Section 4 - Pairs of Angles: - Willets Geometry

Section 4 - Pairs of Angles:
1. ADJACENT ANGLES are two
angles that have a common vertex
and a common side between them.
We can think of them as angles
that are next to each other.
2. If the sum of two angles is 90°,
then the two angles are said to be
COMPLEMENTARY.
Each angle is the COMPLEMENT
of the other
3. If the sum of two angles is 180°,
then the two angles are said
to be SUPPLEMENTARY
Each angle is the SUPPLEMENT
of the other
4. If AB and AC are opposite rays and
AD is any third ray, then
BAD and DAC
are said to be a LINEAR PAIR.
B C
A
42
A
P
D
35°
ABD and DBC are
adjacent angles.
They both have vertex B and
BD is between the other two
rays.
Q
55°
Since mP mQ 90
, then the angles are
complementary
35°
145°
Since mA mB 180 , then the angles are
supplementary
B
D
B A C
AB and AC are opposite rays.
BADand DAC are a linear pair

5. If two angles form a linear pair, then they are supplementary.
6. If two angles are complementary, then both must be acute angles.
7. If two angles are congruent and complementary, then each angle contains 45°.
8. If two angles are congruent and supplementary, then each angle contains 90°
x x
9. The angle formed by the bisectors of two
complementary adjacent angles contains 45°
AE bisects CAD and AF bisects DAB
mEAF 45
10. The angle formed by the bisectors of two
supplementary adjacent angles contains 90°
AE bisects CAD and AF bisects DAB
mEAF 90
BADand DAC are supplementary
x x
x x 180
2x 180 11. The difference between the supplement and the
complement of an angle is always 90°
Angle Supplement Complement Supplement - Complement
40 140 50 140 50 90
23 157 67 157 67 90
66 114 24 114 24 90
81 99 9 99 9 90
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x 90 x x 90
C
2x 90 x 45
A
C
E
E
B
A
D
F
B
D
A
F
B
D
C

12. The non-adjacent angles formed when two straight lines intersect are called
VERTICAL ANGLES.
1
4
2
3
m
13. VERTICAL ANGLE THEOREM: If two angles
are vertical angles, then they are congruent
to each other.
n
1 3 and 2 4
When m and n intersect, four angles are formed.
1and 3 are one pair of vertical angles
2and 4 are another pair of vertical angles
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1
4
2
3
m
n

Assignment: Section 4
1. The sum of two complementary angles is ____ degrees.
2. The sum of two supplementary angles is ____ degrees.
3. Which of the following pairs of angles are complementary?
(a) 20° and 70° (b) 65° and 35° (c) 70° and 110°
4. Which of the following pairs of angles are supplementary?
(a) 30° and 60° (b) 140° and 50° (c) 70° and 110°
5. The sum of an angle and its supplement is ____ degrees.
6. The sum of an angle and its complement is ____ degrees.
7. Find the complement of each angle
(a) 33° (b) 52° (c) 15° (d) 2° (e) 82°
8. Find the supplement of each angle
(a) 33° (b) 52° (c) 15° (d) 2° (e) 82°
9. The complement of an acute angle is always (a) acute (b) obtuse (c) right
10. The supplement of a right angle is always (a) acute (b) obtuse (c) right
11. The supplement of an acute angle is always (a) acute (b) obtuse (c) right
12. The supplement of an obtuse angle is always (a) acute (b) obtuse (c) right
13. If two angles are adjacent angles, then they must have a common _________ and a
common ________ between them.
14. (a) AEB and CED are called
B
E
C (1) complementary angles (2) supplementary angles
(3) adjacent angles (4) vertical angles
A
D
(b) AEB and BEC are called
(1) complementary angles (2) acute angles
(3) a linear pair (4) vertical angles
15. If two straight lines intersect, the vertical angles formed are always __________.
16. The supplement of an angle of 40° contains (a) 40° (b) 50° (c) 140°
17. If two congruent angles are supplementary, then each angle contains
(a) 45° (b) 90° (c) 180°
18. The difference between the supplement and the complement of an angle is
(a) 45° (b) 90° (c) 180°
45

19. In the diagram, m1 67 and m2 23
Consider the following statements:
Which of the three statements above are always true?
(a) i and ii only (b) i and iii only (c) i, ii, and iii
20. If two angles are a linear pair, then they must be
(a) congruent (b) complementary (c) supplementary
21. If an acute angle decreases, then its supplement
(a) increases (b) decreases (c)remains the same
22. If two straight lines intersect, the vertical angles formed are always
(a) complementary (b) supplementary (c) congruent
23. The difference between the supplement and the complement of an angle is
(a) less than 90° (b) greater than 90° (c) 90°
24. The supplement of an acute angle is always a(an) ____________ angle.
25. The supplement of an obtuse angle is always a(an) _____________ angle.
26. The supplement of a right angle is always a(an) ___________ angle.
27. The complement of an acute angle is always a(an) ____________ angle.
28. If two angles are congruent and complementary, then each angle contains _______
degrees.
29. The bisectors of two complementary adjacent angles form an angle of ___ degrees.
30. The bisectors of two supplementary adjacent angles form an angle of ______
degrees.
31. If an angle is congruent to its supplement, then each angle contains _______ degrees.
32. If two straight lines intersect, the _____________ angles formed are congruent.
33. If two angles are complementary, then both must be
(a) acute angles (b) obtuse angles (c) right angles
34. If two angles form a linear pair, then they must be ___________________.
35. Which of the following statements about AEB and BEC is NOT true?
B
E
C
A D
i) 1and2areacute ii) 1and2arecomplementary iii) 1and2areadjacent (a) AEBandBECareadjacent angles
(b) AEBandBECarealinear pair
(c) AEBandBECaresupplementary (d) AEBandBECarevertical angles
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A
1
2
D
B
C