1-4 Angle Measure SWBAT measure and classify angles, identify and use congruent angles and the bisector of an angle. G.4 A ray is a part of a line. It has one endpoint and extends indefinitely in one direction. A B AB If you choose a point on a line, that point determines exactly two rays call opposite rays. l AB C BA and BC are opposite rays

An angle is formed by two noncollinear rays that have a common endpoint. The rays are called sides of the angle. The common endpoint is the vertex. X Y Z Side XZ Side XY Vertex X When naming angles using three letters, the vertex must be the second of the three letters. You can name an angle using a single letter only when there is exactly one angle located at that vertex. The angle shown can be named as <X, <YXZ, <ZXY or <3 3

A B D E F G A. Name all angles that have B as the vertex. B. Name the sides of <5. C. Write another name for <6.

Angles are measured in units called degrees. The degree results from dividing the distance around a circle into 360 parts. 1o= of a turn around a circle To measure an angle, you can u se a protractor. Angle DEF below is a 50 degree angle. We say that the degree measure of <DEF is 50 or m<DEF=50. D E F

Congruent Angles

Right Angle Acute Angle Obtuse Angle Exactly 90o Less than 90o Greater than 90o A B C <ABC is a straight angle. Straight angles are equal to 180o

K L M N P J a. <MJP b. <LJP c. <NJP Classify as right, acute or obutuse. Then, use a protractor to measure the angle to the nearest degree.

If point B is in the interior of  AOC, then m  AOB + m  BOC = m  AOC (Part + Part = Whole) Example 1: Use the angle addition postulate. Find m<1 if m<2 = 56 and m <JKL = J K L

Example Given: m<AOC = 125°, m  AOB = 7x + 1 and m  BOC = -2x + 39, find m  BOC.

A ray that divides an angle into two congruent angles is called an angle bisector. If YW is the angle bisector of <XYZ, then point W lies in the interior of <XYZ and <XYW ≅ <WYZ W X Y Z 80o Just as with segments, when a line, segment or ray divides an angle into smaller angles, the sum of the measures of the smaller angles equals the measure of the largest angle. So in the figure: m<XYW + m<WYZ= m<XYZ

L N M J K In the figure KJ and KM are opposite rays, and KN bisects <JKL. If m<JKN=8x-13 and m<NKL=6x+11, find m<JKN. Step 1: Solve for x Step 2: Use the value of x to find m<JKN

L N M J K Suppose m<JKL=9y+15 and m<JKN=5y+2. Find m<JKL.

1-5 Angle Relationships Mastery Objective: SWBAT identify, apply and create angle pair relationships through the use of theorems and definitions to solve for missing angle measures.

Pairs of Angles: Some pairs of angles are special because of how they are positioned in relationship to each other. Key Concept: Special Angle Pairs Adjacent Angles: 2 angles that lie in the same plane and have a common side, but no common interior points. Examples <1 and <2 are adjacent angles Nonexamples <3 and < ABC are nonadjacent angles A B C 3 B C A

A linear pair is a pair of adjacent angles with noncommon sides that are opposite rays. Example <1 and <2 Nonexample < ADB and < ADC 1 2 A B D C

Vertical Angles are two nonadjacent angles formed by two intersecting lines. Examples <1 and <2 ; <3 and <4 Nonexample <AEB and < DEC A E D B C

Key Concept: Angle Pair Relationships Verical angles are congruent. Examples < ABC ≅ < DBE and <ABD ≅ <CBE A B C E D Complementary Angles are two angles with measures that have a sum of 90. Examples < 1 and <2 are complementary < A is complementary to < B. Supplementary Angles are two angles with measures that have a sum of 180. Examples < 3 and < 4 are supplementary < P and < Q are supplementary The angles in a linear pair are supplementary Example m <1 + m <2 = o 25o A B o 120o Q P 12

Perpendicular Lines: Lines, segments, or rays that form right angles (90o) are perpendicular. Key Concept: Perpendicular Lines Perpendicular lines intersect to form four right angles Perpendicular lines intersect to form congruent adjacent angles. Segments and rays can be perpendicular to lines or other line segments and rays. The right angle symbol in the figure indicates that the lines are perpendicular. A C B D is read is perpendicular to ADCBEx.

Example 3: Perpendicular Lines Find x and y so that PR and SQ are perpendicular. PQ T R W S 2xo (5x+6)o (4y-2)o