Angles
R S T vertex side There are several ways to name this angle. 1) Use the vertex and a point from each side. SRTorTRS 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
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
Classify each angle as acute, obtuse, or right. 110° 90° 40° 50° 130° 75° Obtuse Obtuse Acute Acute Acute Right
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.
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
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 = = 90
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.
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
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 ° A B C 130° D E F m ABC + m DEF = = 180
105° H 75° I Some examples of supplementary angles are shown below. m H + m I = 180 m PHQ + m QHS = ° H 130° Q P S m TZU + m UZV = ° 120° T U V W Z 60° and m TZU + m VZW = 180
Recall that congruent segments have the same ________. measure _______________ also have the same measure. Congruent angles
Definition of Congruent Angles Two angles are congruent iff, they have the same ______________. degree measure 50° B V B V iff m B = m V
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 ZX Z m X = m Z This “arc” notation states that:
When two lines intersect, ____ angles are formed. four There are two pair of nonadjacent angles. These pairs are called _____________. vertical angles
Definition of Vertical Angles Two angles are vertical iff they are two nonadjacent angles formed by a pair of intersecting lines Vertical angles: 1 and 3 2 and 4
Theorem 3-1 Vertical Angle Theorem Vertical angles are congruent m n 1 3 2 4
Find the value of x in the figure: The angles are vertical angles. So, the value of x is 130°. 130° x°
Find the value of x in the figure: The angles are vertical angles. (x – 10) = 125. (x – 10)° 125° x – 10 = 125. x = 135.
Suppose A B and m A = 52. Find the measure of an angle that is supplementary to B. A 52° B 1 B + 1 = 180 1 = 180 – B 1 = 180 – 52 1 = 128°
1) If m 1 = 2x + 3 and the m 2 = 3x + 2, 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
Adjacent, Vertical, Supplementary, and Complementary Angles
Adjacent angles are “side by side” and share a common ray. 45º 15º
These are examples of adjacent angles. 55º 35º 50º130º 80º 45º 85º 20º
These angles are NOT adjacent. 45º55º 50º 100º 35º
When 2 lines intersect, they make vertical angles. 75º 105º
Vertical angles are opposite one another. 75º 105º
Vertical angles are opposite one another. 75º 105º
Vertical angles are congruent (equal). 30º150º 30º
Supplementary angles add up to 180º. 60º120º 40º 140º Adjacent and Supplementary Angles Supplementary Angles but not Adjacent
Complementary angles add up to 90º. 60º 30º 40º 50º Adjacent and Complementary Angles Complementary Angles but not Adjacent
Practice Time!
Directions: Identify each pair of angles as vertical, supplementary, complementary, or none of the above.
#1 60º 120º
#1 60º 120º Supplementary Angles
#2 60º 30º
#2 60º 30º Complementary Angles
#3 75º
#3 75º Vertical Angles
#4 60º 40º
#4 60º 40º None of the above
#5 60º
#5 60º Vertical Angles
#6 45º135º
#6 45º135º Supplementary Angles
#7 65º 25º
#7 65º 25º Complementary Angles
#8 50º 90º
#8 50º 90º None of the above
Directions: Determine the missing angle.
#1 45º?º?º
#1 45º135º
#2 65º ?º?º
#2 65º 25º
#3 35º ?º?º
#3 35º
#4 50º ?º?º
#4 50º 130º
#5 140 º ?º?º
#5 140º
#6 40º ?º?º
#6 40º 50º
Angle Relationships & Parallel Lines Pre-Algebra
Adjacent angles are “side by side” and share a common ray. 45º 15º
These are examples of adjacent angles. 55º 35º 50º130º 80º 45º 85º 20º
These angles are NOT adjacent. 45º55º 50º 100º 35º
Complementary Angles sum to 90° 40° 50°
Complementary angles add up to 90º. 60º 30º 40º 50º Adjacent and Complementary Angles Complementary Angles but not Adjacent
Supplementary Angles sum to 180° 30° 150°
Supplementary angles add up to 180º. 60º120º 40º 140º Adjacent and Supplementary Angles Supplementary Angles but not Adjacent
Vertical Angles are opposite one another. Vertical angles are congruent. 100°
Vertical Angles are opposite one another. Vertical angles are congruent. 80°
Lines l and m are parallel. l || m 120° l m Note the 4 angles that measure 120°. n Line n is a transversal.
Lines l and m are parallel. l || m 60° l m Note the 4 angles that measure 60°. n Line n is a transversal.
Lines l and m are parallel. l || m 60° l m There are many pairs of angles that are supplementary. There are 4 pairs of angles that are vertical. 120° n Line n is a transversal.
If two lines are intersected by a transversal and any of the angle pairs shown below are congruent, then the lines are parallel. This fact is used in the construction of parallel lines.
Practice Time!
1) Find the missing angle. 36° ?°?°
1) Find the missing angle. 36° ?°?° 90 ° – 36 = 54°
2) Find the missing angle. 64° ?°?°
2) Find the missing angle. 64° ?°?° 90 ° – 64° = 26°
3) Solve for x. 3x° 2x°
3) Solve for x. 3x° 2x° 3x° + 2x° = 90° 5x = 90 x =18
4) Solve for x. 2x + 5 x + 25
4) Solve for x. 2x + 5 x + 25 (2x + 5) + (x + 25) = 90 3x + 30 = 90 3x = 60 x = 20
5) Find the missing angle. ?°?° 168°
5) Find the missing angle. ?°?° 168° 180° – 168° = 12°
6) Find the missing angle. 58° ?°?°
6) Find the missing angle. 58° ?°?° 180° – 58° = 122°
7) Solve for x. 4x 5x
7) Solve for x. 4x 5x 4x + 5x = 180 9x = 180 x = 20
8) Solve for x. 2x x + 20
8) Solve for x. 2x x + 20 (2x + 10) + (3x + 20) = 180 5x + 30 = 180 5x = 150 x = 30
9) Lines l and m are parallel. l || m Find the missing angles. 42° l m b°b° d°d° f°f° a ° c°c° e°e° g°g°
9) Lines l and m are parallel. l || m Find the missing angles. 42° l m 138°
10) Lines l and m are parallel. l || m Find the missing angles. 81° l m b°b° d°d° f°f° a ° c°c° e°e° g°g°
10) Lines l and m are parallel. l || m Find the missing angles. 81° l m 99°
11) Find the missing angles. 70 ° b° 70 ° d °65 ° Hint: The 3 angles in a triangle sum to 180°.
11) Find the missing angles. 70 ° 40° 70 ° 75 °65 ° Hint: The 3 angles in a triangle sum to 180°.
12) Find the missing angles. 45 ° b° 50 ° d °75 ° Hint: The 3 angles in a triangle sum to 180°.
12) Find the missing angles. 45 ° 85° 50 ° 20°75 ° Hint: The 3 angles in a triangle sum to 180°.
In the figure a || b. 13. Name the angles congruent to Name all the angles supplementary to If m 1 = 105° what is m 3? 16. If m 5 = 120° what is m 2? 1, 5, 7 1, 3, 5, 7 105° 60°