Angles and types of angles, interior and exterior angles

1. ANGLES

2. After going through this module,
you are expected to:
classify the different kinds of angles; and
derive relationships of geometric figures
using measurements and by inductive
reasoning; supplementary angles,
complementary angles, congruent angles,
vertical angles, adjacent angles, linear
pairs, perpendicular lines, and parallel lines

3. Recall:

4. Activity: Name Me!
Given the figure at the left, name four(4) different
rays you can create from it.

5. What’s New
 The measure of an angle indicates how wide the
opening is between its two sides.
 A protractor is used to find the measure of an angle,
just like a ruler is used to find the length of a segment.

6. What’s New
 To find the measure of an angle using a protractor,
☺ place the center of the protractor over the vertex of
an angle.
☺then align the mark labeled 0 with one side of the angle
and read the scale where the other side of the angle falls.
☺The unit of measurement used for angles is called degree,
denoted by the symbol °.

7. What’s New
 A protractor usually has outer and inner degree scales.
You may use any of these scales depending on the
positions of the angle.
 Using the inner scale of the protractor shown in the
figure below, the measure of is equal to 50
∠𝐺𝐸𝑂
degrees, written as:
𝑚∠𝐺𝐸𝑂 = 50°

8. What’s New
 Now, let us try to draw so that = 80° .
∠𝐿𝐴𝐵 𝑚∠𝐿𝐴𝐵
Use a protractor.
Solution:
o Draw AB
Place the center of protractor
at A with AB coinciding with the
line where the 0° mark is.
o Mark L at 80°
o Draw AL

9. Basic Concepts on Angles
o The two rays are called the sides of the angle.
o The common endpoint is called the vertex.
o The symbol for an angle is .
∠

10. Basic Concepts on Angles
o An angle can be named in three ways, that is, by
using
(1) the number assigned to the angle,
(2) its vertex, or
(3) its vertex and two other points, one from each side of the
angle. The vertex of the angle below is B and its sides are BA
and BC.

11. Basic Concepts on Angles
o The angle below can be name as angle 1 ( 1),
∠
angle B ( ), angle ABC ( ), or angle
∠ ∠
𝐵 𝐴𝐵𝐶
CBA ( ).
∠𝐶𝐵𝐴

12. Basic Concepts on Angles
o The angles are (a) or , (b) ,
∠ ∠ ∠
𝑀𝐴𝑃 𝑃𝐴𝑀 𝑋
and (c) 1.
∠

13. Kinds of Angles
o Angles can be classified according to their
measures. They can be a right angle, an acute
angle, or an obtuse angle.

14. Kinds of Angles
o A right angle is an angle that measures exactly
90°.

15. Kinds of Angles
o An acute angle is an angle that measures less
than 90°.

16. Kinds of Angles
o An obtuse angle is an angle that measures more
than 90° but less than 180°.

17. Kinds of Angles
o You can approximate the measures of angles by
estimating their openings if equal to 90 ° , less
than 90 ° , or greater than 90 ° . These
approximations of angle measures without using
protractors will help you in classifying angles
whether they are right, acute, or obtuse.

18. Example 1
Estimate the measure of each angle. Classify the
angle.

19. Solution
o You can use approximations to determine if the
measure of each angle is less than 90 ° , equal to 90 ° ,
or greater than 90 °.
o a. < 90° ; acute angle
𝑚∠𝐺𝑂𝑇
o b. = 90° ; right angle
𝑚∠𝑀
o c. > 90° ; obtuse angle
𝑚∠𝑆𝐸𝑇

20. Example 2
Refer to the given figure. Measure and classify
each indicated angle.
a. ∠𝑃𝑄𝑅 b. ∠𝑃𝑄𝑇 c. ∠𝑅𝑄𝑆

21. Solution
o a. = 90°; right angle
𝑚∠𝑃𝑄𝑅
o b. = 30° ; acute angle
𝑚∠𝑃𝑄𝑇
o c. = 100° ; obtuse angle
𝑚∠𝑅𝑄𝑆

22. o Refer again to the figure in example 2. Use your
protractor to measure . Angle UQS measures 30°.
∠𝑈𝑄𝑆
Thus, = = 30. Angles PQT and UQS
𝑚∠𝑃𝑄𝑇 𝑚∠𝑈𝑄𝑆
are congruent angles, denoted by ,
∠𝑃𝑄𝑇 ≅ ∠𝑈𝑄𝑆
which is read as “angle PQT is congruent to angle UQS.”

23. ACTIVITY

24. Congruent Angles
o Congruent angles are angles that have the
same measure.
o An angle bisector divides an angle into two
congruent angles.
o Every angle has exactly one angle bisector.

25. Congruent Angles
o Use identical markings to show congruent angles in a
figure.
o If BM is the angle bisector of then
∠𝐴𝐵𝐶 ∠𝐴𝐵𝑀 ≅
. Also, if then BM divides
∠𝑀𝐵𝐶 ∠𝐴𝐵𝑀 ≅ ∠𝑀𝐵𝐶
into two congruent angles. Thus, BM is the angle
∠𝐴𝐵𝐶
bisector of .
∠𝐴𝐵𝐶

26. Congruent Angles
o On the figure below, the identical marks means that
∠𝐴𝐵𝑀 ≅ ∠𝑀𝐵𝐶, hence, BM bisects .
∠𝐴𝐵𝐶

27. Activity
o Name each angle bisector and all congruent angles in
the figure.

28. o Solution
o 1. Since = = 45° , therefore,
𝑚∠𝐶𝐼𝐴 𝑚∠𝐴𝐼𝑁 ∠𝐶𝐼𝐴 ≅
. Hence, IA is an angle bisector of .
∠𝐴𝐼𝑁 ∠𝐶𝐼𝑁

29. o Solution
o 2. Ray AY is not an angle bisector of since
∠𝐾𝐴𝐸
≠ .
𝑚∠𝐾𝐴𝑌 𝑚∠𝑌𝐴𝐸

30. o Solution
o 3. Angle AGN is congruent to angle LEG. Rays GN and GE
are not angle bisectors of since these two
∠𝐴𝐺𝐿
different rays do not divide into two angles.
∠𝐴𝐺𝐿

31. Angle Pairs
o Geometric relationships exist between two angles.
These relationships can be used as bases for
classifying angle pairs.
o Angle pairs include adjacent angles,
complementary angles, supplementary angles,
linear pairs, and vertical angles.

32. Adjacent Angles
o Look at and . What do they have in
∠ ∠
𝐴𝐵𝐷 𝐷𝐵𝐶
common?

33. Adjacent Angles
o Angles ABD and DBC are adjacent angles with B as
their common vertex and ray BD as their common
side.

34. Adjacent Angles
o Adjacent angles are angles on the same plane
that have a common side and a common vertex
but no common interior points.

35. Complementary Angles
o Look at and . What is the sum of their
∠𝐹𝐸𝐺 ∠𝐺𝐸𝐷
measures? Compare this sum to that of and
∠𝑇𝑈𝑉
.
∠𝑋𝑌𝑍
o What can you say about the sum of two pairs of angles?

36. Complementary Angles
o On the figures below, Angles FEG and GED in (a) are
adjacent complementary angles.
o Angles TUV and XYZ in (b) are nonadjacent
complementary angles.

37. Complementary Angles
o Since + = 60° + 30° = 90° , then
∠ ∠
and are complementary angles.
∠ ∠

38. Complementary Angles
o Complementary angles are two angles whose
measures have a sum of 90°.

39. Complementary Angles
o Angle GED is the complement of .
∠𝐹𝐸𝐺
o Also, is the complement of . Angles GED
∠𝐹𝐸𝐺 ∠𝐺𝐸𝐷
and FEG have a common side EG, that is, they are also
adjacent angles.
o Therefore, and are adjacent
∠𝐺𝐸𝐷 ∠𝐹𝐸𝐺
complementary angles.

40. Complementary Angles
o On the other hand, in figure (b),
𝑚∠𝑇𝑈𝑉 + = 45° + 45° = 90°
𝑚∠𝑋𝑌𝑍
o Hence, and are complementary.
∠𝑇𝑈𝑉 ∠𝑋𝑌𝑍
o They have no common side so they are non-adjacent
complementary angles.

41. Complementary Angles
o Example 1
Find the measure of the complement of an angle
whose measure is 55 ° .
o Solution
Let = the measure of the complement of an angle.
𝑥
Hence,
𝑥 + 55 ° = 90 °
𝑥 = 90 ° − 55 °
𝑥 = 35 °

42. Complementary Angles
o Example 1
Check your answer.
35° + 55° = 90° True
Hence, the measure of the complement of a 55 °
angle is a 35 ° angle.

43. Complementary Angles
o Example 2
Find the measures of the complementary angle of a
12 ° angle less than twice the other.
o Solution
Let = the measure of the first angle,
𝑥
2 − 12 = the measure of the second angle.
𝑥

44. Complementary Angles
o Example 2
Since the two angles are complementary
𝑥 + (2 − 12) = 90
𝑥
3 − 12 = 90
𝑥 Combine similar terms.
3 = 90 + 12
𝑥
3 = 102
𝑥
3 = 102
𝑥 Divide both sides by 3
3 3
𝑥 = 34

45. Complementary Angles
o Example 2
It follows that the second angle measures
2 − 12
𝑥
2(34) − 12
68 − 12 = 56
Thus, the two complementary angles measure 34° and 56°,
respectively.

46. Activity
o 1. Find the measure of the complement of an angle
whose measure is 25 ° .
o 2. Find the measures of the complementary angle of a
10 ° angle less than twice the other.

59. Supplementary Angles
o Look at the following angles pairs: and ,
∠𝑀𝑁𝑂 ∠𝑂𝑁𝑃
and and . What is the sum of the measures
∠𝐴𝐵𝐶 ∠𝑅𝑆𝑇
of each angle pair?

60. Supplementary Angles
o In Figure(a), Angles MNO and ONP are adjacent
supplementary.

61. Supplementary Angles
o While in (b), Angles ABC and RST are nonadjacent
supplementary angles. The sum of the measures of
and is + = 135° +
∠𝑀𝑁𝑂 ∠𝑂𝑁𝑃 𝑚∠𝑀𝑁𝑂 𝑚∠𝑂𝑁𝑃
45° = 180 ° . Hence, and are
∠𝑀𝑁𝑂 ∠𝑂𝑁𝑃
supplementary angles. These two angles are supplements
of each other.

62. Supplementary Angles
o Notice also that these angles are adjacent (since they
have a common side NO) and, at the same time,
supplementary. On the other hand, in figure (b),
𝑚∠𝐴𝐵𝐶 + = 60 + 120 = 180
𝑚∠𝑅𝑆𝑇

63. Supplementary Angles
o Hence, and are also supplementary.
∠𝐴𝐵𝐶 ∠𝑅𝑆𝑇
Notice also that these two angles are supplementary but
not adjacent.

64. Supplementary Angles
o Supplementary angles are two angles whose measures
have a sum of 180 °

65. Supplementary Angles
o Example
The measure of the supplement of an angle is 25 °
more than 4 times the measure of the angle. Find the
measure of each angle.
Solution
Let = the measure of an angle
𝑥
180 − = the measure of its supplement.
𝑥

66. Supplementary Angles
o Example
Then solve for in the equation 180 − = 4 + 25.
𝑥 𝑥 𝑥
180 − = 4 + 25
𝑥 𝑥
− −
𝑥 4 = 25 − 180 Subtract 4 and 180 from both
𝑥 𝑥
sides. −5 = −155
𝑥 Divide both sides
by -5
−5 −5
𝑥 = 31

67. Supplementary Angles
o Example
Then substitute 31 for in 180 − to get the measure of
𝑥 𝑥
the supplement of the angle.
180 − = 180 − 31 = 149
𝑥
Check your answer.
31 + 149 = 180 True
Therefore, the angle measures 31° and its supplement
measures 149°
.

68. Linear Pairs
o Look at and . How will you classify this
∠𝑉𝑋𝑌 ∠𝑋𝑌𝑍
angle pair?

69. Linear Pairs
o In the figure below, Angles VXY and YXZ are adjacent
and supplementary angles with a common side XY.
o You call and a linear pair.
∠𝑉𝑋𝑌 ∠𝑌𝑋𝑍

70. Linear Pairs
o A linear pair consists of two adjacent angles whose
noncommon sides are opposite rays.

71. Linear Pairs
o Example
Find the measure of each angle in the linear pair.

72. Linear Pairs
o Solution
o Since and form a linear pair, they are
∠𝑃𝐴𝑀 ∠𝑀𝐴𝑅
supplementary. Thus,
(3 − 15) + ( + 5) = 180
𝑥 𝑥
4 − 10 = 180 Combine similar terms.
𝑥
4 = 180 + 10
𝑥 Add 10 to both sides. 4
4 = 190
𝑥
4 = 190
𝑥 Divide both sides by 4.
4 4
𝑥 =47.5

73. Linear Pairs
o Solution
o Substituting 47.4 for , the two angles measure
𝑥
𝑥 + 5 = 47.5 + 5 and 3 − 15 = 3(47.5) − 15
𝑥
𝑥 = 52.5 =
𝑥
127.5

74. Vertical Angles
o When two lines intersect, they form a pair of nonadjacent
angles called vertical angles
o Vertical angles are two nonadjacent angles formed by
two intersecting lines.
o Vertical angles are congruent.

75. Vertical Angles

76. Vertical Angles

77. Perpendicular Lines
o When two lines intersect to form right angles, the two lines
are said to be perpendicular to each other

78. Perpendicular Lines
o Since = 90° , therefore, = 90° . In the
𝑚∠𝐴𝑂𝐶 𝑚∠𝐵𝑂𝐶
same manner, = 90° and = 90° . Thus,
𝑚∠𝐴𝑂𝐷 𝑚∠𝐵𝑂𝐷
perpendicular lines form four right angles.

79. Parallel Lines
o Parallel lines are lines that do not intersect no matter how
far they are extended. Lines m and n do not intersect,
and are parallel. You write this as . The symbol is
𝑚 ∥ 𝑛 ∥
read as “is parallel to”.

80. QUIZ

81. QUIZ