1Angles § 3.1 Angles § 3.2 Angle Measure § 3.3 The Angle Addition Postulate§ 3.4 Adjacent Angles and Linear Pairs of Angles§ 3.5 Complementary and Supplementary Angles§ 3.6 Congruent Angles§ 3.7 Perpendicular Lines
2Vocabulary What You'll Learn AnglesWhat You'll LearnYou will learn to name and identify parts of an angle.Vocabulary1) Opposite Rays2) Straight Angle3) Angle4) Vertex5) Sides6) Interior7) Exterior
3XY and XZ are ____________. opposite rays AnglesOpposite rays___________ are two rays that are part of a the same line and have only their endpoints in common.XYZXY and XZ are ____________.opposite raysThe figure formed by opposite rays is also referred to as a ____________.straight angle
4There is another case where two rays can have a common endpoint. AnglesThere is another case where two rays can have a common endpoint.This figure is called an _____.angleSome parts of angles have special names.SThe common endpoint is called the ______,vertexand the two rays that make up the sides of the angle are called the sides of the angle.sideRTsidevertex
5There are several ways to name this angle. AnglesThere are several ways to name this angle.1) Use the vertex and a point from each side.SRTorTRSSThe vertex letter is always in the middle.2) Use the vertex only.sideRIf there is only one angle at a vertex, then the angle can be named with that vertex.1RTsidevertex3) Use a number.1
6AnglesDefinition of AngleAn angle is a figure formed by two noncollinear rays that have a common endpoint.EDF2Symbols:DEFFEDE2
71) Name the angle in four ways. Angles1) Name the angle in four ways.BA1CABCCBAB12) Identify the vertex and sides of this angle.vertex:Point Bsides:BA andBC
81) Name all angles having W as their vertex. 21W2XWZY2) What are other names for ?1ZXWY orYWX3) Is there an angle that can be named ?WNo!
9An angle separates a plane into three parts: 1) the ______interiorexterior2) the ______exteriorWYZAinterior3) the _________angle itselfIn the figure shown, point B and all other points in the blue region are in the interior of the angle.BPoint A and all other points in the green region are in the exterior of the angle.Points Y, W, and Z are on the angle.
10AnglesIs point B in the interior of the angle, exterior of the angle, or on the angle?PGExteriorBIs point G in the interior of the angle, exterior of the angle, or on the angle?On the angleIs point P in the interior of the angle, exterior of the angle, or on the angle?Interior
11Vocabulary What You'll Learn §3.2 Angle MeasureWhat You'll LearnYou will learn to measure, draw, and classify angles.Vocabulary1) Degrees2) Protractor3) Right Angle4) Acute Angle5) Obtuse Angle
12In geometry, angles are measured in units called _______. degrees §3.2 Angle MeasureIn geometry, angles are measured in units called _______.degreesThe symbol for degree is °.QPR75°In the figure to the right, the angle is 75 degrees.In notation, there is no degree symbol with 75because the measure of an angle is a realnumber with no unit of measure.m PQR = 75
13Angles Measure Postulate §3.2 Angle MeasurePostulate 3-1Angles Measure PostulateFor every angle, there is a unique positive number between __ and ____ called the degree measure of the angle.180BACn°m ABC = nand 0 < n < 180
14Use a protractor to measure SRQ. §3.2 Angle MeasureYou can use a _________ to measure angles and sketch angles of given measure.protractorUse a protractor to measure SRQ.1) Place the center point of the protractor on vertex R Align the straightedge with side RS.QRS2) Use the scale that begins with at RS Read where the other side of the angle, RQ, crosses this scale.
15Find the measurement of: m SRH 70 §3.2 Angle MeasureFind the measurement of:m SRH70m SRQ =180m QRG =180 – 150= 30m SRJ =45m GRJ =150 – 45= 105m SRG =150HJGSQR
16Use a protractor to draw an angle having a measure of 135. §3.2 Angle MeasureUse a protractor to draw an angle having a measure of 135.1) Draw AB3) Locate and draw point C at the mark labeled Draw AC.2) Place the center point of the protractor on A. Align the mark labeled 0 with the ray.CAB
17obtuse angle 90 < m A < 180 right angle m A = 90 §3.2 Angle MeasureOnce the measure of an angle is known, the angle can be classified as one of three types of angles. These types are defined in relation to a right angle.Types of AnglesAAAobtuse angle 90 < m A < 180right angle m A = 90acute angle 0 < m A < 90
18Classify each angle as acute, obtuse, or right. §3.2 Angle MeasureClassify each angle as acute, obtuse, or right.110°90°40°ObtuseRightAcute75°50°130°AcuteObtuseAcute
19The measure of B is 138. Solve for x. §3.2 Angle Measure5x - 7BThe measure of B is 138. Solve for x.9y + 4HThe measure of H is 67. Solve for y.Given: (What do you know?)Given: (What do you know?)B = 5x – 7 and B = 138H = 9y and H = 675x – 7 = 1389y + 4 = 67Check!Check!5x = 1459y = 635(29) -7 = ?9(7) + 4 = ?x = 29y = 7= ?= ?138 = 13867 = 67
22§3.3 The Angle Addition Postulate What You'll LearnYou will learn to find the measure of an angle and the bisector of an angle.VocabularyNOTHING NEW!
23§3.3 The Angle Addition Postulate 1) Draw an acute, an obtuse, or a right angle Label the angle RST.RTS45°2) Draw and label a point X in the interior of the angle. Then draw SX.X75°30°3) For each angle, find mRSX, mXST, and RST.
24§3.3 The Angle Addition Postulate 1) How does the sum of mRSX and mXST compare to mRST ?Their sum is equal to the measure of RST .mXST = 30+ mRSX = 45= mRST = 75RTS2) Make a conjecture about the relationship between the two smaller angles and the larger angle.45°XThe sum of the measures of the two smaller angles is equal to the measure of the larger angle.The Angle Addition Postulate (Video)75°30°
25§3.3 The Angle Addition Postulate Angle Addition Postulate For any angle PQR, if A is in the interior of PQR, then mPQA + mAQR = mPQR.21ARPQm1 + m2 = mPQR.There are two equations that can be derived using Postulate 3 – 3.m1 = mPQR – m2These equations are true no matter where A is located in the interior of PQR.m2 = mPQR – m1
27§3.3 The Angle Addition Postulate Find mABC and mCBD if mABD = 120.mABC + mCBD = mABDPostulate 3 – 3.2x + (5x – 6) = 120Substitution7x – 6 = 120Combine like terms7x = 126Add 6 to both sidesx = 18Divide each side by 7= 1202x°(5x – 6)°BDCAmABC = 2xmCBD = 5x – 6mABC = 2(18)mCBD = 5(18) – 6mABC = 36mCBD = 90 – 6mCBD = 84
28§3.3 The Angle Addition Postulate Just as every segment has a midpoint that bisects the segment, every anglehas a ___ that bisects the angle.rayThis ray is called an ____________ .angle bisector
29§3.3 The Angle Addition Postulate Definition ofan Angle BisectorThe bisector of an angle is the ray with its endpoint at the vertex of the angle, extending into the interior of the angle.The bisector separates the angle into two angles of equal measure.21ARPQis the bisector of PQR.m1 = m2
30§3.3 The Angle Addition Postulate If bisects CAN and mCAN = 130, find 1 and 2.Since bisects CAN, 1 = 2.12ACNT1 + 2 = CANPostulate 3 - 31 + 2 = 130Replace CAN with 1301 + 1 = 130Replace 2 with 12(1) = 130Combine like terms(1) = 65Divide each side by 2Since 1 = 2, 2 = 65
32Adjacent Angles and Linear Pairs of Angles What You'll LearnYou will learn to identify and use adjacent angles and linear pairs of angles.ACBWhen you “split” an angle, you create two angles.DThe two angles are called_____________adjacent angles21adjacent = next to, joining.1 and 2 are examples of adjacent angles. They share a common ray.Name the ray that 1 and 2 have in common. ____
33Adjacent Angles and Linear Pairs of Angles Definition ofAdjacentAnglesAdjacent angles are angles that:A) share a common sideB) have the same vertex, andC) have no interior points in commonMJNR121 and 2 are adjacentwith the same vertex R andcommon side
34Adjacent Angles and Linear Pairs of Angles Determine whether 1 and 2 are adjacent angles.No. They have a common vertex B, but_____________12Bno common sideYes. They have the same vertex G and a common side with no interior points in common.12GN12JLNo. They do not have a common vertex or ____________a common sideThe side of 1 is ____The side of 2 is ____
35Adjacent Angles and Linear Pairs of Angles Determine whether 1 and 2 are adjacent angles.No.12Yes.12XDZIn this example, the noncommon sides of the adjacent angles form a___________.straight lineThese angles are called a _________linear pair
36Adjacent Angles and Linear Pairs of Angles Definition ofLinear PairsTwo angles form a linear pair if and only if (iff):A) they are adjacent andB) their noncommon sides are opposite raysCADB121 and 2 are a linear pair.Note:
37Adjacent Angles and Linear Pairs of Angles In the figure, and are opposite rays.12M43EHTAC1) Name the angle that forms alinear pair with 1.ACEACE and 1 have a common side ,the same vertex C, and opposite raysand2) Do 3 and TCM form a linear pair? Justify your answer.No. Their noncommon sides are not opposite rays.
39§3.5 Complementary and Supplementary Angles What You'll LearnYou will learn to identify and use Complementary andSupplementary angles
40§3.5 Complementary and Supplementary Angles Definition ofComplementaryAnglesTwo angles are complementary if and only if (iff) the sum of their degree measure is 90.60°DEF30°ABCmABC + mDEF = = 90
41§3.5 Complementary and Supplementary Angles If two angles are complementary, each angle is a complement of the other.ABC is the complement of DEF and DEF is the complement of ABC.60°DEF30°ABCComplementary angles DO NOT need to have a common side or even thesame vertex.
42§3.5 Complementary and Supplementary Angles Some examples of complementary angles are shown below.75°ImH + mI = 9015°H50°H40°QPSmPHQ + mQHS = 9030°60°TUVWZmTZU + mVZW = 90
43§3.5 Complementary and Supplementary Angles If the sum of the measure of two angles is 180, they form a special pair ofangles called supplementary angles.Definition ofSupplementaryAnglesTwo angles are supplementary if and only if (iff) the sum of their degree measure is 180.130°DEF50°ABCmABC + mDEF = = 180
44§3.5 Complementary and Supplementary Angles Some examples of supplementary angles are shown below.105°H75°ImH + mI = 18050°H130°QPSmPHQ + mQHS = 18060°120°TUVWZmTZU + mUZV = 180andmTZU + mVZW = 180
46What You'll Learn You will learn to identify and use congruent and §3.6 Congruent AnglesWhat You'll LearnYou will learn to identify and use congruent andvertical angles.Recall that congruent segments have the same ________.measure_______________ also have the same measure.Congruent angles
47Two angles are congruent iff, they have the same ______________. §3.6 Congruent AnglesDefinition ofCongruentAnglesTwo angles are congruent iff, they have the same______________.degree measureB V iff50°VmB = mV50°B
48To show that 1 is congruent to 2, we use ____. arcs §3.6 Congruent AnglesTo show that 1 is congruent to 2, we use ____.arcs12To show that there is a second set of congruent angles, X and Z, we use double arcs.This “arc” notation states that:ZXX ZmX = mZ
49When two lines intersect, ____ angles are formed. four §3.6 Congruent AnglesWhen two lines intersect, ____ angles are formed.fourThere are two pair of nonadjacent angles.These pairs are called _____________.vertical angles1423
50Two angles are vertical iff they are two nonadjacent §3.6 Congruent AnglesDefinition ofVerticalAnglesTwo angles are vertical iff they are two nonadjacentangles formed by a pair of intersecting lines.Vertical angles:1 and 31422 and 43
51Hands-On 1) On a sheet of paper, construct two intersecting lines §3.6 Congruent Angles1) On a sheet of paper, construct two intersecting linesthat are not perpendicular.2) With a protractor, measure each angle formed.3) Make a conjecture about vertical angles.1234Consider:A. 1 is supplementary to 4.m1 + m4 = 180Hands-OnB. 3 is supplementary to 4.m3 + m4 = 180Therefore, it can be shown that1 3Likewise, it can be shown that2 4
521) If m1 = 4x + 3 and the m3 = 2x + 11, then find the m3 §3.6 Congruent Angles12341) If m1 = 4x + 3 and the m3 = 2x + 11, then find the m3x = 4; 3 = 19°2) If m2 = x + 9 and the m3 = 2x + 3, then find the m4x = 56; 4 = 65°3) If m2 = 6x - 1 and the m4 = 4x + 17, then find the m3x = 9; 3 = 127°4) If m1 = 9x - 7 and the m3 = 6x + 23, then find the m4x = 10; 4 = 97°
53Vertical angles are congruent. §3.6 Congruent AnglesTheorem 3-1Vertical AngleTheoremVertical angles are congruent.nm21 3312 44
54Find the value of x in the figure: §3.6 Congruent AnglesFind the value of x in the figure:The angles are vertical angles.130°x°So, the value of x is 130°.
55Find the value of x in the figure: §3.6 Congruent AnglesFind the value of x in the figure:The angles are vertical angles.(x – 10) = 125.(x – 10)°x – 10 = 125.125°x = 135.
56Suppose two angles are congruent. §3.6 Congruent AnglesSuppose two angles are congruent.What do you think is true about their complements?1 21 + x = 902 + y = 90x is the complementof 1y is the complementof 2x = 90 - 1y = 90 - 2Because 1 2, a “substitution” is made.x = 90 - 1y = 90 - 1x = yx yIf two angles are congruent, their complements are congruent.
57If two angles are congruent, then their complements are _________. §3.6 Congruent AnglesTheorem 3-2If two angles are congruent, then their complements are_________.congruentThe measure of angles complementary to A and Bis 30.AB60°A BTheorem 3-3If two angles are congruent, then their supplements are_________.congruentThe measure of angles supplementary to 1 and 4is 110.432170°110°110°70°4 1
58If two angles are complementary to the same angle, §3.6 Congruent AnglesTheorem 3-4If two angles are complementary to the same angle,then they are _________.congruent3 is complementary to 45 is complementary to 4435 35Theorem 3-5If two angles are supplementary to the same angle,then they are _________.congruent3121 is supplementary to 23 is supplementary to 21 3
59Find the measure of an angle that is supplementary to B. §3.6 Congruent AnglesSuppose A B and mA = 52.Find the measure of an angle that is supplementary to B.A52°B52°1B + 1 = 1801 = 180 – B1 = 180 – 521 = 128°
60If 1 is complementary to 3, 2 is complementary to 3, and m3 = 25, §3.6 Congruent AnglesIf 1 is complementary to 3,2 is complementary to 3,and m3 = 25,What are m1 and m2 ?m1 + m3 = Definition of complementary angles.m1 = 90 - m Subtract m3 from both sides.m1 = Substitute 25 in for m3.m1 = Simplify the right side.You solve for m2m2 + m3 = Definition of complementary angles.m2 = 90 - m Subtract m3 from both sides.m2 = Substitute 25 in for m3.m2 = Simplify the right side.
611) If m1 = 2x + 3 and the m3 = 3x - 14, then find the m3 §3.6 Congruent AnglesABCDEGH12341) If m1 = 2x + 3 and the m3 = 3x - 14, then find the m3x = 17; 3 = 37°2) If mABD = 4x + 5 and the mDBC = 2x + 1, then find the mEBCx = 29; EBC = 121°3) If m1 = 4x and the m3 = 2x + 19, then find the m4x = 16; 4 = 39°4) If mEBG = 7x and the mEBH = 2x + 7, then find the m1x = 18; 1 = 43°
62Suppose you draw two angles that are congruent and supplementary. What is true about the angles?
63All right angles are _________. congruent §3.6 Congruent AnglesTheorem 3-6If two angles are congruent and supplementary then each is a __________.right angle1 is supplementary to 2121 and 2 = 90Theorem 3-7All right angles are _________.congruentCBAA B C
64If 1 is supplementary to 4, 3 is supplementary to 4, and §3.6 Congruent AnglesIf 1 is supplementary to 4, 3 is supplementary to 4, andm 1 = 64, what are m 3 and m 4?ADCBE12341 3They are vertical angles.m 1 = m3m 3 = 643 is supplementary to 4Givenm3 + m4 = 180Definition of supplementary.64 + m4 = 180m4 = 180 – 64m4 = 116
66§3.7 Perpendicular LinesWhat You'll LearnYou will learn to identify, use properties of, and constructperpendicular lines and segments.
67Lines that intersect at an angle of 90 degrees are _________________. §3.7 Perpendicular LinesLines that intersect at an angle of 90 degrees are _________________.perpendicular linesIn the figure below, lines are perpendicular.ADCB1234
68Perpendicular lines are lines that intersect to form a right angle. Definition ofPerpendicularLinesPerpendicular lines are lines that intersect to form aright angle.mn
69In the figure below, l m. The following statements are true. §3.7 Perpendicular LinesIn the figure below, l m. The following statements are true.m2134l1) 1 is a right angle.Definition of Perpendicular Lines2) 1 3.Vertical angles are congruent3) 1 and 4 form a linear pair.Definition of Linear Pair4) 1 and 4 are supplementary.Linear pairs are supplementary5) 4 is a right angle.m = 180, m4 = 906) 2 is a right angle.Vertical angles are congruent
70If two lines are perpendicular, then they form four right angles. §3.7 Perpendicular LinesTheorem 3-8If two lines are perpendicular, then they form four right angles.1342ab
711) PRN is an acute angle. False. 2) 4 8 True §3.7 Perpendicular Lines1) PRN is an acute angle.False.2) 4 8True
72If a line m is in a plane and T is a point in m, then there §3.7 Perpendicular LinesTheorem 3-9If a line m is in a plane and T is a point in m, then thereexists exactly ___ line in that plane that is perpendicular tom at T.onemT