9. OBJECTIVES:
At the end of this lesson, you will be able to:
1) Define angle pairs;
2) Derive the relationship of geometric figures using
inductive reasoning: angle pairs;
3) Solve problems involving angle pairs.
10. GEOMETRIC FIGURES
A geometric figure is any
combination of points,
lines, or planes. Geometric
figures are often classified
as space figure, plane
figure, lines, line
segments, rays, and points
depending on
the dimensions of the
figure.
11. ANGLE PAIRS
Angle pairs are angles that appear in twos to display a
certain geometrical property.
Adjacent Angles
Vertical Angles
Complementary Angles
Supplementary Angles
Linear Pair of Angles
12. To denote the measure of an angle we write an
โmโ in front of the symbol for the angle.
๐จ๐๐๐๐ ๐จ๐๐๐๐ ๐น๐๐๐๐ ๐จ๐๐๐๐
๐ถ๐๐๐๐๐ ๐จ๐๐๐๐
๐บ๐๐๐๐๐๐๐ ๐จ๐๐๐๐
Here are some common angles and their measurements.
4
4 180
m๏ ๏ฝ ๏ฐ
2 90
m๏ ๏ฝ ๏ฐ
2
1
1 45
m๏ ๏ฝ ๏ฐ
3 135
m๏ ๏ฝ ๏ฐ
3
โthe measure of angle 1
is equal to 45 degreesโ
13. Congruent Angles
โข So, two angles are congruent if and only if
they have the same measure.
โ ๐ด๐ต๐ถ โ โ XYZ if and only if ๐โ ๐ด๐ต๐ถ = ๐โ ๐๐๐.
Means
Congruent
Means Equal
โข So, the angles are congruent.
๐ด
๐
๐ธ๐ฅ๐๐๐๐๐:
๐ต
๐ถ
๐
๐
30ยฐ
30ยฐ
15. Adjacent Angles
Adjacent angles share a common point(vertex) and common
ray(side) but no interior point in common.
Adjacent angles are โside by sideโ and share a common ray.
45ยบ
๐ด
25ยบ
๐ต
๐ถ
๐ท
๐โ๐๐๐๐๐๐๐, โ ๐จ๐ฉ๐ช ๐๐๐ โ ๐ช๐ฉ๐ซ
๐๐๐ ๐จ๐ซ๐ฑ๐จ๐ช๐ฌ๐ต๐ป ๐จ๐ต๐ฎ๐ณ๐ฌ๐บ.
18. Vertical Angles
Two opposite angles formed by intersecting lines and have no
common sides but share a common vertex.
Four angles are formed at the point of
intersection.
โ ๐จ๐ท๐ช
โ ๐จ๐ท๐ฉ
โ ๐ฉ๐ท๐ซ
โ ๐ช๐ท๐ซ
โ
โ
Vertical angles are congruent.
Point of intersection โPโ is the
common vertex of the four
angle.
๐ต
๐ด
๐ถ
๐ท
๐
Common
Vertex
105ยบ
105ยบ
75ยบ
75ยบ
20. 145ยบ
35ยบ
๐บ๐๐๐๐๐๐๐:
2๐ฅ + 3 = 105
2๐ฅ = 105 โ 3
2๐ฅ = 102
2 2
๐ = ๐๐
๐๐๐ยฐ
Example :Given that the pair of angles are vertical angles. Find
for the angle measure and the value of the variable.
2) 105ยฐ
(2๐ฅ + 3)ยฐ
1)
35ยฐ
145ยฐ
?
?
Vertical Angles
2๐ฅ + 3; ๐ฅ = 51
2 51 + 3 = ๐๐๐ยฐ
21. Complementary Angles
If the sum of two angles is ๐๐ยฐ, then they are called
complementary angles.
70ยบ
20ยบ
๐
๐ ๐
๐
๐ด
60ยบ
๐ต
๐ถ
๐ธ
30ยบ
๐น
๐บ
๏ ABC and ๏EFG are
complementary angles.
๐๏๐จ๐ฉ๐ช + ๐๏๐ฌ๐ญ๐ฎ = ๐๐ยฐ
60ยฐ + 30ยฐ = 90ยฐ
Not
adjacent
angles.
๐๏๐พ๐ฟ๐ + ๐๏๐๐ฟ๐ = ๐๐ยฐ
70ยฐ + 20ยฐ = 90ยฐ
๏๐๐๐ and ๏YXZ are
complementary angles.
Adjacent
angles.
22. Example :Given that the pair of angles are complementary. Find
for the angle measure and the value of the variable.
Complement
16ยบ
๐บ๐๐๐๐๐๐๐:
3๐ฅ + 30 = 90
3๐ฅ = 90 โ 30
3๐ฅ = 60
3 3
๐ = ๐๐
2)
(3๐ฅ)ยฐ
30ยฐ
๐๐ยฐ
1)
74ยบ
?
Complementary Angles
3๐ฅ; ๐ฅ = 20
3 20 = ๐๐ยฐ
23. Supplementary Angles
If the sum of two angles is 18๐ยฐ, then they are called
complementary angles.
๐ด
120ยบ
๐ต ๐ถ
๐ธ
60ยบ
๐น ๐บ
๏ ABC and ๏EFG are
supplementary angles.
๐๏๐จ๐ฉ๐ช + ๐๏๐ฌ๐ญ๐ฎ = ๐๐๐ยฐ
120ยฐ + 60ยฐ = 180ยฐ
24. Linear Pair of Angles
Two angles are linear pair if they are adjacent and
supplementary.
55ยบ
๐ด
125ยบ
๐ ๐ถ
๐ท
๏ APC and ๏APD are
supplementary angles
and are also adjacent
angles .
๐๏๐จ๐ท๐ช + ๐๏๐จ๐ท๐ซ = ๐๐๐ยฐ
125ยฐ + 55ยฐ = 180ยฐ
Therefore, ๏ APC and
๏APD are linear pair of
angles.
25. Supplementary & Linear Pair of Angles
1)
Supplement
37ยบ
๐บ๐๐๐๐๐๐๐:
5๐ฅ + 5 + 45 = 180
5๐ฅ = 130
5๐ฅ = 180 โ 50
5 5
๐ = ๐๐
143ยบ
?
๐๐๐ยฐ
Example :Given that the pair of angles are supplementary. Find
for the angle measure and the value of the variable.
5๐ฅ + 50 = 180
2)
(5๐ฅ + 5)ยฐ
45ยฐ
5๐ฅ + 5; ๐ฅ = 26
5 26 + 5 = ๐๐๐ยฐ