Angles • § 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
Vocabulary Angles What You'll Learn You will learn to name and identify parts of an angle. 1) Opposite Rays 2) Straight Angle 3)Angle 4) Vertex 5) Sides 6) Interior 7) Exterior
Z Y XY and XZ are ____________. X Angles Opposite rays ___________ are two rays that are part of a the same line and have only theirendpoints in common. opposite rays straight angle The figure formed by opposite rays is also referred to as a ____________. Straight Angle (Video)
S vertex T Angles There is another case where two rays can have a common endpoint. angle This figure is called an _____. Some parts of angles have special names. The common endpoint is called the ______, vertex and the two rays that make up the sides ofthe angle are called the sides of the angle. side R side
S vertex SRT R TRS 1 T Angles There are several ways to name this angle. 1) Use the vertex and a point from each side. or The vertex letter is always in the middle. side 2) Use the vertex only. 1 If there is only one angle at a vertex, then theangle can be named with that vertex. R side 3) Use a number.
D 2 F DEF 2 E FED E Angles Symbols: Naming Angles (Video)
C A 1 B ABC 1 B CBA BA and BC Angles 1) Name the angle in four ways. 2) Identify the vertex and sides of this angle. vertex: Point B sides:
2) What are other names for ? 3) Is there an angle that can be named ? 2 1 XWY or 1 W XWZ YWX Angles 1) Name all angles having W as their vertex. X W 1 2 Y Z No!
exterior W Y Z interior B A Angles An angle separates a plane into three parts: interior 1) the ______ exterior 2) the ______ angle itself 3) the _________ In the figure shown, point B and all other points in the blue region are in the interiorof the angle. Point A and all other points in the greenregion are in the exterior of the angle. Points Y, W, and Z are on the angle.
P G Angles Is point B in the interior of the angle, exterior of the angle, or on the angle? Exterior B 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
Vocabulary §3.2 Angle Measure What You'll Learn You will learn to measure, draw, and classify angles. 1) Degrees 2) Protractor 3)Right Angle 4) Acute Angle 5) Obtuse Angle
P 75° Q R m PQR = 75 §3.2 Angle Measure degrees In geometry, angles are measured in units called _______. The symbol for degree is °. 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.
A n° C m ABC = n and 0 < n < 180 B §3.2 Angle Measure 0 180
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. Q R S §3.2 Angle Measure protractor You can use a _________ to measure angles and sketch angles of givenmeasure.
m SRG = m GRJ = m QRG = m SRH m SRJ = m SRQ = H J G Q S R §3.2 Angle Measure 70 Find the measurement of: 180 – 150 = 30 180 45 150 – 45 = 105 150
1) Draw AB 3) Locate and draw point C at the mark labeled 135. Draw AC. C A B §3.2 Angle Measure Use a protractor to draw an angle having a measure of 135. 2) Place the center point of the protractor on A. Align the mark labeled 0 with the ray.
A A A obtuse angle 90 < m A < 180 acute angle 0 < m A < 90 right angle m A = 90 §3.2 Angle Measure Once the measure of an angle is known, the angle can be classified as oneof three types of angles. These types are defined in relation to a right angle. Angle Classification (Video)
40° 110° 90° 50° 75° 130° §3.2 Angle Measure Classify each angle as acute, obtuse, or right. Acute Obtuse Right Obtuse Acute Acute
The measure of H is 67.Solve for y. The measure of B is 138.Solve for x. H 9y + 4 5x - 7 B B = 5x – 7 and B = 138 H = 9y + 4 and H = 67 §3.2 Angle Measure Given: (What do you know?) Given: (What do you know?) 9y + 4 = 67 5x – 7 = 138 Check! Check! 9y = 63 5x = 145 9(7) + 4 = ? 5(29) -7 = ? y = 7 x = 29 63 + 4 = ? 145 -7 = ? 67 = 67 138 = 138
Is m a larger than m b ? ? ? ? 60° 60°
Vocabulary §3.3 The Angle Addition Postulate What You'll Learn You will learn to find the measure of an angle and the bisectorof an angle. NOTHING NEW!
R X 2) Draw and label a point X in the interior of the angle. Then draw SX. S T §3.3 The Angle Addition Postulate 1) Draw an acute, an obtuse, or a right angle. Label the angle RST. 45° 75° 30° 3) For each angle, find mRSX, mXST, and RST.
R X S T §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 = 75 2) Make a conjecture about the relationship between the two smaller angles and the larger angle. 45° The sum of the measures of the twosmaller angles is equal to the measureof the larger angle. The Angle Addition Postulate (Video) 75° 30°
P 1 Q A 2 R §3.3 The Angle Addition Postulate m1 + m2 = mPQR. There are two equations that can be derived using Postulate 3 – 3. m1 = mPQR –m2 These equations are true no matter where A is locatedin the interior of PQR. m2 = mPQR –m1
X 1 Y W 2 Z §3.3 The Angle Addition Postulate Find m2 if mXYZ = 86 and m1 = 22. Postulate 3 – 3. m2 + m1 = mXYZ m2 = mXYZ –m1 m2 = 86 – 22 m2 = 64
C D (5x – 6)° 2x° B A §3.3 The Angle Addition Postulate Find mABC and mCBD if mABD = 120. mABC + mCBD = mABD Postulate 3 – 3. Substitution 2x + (5x – 6) = 120 7x – 6 = 120 Combine like terms 7x = 126 Add 6 to both sides x = 18 Divide each side by 7 36 + 84 = 120 mCBD = 5x – 6 mABC = 2x mCBD = 5(18) – 6 mABC = 2(18) mCBD = 90 – 6 mABC = 36 mCBD = 84
§3.3 The Angle Addition Postulate Just as every segment has a midpoint that bisects the segment, every angle has a ___ that bisects the angle. ray angle bisector This ray is called an ____________ .
is the bisector of PQR. P 1 Q A 2 R §3.3 The Angle Addition Postulate m1 = m2
Since bisects CAN, 1 = 2. N T 2 1 A C §3.3 The Angle Addition Postulate If bisects CAN and mCAN = 130, find 1 and 2. 1 + 2 = CAN Postulate 3 - 3 Replace CAN with 130 1 + 2 = 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
A B D C Adjacent Angles and Linear Pairs of Angles What You'll Learn You will learn to identify and use adjacent angles and linear pairs of angles. When you “split” an angle, you create two angles. The two angles are called _____________ adjacent angles 2 1 adjacent = 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. ____
J 2 common side R M 1 1 and 2 are adjacent with the same vertex R and N Adjacent Angles and Linear Pairs of Angles Adjacent angles are angles that: A) share a common side B) have the same vertex, and C) have no interior points in common
B 2 1 1 2 G N L 1 J 2 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 Yes. They have the same vertex G and a common side with no interior points in common. No. They do not have a common vertex or ____________ a common side The side of 1 is ____ The side of 2 is ____
1 2 1 2 Z D X Adjacent Angles and Linear Pairs of Angles Determine whether 1 and 2 are adjacent angles. No. Yes. In this example, the noncommon sides of the adjacent angles form a ___________. straight line linear pair These angles are called a _________
D A B 2 1 C Note: Adjacent Angles and Linear Pairs of Angles Two angles form a linear pair if and only if (iff): A) they are adjacent and B) their noncommon sides are opposite rays 1 and 2 are a linear pair.
In the figure, and are opposite rays. H T E 3 A 4 2 1 C ACE and 1 have a common side , the same vertex C, and opposite rays and M Adjacent Angles and Linear Pairs of Angles 1) Name the angle that forms a linear pair with 1. ACE 2) Do 3 and TCM form a linear pair? Justify your answer. No. Their noncommon sides are not opposite rays.
§3.5 Complementary and Supplementary Angles What You'll Learn You will learn to identify and use Complementary and Supplementary angles
E D A 60° 30° F B C §3.5 Complementary and Supplementary Angles Two angles are complementary if and only if (iff) the sum of their degree measure is 90. mABC + mDEF = 30 + 60 = 90
E D A 60° 30° F B C §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. Complementary angles DO NOT need to have a common side or even the same vertex.
I 75° 15° H P Q 40° 50° H S U V 60° T 30° Z W §3.5 Complementary and Supplementary Angles Some examples of complementary angles are shown below. mH + mI = 90 mPHQ + mQHS = 90 mTZU + mVZW = 90
D C 130° 50° E B F A §3.5 Complementary and 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. mABC + mDEF = 50 + 130 = 180
I 75° 105° H Q 130° 50° H S P U V 60° 120° 60° Z W T §3.5 Complementary and Supplementary Angles Some examples of supplementary angles are shown below. mH + mI = 180 mPHQ + mQHS = 180 mTZU + mUZV = 180 and mTZU + mVZW = 180
§3.6 Congruent Angles What You'll Learn You will learn to identify and use congruent and vertical angles. Recall that congruent segments have the same ________. measure Congruent angles _______________ also have the same measure.
50° 50° B V §3.6 Congruent Angles Two angles are congruent iff, they have the same ______________. degree measure B V iff mB = mV
1 2 X Z §3.6 Congruent Angles arcs To show that 1 is congruent to 2, we use ____. To show that there is a second set of congruent angles, X and Z, we use double arcs. This “arc” notation states that: X Z mX = mZ
§3.6 Congruent Angles four When two lines intersect, ____ angles are formed. There are two pair of nonadjacent angles. vertical angles These pairs are called _____________. 1 4 2 3
§3.6 Congruent Angles Two angles are vertical iff they are two nonadjacent angles formed by a pair of intersecting lines. Vertical angles: 1 and 3 1 4 2 2 and 4 3