Chapte2 Intro to Proofs (Segments and Angles

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Slide 1:

Angles § 3.1 Angles § 3.4 Adjacent Angles and Linear Pairs of Angles § 3.3 The Angle Addition Postulate § 3.2 Angle Measure § 3.6 Congruent Angles § 3.5 Complementary and Supplementary Angles § 3.7 Perpendicular Lines

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Angles You will learn to name and identify parts of an angle . What You'll Learn Vocabulary 1) Opposite Rays 2) Straight Angle 3) Angle 4) Vertex 5) Sides 6) Interior 7) Exterior

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Angles ___________ are two rays that are part of a the same line and have only their endpoints in common. Opposite rays X Y Z XY and XZ are ____________. opposite rays The figure formed by opposite rays is also referred to as a ____________. straight angle Straight Angle (Video)

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Angles There is another case where two rays can have a common endpoint. R S T This figure is called an _____. angle Some parts of angles have special names. The common endpoint is called the ______, vertex vertex and the two rays that make up the sides of the angle are called the sides of the angle. side side

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Angles R S T vertex side side There are several ways to name this angle. 1) Use the vertex and a point from each side. SRT or TRS The vertex letter is always in the middle. 2) Use the vertex only. R If there is only one angle at a vertex, then the angle can be named with that vertex. 3) Use a number. 1 1

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Angles Definition of Angle An angle is a figure formed by two noncollinear rays that have a common endpoint. E D F 2 Symbols: DEF 2 E FED Naming Angles (Video)

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Angles B A 1 C 1) Name the angle in four ways. ABC 1 B CBA 2) Identify the vertex and sides of this angle. Point B BA and BC vertex: sides:

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Angles W Y X 1) Name all angles having W as their vertex. 1 2 Z 1 2 2) What are other names for ? 1 XWY or YWX 3) Is there an angle that can be named ? W No! XWZ

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Angles An angle separates a plane into three parts: 1) the ______ 2) the ______ 3) the _________ interior exterior angle itself exterior interior W Y Z A B In the figure shown, point B and all other points in the blue region are in the interior of the angle. Point 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.

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Angles B Is point B in the interior of the angle, exterior of the angle, or on the angle? Exterior G Is point G in the interior of the angle, exterior of the angle, or on the angle? On the angle Is point P in the interior of the angle, exterior of the angle, or on the angle? Interior P

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§3.2 Angle Measure You will learn to measure, draw, and classify angles. What You'll Learn Vocabulary 1) Degrees 2) Protractor 3) Right Angle 4) Acute Angle 5) Obtuse Angle

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In geometry, angles are measured in units called _______. degrees The symbol for degree is ° . Q P R 75 ° In the figure to the right, the angle is 75 degrees. In notation , there is no degree symbol with 75 because the measure of an angle is a real number with no unit of measure. m PQR = 75 §3.2 Angle Measure

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Postulate 3-1 Angles Measure Postulate For every angle, there is a unique positive number between __ and ____ called the degree measure of the angle. B A C n ° 0 180 m ABC = n and 0 < n < 180 §3.2 Angle Measure

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You can use a _________ to measure angles and sketch angles of given measure. protractor Q R S Use a protractor to measure SRQ. 1) Place the center point of the protractor on vertex R. Align the straightedge with side RS. 2) Use the scale that begins with 0 at RS. Read where the other side of the angle, RQ, crosses this scale. §3.2 Angle Measure

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J H G S Q R m SRQ = Find the measurement of: m S RJ = m SRG = m QRG = m GRJ = 180 45 150 70 180 – 150 = 30 150 – 45 = 105 m SRH §3.2 Angle Measure

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Use a protractor to draw an angle having a measure of 135. 1) Draw AB 2) Place the center point of the protractor on A. Align the mark labeled 0 with the ray. 3) Locate and draw point C at the mark labeled 135. Draw AC. C A B §3.2 Angle Measure

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Once 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 Angles A right angle m A = 90 acute angle 0 < m A < 90 A obtuse angle 90 < m A < 180 A §3.2 Angle Measure Angle Classification (Video)

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Classify each angle as acute , obtuse , or right . 110 ° 90 ° 40 ° 50 ° 130 ° 75 ° Obtuse Obtuse Acute Acute Acute Right §3.2 Angle Measure

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5x - 7 B The measure of B is 138. Solve for x. 9y + 4 H The measure of H is 67. Solve for y. B = 5x – 7 and B = 138 Given: (What do you know?) 5x – 7 = 138 5x = 145 x = 29 5(29) -7 = ? 145 -7 = ? 138 = 138 Check! H = 9y + 4 and H = 67 Given: (What do you know?) 9y + 4 = 67 9y = 63 y = 7 9(7) + 4 = ? 63 + 4 = ? 67 = 67 Check! §3.2 Angle Measure

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? ? ? Is m a larger than m b ? 60 ° 60 °

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End of Lesson

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§3.3 The Angle Addition Postulate You will learn to find the measure of an angle and the bisector of an angle. What You'll Learn Vocabulary NOTHING NEW!

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1) Draw an acute, an obtuse, or a right angle. Label the angle RST. R T S 2) Draw and label a point X in the interior of the angle. Then draw SX. X 3) For each angle, find m RSX, mXST, and RST. 30 ° 45 ° 75 ° §3.3 The Angle Addition Postulate

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R T S X 30 ° 45 ° 75 ° = m RST = 75 m XST = 30 + m RSX = 45 1) How does the sum of m RSX and m XST compare to m RST ? 2) Make a conjecture about the relationship between the two smaller angles and the larger angle. Their sum is equal to the measure of RST . The sum of the measures of the two smaller angles is equal to the measure of the larger angle. The Angle Addition Postulate (Video) §3.3 The Angle Addition Postulate

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Postulate 3-3 Angle Addition Postulate For any angle PQR, if A is in the interior of PQR, then m  PQA + m AQR = m PQR. 2 1 A R P Q m  1 + m 2 = m PQR. There are two equations that can be derived using Postulate 3 – 3. m  1 = m PQR – m  2 m 2 = m PQR – m  1 These equations are true no matter where A is located in the interior of PQR. §3.3 The Angle Addition Postulate

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2 1 Y Z X W Find m 2 if m XYZ = 86 and m 1 = 22. m 2 = m XYZ – m  1 m2 = 86 – 22 m 2 = 64 m 2 + m 1 = m XYZ Postulate 3 – 3. §3.3 The Angle Addition Postulate

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2x ° (5x – 6) ° B D C A Find m  ABC and m CBD if m ABD = 120. m  ABC + m CBD = m ABD Postulate 3 – 3. 2x + (5x – 6) = 120 Substitution 7x – 6 = 120 Combine like terms 7x = 126 x = 18 Add 6 to both sides Divide each side by 7 m ABC = 2x m ABC = 2( 18 ) m ABC = 36 m CBD = 5x – 6 m CBD = 5( 18 ) – 6 m CBD = 90 – 6 m CBD = 84 36 + 84 = 120 §3.3 The Angle Addition Postulate

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Just as every segment has a midpoint that bisects the segment, every angle has a ___ that bisects the angle. ray This ray is called an ____________ . angle bisector §3.3 The Angle Addition Postulate

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Definition of an Angle Bisector The 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. 2 1 A R P Q m  1 = m 2 is the bisector of PQR. §3.3 The Angle Addition Postulate

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If bisects CAN and mCAN = 130, find 1 and 2. Since bisects CAN, 1 = 2. 1 + 2 = CAN Postulate 3 - 3 1 + 2 = 130 Replace CAN with 130 1 + 1 = 130 Replace 2 with 1 2(1) = 130 Combine like terms (1) = 65 Divide each side by 2 Since 1 = 2, 2 = 65 1 2 A C N T §3.3 The Angle Addition Postulate

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End of Lesson

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Adjacent Angles and Linear Pairs of Angles You will learn to identify and use adjacent angles and linear pairs of angles. What You'll Learn When you “split” an angle, you create two angles. D A C B 1 2 The two angles are called _____________ adjacent angles 1 and 2 are examples of adjacent angles. They share a common ray. Name the ray that 1 and 2 have in common. ____ adjacent = next to, joining.

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Adjacent Angles and Linear Pairs of Angles Definition of Adjacent Angles Adjacent angles are angles that: M J N R 1 2 1 and 2 are adjacent with the same vertex R and common side A) share a common side B) have the same vertex, and C) have no interior points in common

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Adjacent Angles and Linear Pairs of Angles Determine whether 1 and 2 are adjacent angles. No . They have a common vertex B, but _____________ no common side 1 2 B 1 2 G Yes . They have the same vertex G and a common side with no interior points in common. N 1 2 J L No . They do not have a common vertex or ____________ a common side The side of 1 is ____ The side of 2 is ____

Slide 35:

Adjacent Angles and Linear Pairs of Angles Determine whether 1 and 2 are adjacent angles. No . 2 1 Yes . 1 2 X D Z In this example, the noncommon sides of the adjacent angles form a ___________. straight line These angles are called a _________ linear pair

Slide 36:

Adjacent Angles and Linear Pairs of Angles Definition of Linear Pairs Two angles form a linear pair if and only if (iff): 1 and 2 are a linear pair. A) they are adjacent and B) their noncommon sides are opposite rays C A D B 1 2 Note:

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Adjacent Angles and Linear Pairs of Angles In the figure, and are opposite rays. 1 2 M 4 3 E H T A C 1) Name the angle that forms a linear pair with 1. ACE ACE and 1 have a common side , the same vertex C, and opposite rays and 2) Do 3 and TCM form a linear pair? Justify your answer. No . Their noncommon sides are not opposite rays.

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End of Lesson

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§3.5 Complementary and Supplementary Angles You will learn to identify and use Complementary and Supplementary angles What You'll Learn

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Definition of Complementary Angles 30 ° A B C 60 ° D E F Two angles are complementary if and only if (iff) the sum of their degree measure is 90 . m ABC + mDEF = 30 + 60 = 90 §3.5 Complementary and Supplementary Angles

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30 ° A B C 60 ° D E F 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. Complementary angles DO NOT need to have a common side or even the same vertex. §3.5 Complementary and Supplementary Angles

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15 ° H 75 ° I Some examples of complementary angles are shown below. m H + m I = 90 m PHQ + mQHS = 90 50 ° H 40 ° Q P S 30 ° 60 ° T U V W Z m TZU + mVZW = 90 §3.5 Complementary and Supplementary Angles

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Definition of Supplementary Angles If the sum of the measure of two angles is 180, they form a special pair of angles called supplementary angles . Two angles are supplementary if and only if (iff) the sum of their degree measure is 180 . 50 ° A B C 130 ° D E F m ABC + mDEF = 50 + 130 = 180 §3.5 Complementary and Supplementary Angles

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105 ° H 75 ° I Some examples of supplementary angles are shown below. m H + mI = 180 m PHQ + mQHS = 180 50 ° H 130 ° Q P S m TZU + mUZV = 180 60 ° 120 ° T U V W Z 60 ° and m TZU + mVZW = 180 §3.5 Complementary and Supplementary Angles

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End of Lesson

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§3.6 Congruent Angles You will learn to identify and use congruent and vertical angles. What You'll Learn Recall that congruent segments have the same ________. measure _______________ also have the same measure. Congruent angles

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Definition of Congruent Angles Two angles are congruent iff, they have the same ______________. degree measure 50 ° B 50 ° V  B   V iff m  B = m  V §3.6 Congruent Angles

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1 2 To show that 1 is congruent to 2, we use ____. arcs Z X To show that there is a second set of congruent angles, X and Z, we use double arcs.  X   Z mX = mZ This “arc” notation states that: §3.6 Congruent Angles

Slide 49:

When two lines intersect, ____ angles are formed. four 1 2 3 4 There are two pair of nonadjacent angles. These pairs are called _____________. vertical angles §3.6 Congruent Angles

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Definition of Vertical Angles Two angles are vertical iff they are two nonadjacent angles formed by a pair of intersecting lines. 1 2 3 4 Vertical angles:  1 and  3  2 and  4 §3.6 Congruent Angles

Slide 51:

Hands-On 1) On a sheet of paper, construct two intersecting lines that are not perpendicular. 2) With a protractor, measure each angle formed. 1 2 3 4 3) Make a conjecture about vertical angles. Consider: A. 1 is supplementary to 4. m  1 + m4 = 180 B. 3 is supplementary to 4. m  3 + m4 = 180 Therefore, it can be shown that  1   3 Likewise, it can be shown that 2   4 §3.6 Congruent Angles

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1) If m 1 = 4x + 3 and the m3 = 2x + 11, then find the m3 1 2 3 4 2) If m 2 = x + 9 and the m3 = 2x + 3, then find the m4 3) If m 2 = 6x - 1 and the m4 = 4x + 17, then find the m3 4) If m 1 = 9x - 7 and the m3 = 6x + 23, then find the m4 x = 4; 3 = 19 ° x = 56; 4 = 65° x = 9; 3 = 127° x = 10; 4 = 97° §3.6 Congruent Angles

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Theorem 3-1 Vertical Angle Theorem Vertical angles are congruent. 1 4 3 2 m n 1  3 2  4 §3.6 Congruent Angles

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Find the value of x in the figure: The angles are vertical angles. So, the value of x is 130 °. 130 ° x° §3.6 Congruent Angles

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Find the value of x in the figure: The angles are vertical angles. (x – 10) = 125. (x – 10) ° 125° x – 10 = 125. x = 135. §3.6 Congruent Angles

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Suppose two angles are congruent. What do you think is true about their complements ? 1  2 1 + x = 90 2 + y = 90 x = 90 - 1 y = 90 - 2 x = y x = 90 - 1 y = 90 - 1 Because 1  2, a “substitution” is made. x is the complement of  1 y is the complement of  2 If two angles are congruent, their complements are congruent. x  y §3.6 Congruent Angles

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Theorem 3-2 If two angles are congruent, then their complements are _________. The measure of angles complementary to A and B is 30. A B 60° 60° A  B Theorem 3-3 If two angles are congruent, then their supplements are _________. The measure of angles supplementary to 1 and 4 is 110. 70 ° 70 ° 4 3 2 1 110 ° 110 ° 4  1 congruent congruent §3.6 Congruent Angles

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Theorem 3-4 If two angles are complementary to the same angle, then they are _________. 3 is complementary to 4 3 5 Theorem 3-5 If two angles are supplementary to the same angle, then they are _________. congruent congruent 4 5 is complementary to 4 5  3 3 1 2 1 is supplementary to 2 3 is supplementary to 2 1  3 §3.6 Congruent Angles

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Suppose A  B and m A = 52. Find the measure of an angle that is supplementary to B. A 52° B 52° 1 B + 1 = 180 1 = 180 – B 1 = 180 – 52 1 = 128 ° §3.6 Congruent Angles

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If 1 is complementary to 3, 2 is complementary to 3, and m 3 = 25, What are m 1 and m 2 ? m 1 + m 3 = 90 Definition of complementary angles. m 1 = 90 - m 3 Subtract m3 from both sides. m 1 = 90 - 25 Substitute 25 in for m3. m 1 = 6 5 Simplify the right side. m 2 + m 3 = 90 Definition of complementary angles. m 2 = 90 - m 3 Subtract m3 from both sides. m 2 = 90 - 25 Substitute 25 in for m3. m 2 = 6 5 Simplify the right side. You solve for m 2 §3.6 Congruent Angles

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1) If m 1 = 2x + 3 and the m3 = 3x - 14, then find the m3 2) If m ABD = 4x + 5 and the mDBC = 2x + 1, then find the mEBC 3) If m 1 = 4x - 13 and the m3 = 2x + 19, then find the m4 4) If m EBG = 7x + 11 and the mEBH = 2x + 7, then find the m1 x = 17; 3 = 37 ° x = 29; EBC = 121° x = 16; 4 = 39° x = 18; 1 = 43° A B C D E G H 1 2 3 4 §3.6 Congruent Angles

Slide 62:

Suppose you draw two angles that are congruent and supplementary. What is true about the angles?

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Theorem 3-6 If two angles are congruent and supplementary then each is a __________. 1 is supplementary to 2 1 2 Theorem 3-7 All right angles are _________. right angle congruent 1 and 2 = 90 C B A A  B  C §3.6 Congruent Angles

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A D C B E 1 2 3 4 If 1 is supplementary to 4, 3 is supplementary to 4, and m 1 = 64, what are m 3 and m 4? 1  3 They are vertical angles. m 1 = m 3 m 3 = 64 3 is supplementary to 4 m 3 + m 4 = 180 Definition of supplementary. 64 + m 4 = 180 m 4 = 180 – 64 m 4 = 116 Given §3.6 Congruent Angles

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End of Lesson

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§3.7 Perpendicular Lines You will learn to identify, use properties of, and construct perpendicular lines and segments . What You'll Learn

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§3.7 Perpendicular Lines Lines that intersect at an angle of 90 degrees are _________________. perpendicular lines In the figure below, lines are perpendicular. A D C B 1 2 3 4

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§3.7 Perpendicular Lines Definition of Perpendicular Lines Perpendicular lines are lines that intersect to form a right angle. m n

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1 3 4 2 §3.7 Perpendicular Lines m l In the figure below, l  m . The following statements are true. 1) 1 is a right angle. 2) 1  3. 3) 1 and 4 form a linear pair. 4) 1 and 4 are supplementary. 5) 4 is a right angle. 6) 2 is a right angle. Definition of Perpendicular Lines Vertical angles are congruent Definition of Linear Pair Linear pairs are supplementary m 4 + 90 = 180, m4 = 90 Vertical angles are congruent

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§3.7 Perpendicular Lines Theorem 3-8 1 3 4 2 a b If two lines are perpendicular, then they form four right angles.

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§3.7 Perpendicular Lines 1) PRN is an acute angle. False. 2) 4  8 True

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§3.7 Perpendicular Lines Theorem 3-9 If a line m is in a plane and T is a point in m , then there exists exactly ___ line in that plane that is perpendicular to m at T. one m T

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End of Lesson