Download 2.5 Proving Angles Congruent

Proving Angles
Congruent
Proving Angles Congruent

Vertical Angles: Two angles whose sides form
two pairs of opposite rays; form two pairs of
congruent angles
1
2
4
<1 and <3 are
Vertical angles
<2 and <4 are
Vertical angles
3
Proving Angles Congruent

Adjacent Angles: Two coplanar angles that share
a side and a vertex
1
2
<1 and <2 are
Adjacent Angles
1
2
2.5 Proving Angles Congruent

Complementary Angles: Two angles whose
measures have a sum of 90°
50°
2
40°
1

Supplementary Angles: Two angles whose
measures have a sum of 180°
105°
3
4
75°
Identifying Angle Pairs
In the diagram identify pairs of numbered angles
that are related as follows:
a. Complementary
b.
Supplementary
c.
Vertical
d.
Adjacent
1
5
2
4
3
Making Conclusions
Whether you draw a diagram or use a given
diagram, you can make some conclusions directly
from the diagrams. You CAN conclude that
angles are



Adjacent angles
Adjacent supplementary angles
Vertical angles
Making Conclusions
Unless there are markings that give this
information, you CANNOT assume



Angles or segments are congruent
An angle is a right angle
Lines are parallel or perpendicular
Theorems About Angles
Theorem 2-1
Vertical Angles Theorem
Vertical Angles are Congruent
Theorem 2-2
Congruent Supplements
If two angles are supplements of the same angle
or congruent angles, then the two angles are
congruent
Theorems About Angles
Theorem 2-3
Congruent Complements
If two angles are complements of the same angle
or congruent angles, then the two angles are
congruent
Theorem 2-4
Theorem 2-5
All right angles are congruent
If two angles are congruent and
supplementary, each is a right angle
Proving Theorems
Paragraph Proof: Written as sentences in a
paragraph
1
Given: <1 and <2 are
vertical angles
Prove: <1 = <2
3
2
Paragraph Proof: By the Angle Addition Postulate, m<1 +
m<3 = 180 and m<2 + m<3 = 180. By substitution, m<1
+ m<3 = m<2 + m<3. Subtract m<3 from each side. You
get m<1 = m<2, which is what you are trying to prove.
Proving Theorems
Given:
Prove:
<1 and <2 are supplementary
<3 and <2 are supplementary
<1 = <3
Proof: By the definition of supplementary angles,
m<___ + m<____ = _____ and m<___ +
m<___ = ____. By substitution, m<___ +
m<___ = m<___ + m<___. Subtract m<2 from
each side. You get __________.