Angle Relationships Terms l CONGRUENT ANGLES ANGLES Two

Angle Relationships

Terms l CONGRUENT ANGLES: ANGLES Two angles are congruent angles if and only if they have the same measure. 40 o

Terms § A Pair of Complementary Angles – 2 angles whose measures have the sum of 90 o § A Pair of Supplementary Angles – 2 angles whose measures have the sum of 180 o Are complementary (/supplementary angles) adjacent angles? l ADJACENT ANGLES: ANGLES Angles that share a vertex and a side and whose interiors do not overlap B D Side Vertex A C Side

Terms § A Pair of Vertical Angles – angles formed by 2 intersecting lines; they share a common vertex but not a common side – If AB and CD intersect at point P so that point P is between points A and B and also between points C and D, then APC and BPD are a pair of vertical angles. APD and BPC are also a pair of vertical angles. D A P C B

Terms § A Linear Pair of Angles -– adjacent angles whose noncommon sides are opposite rays -– If X, Y, Z are consecutive collinear points and W is a point not on XZ, then XYW and WYZ form a linear pair of angles. v. Are supplementary angles a linear pair of angles? W X Y Z v. How many linear pair of angles are formed when 2 lines intersect?

Theorems l Linear Pair Thm - If two angles form a linear pair, then they are supplementary. l Vertical Angles Thm - If two angles are vertical angles, then they are congruent.

a=c=68; b=112 a=127 a=c=35; b=40; d=70 a=b=90; c=42; d=48; e=132 a=c=20; b=d=70; e=110 a=70; b=55; c=25

Special Angles formed by a Transversal

Perpendicular Lines ( ) – two lines that intersect to form a right angle Parallel Lines (//) – 2 or more lines that are coplanar and that do not intersect Skew Lines – lines that are not coplanar and that do not intersect Why is ‘coplanar’ not in the definition of lines? Parallel Lines Perpendicular Lines

l TRANSVERSAL - A line that intersects 2 or more coplanar lines at different points l Which of the following has a transversal? l 2 l 1 l 3 l 2 l 3 l 1

DEFINITION s 3, 4, 5 & 6 are INTERIOR ANGLES 5 6 l s 1, 2, 7 & 8 are 7 8 EXTERIOR ANGLES ALTERNATE interior angles (AIA) 1 2 3 4 l l - 2 non-adjacent interior angles on opposite sides of the transversal l Example: s 3 & 6 and s 4 & 5

DEFINITION s 3, 4, 5 & 6 are INTERIOR ANGLES 5 6 l s 1, 2, 7 & 8 are 7 8 EXTERIOR ANGLES ALTERNATE exterior angles (AEA) 1 2 3 4 l l - 2 non-adjacent exterior angles on opposite sides of the transversal l Example: s 1 & 8 and s 2 & 7

DEFINITION 1 2 3 4 5 6 7 8 l l CONSECUTIVE interior angles (CIA) - 2 interior angles on the same side of the transversal Example: s 3 & 5 and s 4 & 6

DEFINITION 1 2 3 4 5 6 7 8 l l CORRESPONDING angles (CA) - 2 non-adjacent angles on the same side of the transversal such that one is an exterior angle and the other is an interior angle Example: s 1 & 5, s 3 & 7, s 2 & 6, s 4 & 8

Exercise 1 5 9 2 6 3 Identify each angle pair as AIA, AEA, CIA or none of these. 4 7 8 10 11 12 13 14 15 16 *Identify the transversal. a. 13 & 5 CA b. 12 & 7 AIA c. 10 & 7 none d. 3 & 1 CA e. 3 & 16 AEA f. 13 & 4 none g. 10 & 1 none h. 10 & 11 CIA

Parallel Line Properties

PROPERTIES of // Lines l 1 2 3 4 5 7 6 8 l What if our transversal is intersecting 2 // lines? What relationships can we observe between: • CA? • AIA? • AEA? • CIA? congruent supplementary

Parallel Line Theorems ÌF TWO LINES ARE PARALLEL… l CA Theorem - …then CORRESPONDING ANGLES are CONGRUENT. l AIA Theorem - …. then ALTERNATE INTERIOR ANGLES are CONGRUENT. l AEA Theorem - …then ALTERNATE EXTERIOR ANGLES are CONGRUENT. l CIA Theorem - …then CONSECUTIVE INTERIOR ANGLES are SUPPLEMENTARY. * Prove algebraically.

Practice (Source: DG by Serra) 2. a=b=c=54 b=d=65; a=c=115

Practice (Source: DG by Serra) 4. 3. a=72; b=126

Practice 5. 5 x + 2 = 182 – 4 x 9 x = 180 x = 20 182 – 4 x 102 = 4 y + 2 y=25

Practice 6. 7.

Practice 8.

Practice What’s wrong with this picture? Explain. 9.

Practice 10. m=125 11. m=38

HOMEWORK a=102; b=78; c=f=58; d=122; e=26

Parallel Line Properties (Part II)

PROPERTIES of // Lines l Is the converse of the // Line Thm true? If 2 lines are cut by a transversal to form pairs of congruent CA, congruent AIA, and congruent AEA, then the lines are parallel.

// by the Converse of AIA Thm Not // (CIAs are not supplementary) What is b so that the 2 lines are parallel? 4 x – 12 3 x + 2 b 4 x – 12 = 3 x + 2 x = 14 4 x – 12 44 b=136 Not //

Practice

Practice Determine which lines are parallel.