides. Types of Triangles: Equilateral all 3 sides and angles are congruent. Also called Equiangular. Isosceles Triangle— 2 or 3 sides/angles are congruent Scalene— no sides or angles are congruent Triangle Parts: • 3 vertices: A, B, C. • 3 line segments: __ __ __ BC , CA & AB • Any 2 line segments in a triangle that share a common endpoint are called adjacent sides. __ CB is the opposite side of endpoint A C B __ __ CA and BA share endpoint A, so they are adjacent sides. __ __ CB and BA share endpoint B, so they are adjacent sides. __ __ BC and CA share endpoint C, so they are adjacent sides. A EQUILATERAL is EQUIANGULAR • 3 congruent 60◦ acute angles (“equiangular”). • 3 congruent line segments. (“equilateral” ). • Acute. • Can’t be right • Also isosceles because at least 2 sides/angles are congruent. • Each line segment is a side. Isosceles Triangles need only 2 congruent sides/angles. • The 2 congruent sides are called legs. Green dashes means congruent. • The noncongruent side is called the base. No green dashes. • The 2 congruent angles are called base angles. Blue arc means congruent. • The noncongruent angle is called the vertex angle. No blue arcs. Vertex Angle leg leg Base Base Base Angle Angle ISOCELES Right Triangle 45◦ 45◦ base • The red side is both a hypotenuse and a base. angle • The boxed 90◦ right leg angle is the vertex angle. Hypotenuse base leg ◦ 45 Base angle • A right triangle with 2 congruent sides/angles, is an isosceles right triangle. • its base angles are each 90◦ vertex • Green dashes mean congruent. • Blue arcs mean congruent. Acute Triangle: an acute triangle has 3 acute angles. Could be equilateral/equiangular (all = 60◦). Could be isosceles ∆ or scalene ∆ but never a right and never an obtuse ∆. m ABC = 70.26 m CAB = 41.76 m BCA = 67.97 B C A Right Triangle • 1 right angle Obtuse Triangle: has one obtuse angle. Right and Obtuse triangles are never acute or equilateral, but can be isosceles (have 2 congruent sides/angles). Right Triangle hypotenuse leg • Red represents the hypotenuse of a right triangle. • The blue sides that form the right angle leg are the legs. What is this? leg base leg What is this? An isosceles triangle. leg base leg Quick Algebra Review Commutative Property Commutative Property of Addition: a + b = b + a Commutative Property of Multiplication: ab = ba Examples 2+3=5=3+2 3• 4 = 12 = 4 • 3 The commutative property does not work for subtraction or division!!!!!!!! Associative Property Associative property of Addition: (a + b) + c = a + (b + c) Associative Property of Multiplication: (ab) c = a (bc) Examples (1 + 2) + 3 = 1 + (2 + 3) (2 • 3) • 4 = 2 • (3 • 4) The associative property does not work for subtraction or division!!!!! Identity Properties 1) Additive Identity a+0=a 2) Multiplicative Identity a•1=a Inverse Properties 1) Additive Inverse (Opposite) a + (-a) = 0 2) Multiplicative Inverse (Reciprocal) 1 a 1 a Multiplicative Property of Zero a•0=0 (If you multiply by 0, the answer is 0.) The Distributive Property Any factor outside of expression enclosed within grouping symbols, must be multiplied by each term inside the grouping symbols. Outside left a(b + c) = ab + ac a(b - c) = ab – ac or Outside right (b + c)a = ba + ca (b - c)a = ba - ca Time permitting start homework