Download Notes Section 1.5: Describe Angle Pair Relationships

Honors Geometry
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Notes Section 1.5: Describe Angle Pair Relationships
1. Def: Two angles are complementary angles if the sum of their measures is 90. That is, if <A and <B are
complements, then m<A + m<B = 90.
2. Def: Two angles are supplementary angles if the sum of their measures is 180. That is, if <A and <B are
supplements, then m<A + m<B = 180.
3. Example: Find the complement and supplement (if possible):
COMPLEMENT
SUPPLEMENT
a. 48
42
132
b. 95
none
85
c. 90
none
90
d. x
(90 – x)
(180 – x)
e. (2x – 11)
(101 – 2x)
(191 – 2x)
4. T / F
If <A is the complement of <B and <B is the supplement of <C, then <C is obtuse.
33
5. Example: If <A is the complement of <B and <B is the supplement of <C and m<C = 123, find m<A =______
6. Example: If <A and <B are complements and <A’s measure is ¼ the measure of <B, find the supplement of <B:
A = ¼ B; A + B = 90  ¼ B + B = 90  B + 4B = 360  5B = 360  B = 72  Supplement is 108
7. Def: Two angles are adjacent angles if they share a vertex point, a side, and no interior points.
<MEG and <GEO are
adjacent angles
8. Question: Suppose the phrase “and no interior points” was removed from the definition of adjacent angles.
What problem does this present? Without this, an angle could be adjacent to itself, or adjacent angles could
overlap, such as <OEG and <OEM – these share a vertex point and a side but clearly are not adjacent.
9. Example: Given the diagram below, prove that <1 and <2 are complementary
From the diagram, <ABC is right, meaning
m<ABC = 90. The angle addition postulate
tells us m<1 + m<2 = m<ABC. Substituting,
m<1 + m<2 = 90. Therefore, <1 and <2 are
complementary.
A
D
1
B
2
C
10. In your own words, state the theorem we just proved:
If two adjacent angles’ noncommon sides form a right angle, then those adjacent angles are complementary.
11. T/F
If two angles are complementary, then their non-shared sides must form a right angle.
Complementary angles do not have to be adjacent.
12. Def: If two adjacent angles are such that their noncommon sides are opposite rays, then the angles are said to
form a linear pair.
1
2
13. Linear Pair Postulate: If two angles form a linear pair, then they are supplementary.

Therefore, we can conclude from the diagram above, that <1 and <2 are supplementary by the linear pair
postulate, and that m<1 + m<2 = 180 by the definition of supplementary.

Note a linear pair consists of two angles – proving that m<1 + m<2 + m<3 = 180 in the diagram below would
be difficult!
1
2
3
14. Def: Two angles are vertical angles if their sides form two pairs of opposite rays.

<1 and <3 are vertical angles

<2 and <4 are vertical angles
1
4
2
3
15. Vertical Angle Theorem: If two angles form vertical angles, then they are congruent, or their measures are
equal. That is, if <1 and <3 are vertical angles, then <1  <3 and m<1 = m<3.
Proof: (refer to diagram above)
Because they form linear pairs, <1 & <2 are supplementary, as are <2 and & <3. By the definition of
supplementary, m<1 + m<2 = 180 and m<2 + m<3 = 180. Substituting, m<1 + m<2 = m<2 + m<3. Subtracting
m<2 from both sides of the last equation gives us m<1 = m<3  <1  <3. In a similar fashion, <2  <4.