BELL RINGER
2-5 PROVING
ANGLES
CONGRUENT
VERTICAL ANGLES

Two angles whose sides are opposite rays
3
1
2
4
ADJACENT ANGLES
 Two
coplanar angles with a common side,
a common vertex, and no common interior
points
3
4
1
2
COMPLEMENTARY ANGLES
 Two
angles that ADD up to 90˚
 Each angle is called the complement of
the other
50°
1
2
A
B
40°
SUPPLEMENTARY ANGLES
Two angles that ADD up to 180°
 Each angle is called the supplement
of the other

105°
75°
3
4
EXAMPLES!

1. Identify Angle Pairs
a.
Complementary
b.
Supplementary
c.
Vertical
2
1
d.
Adjacent
5
3
4

2. What can you tell
from the picture?
a.
About ∠1 and ∠2?
b.
What type of angles are
∠2 and ∠3?
3
4
c.
What type of angles are
∠4 and ∠5?
d.
What type of angles are
∠1 and ∠4?
5
2
1
VERTICAL ANGLES THEOREM (2-1)
A statement that is proven (true) is called a
THEOREM
 The steps taken to show a theorem is true is
called a PROOF
 Vertical angles are congruent (≅)
 ∠1 ≅ ∠2 and ∠3 ≅ ∠4

3
1
2
4
PROOFS

In a proof, a “GIVEN” list shows what you know
from the hypothesis of the theorem. You must
prove the conclusion of the theorem.
PROVING THEOREM 2-1
Given: ∠1 and ∠2 are vertical
angles.
Prove: ∠1 ≅ ∠2
3
1
2
Proof:
1. We know by the definition of
___________________________________ that m∠1 + m∠3 =
180° and m∠2 + m∠3 = _________
2. By substitution, m∠1 + m∠3 ________ m∠2 + m∠3
3. Subtract m∠3 from both sides.
4. You get m∠1 ______ m∠2 or ∠1 ______ ∠2
USING THE VERTICAL ANGLES THEOREM
Find the value of x.
 Find the value of each angle.

4x
(3x + 35)
CONGRUENT SUPPLEMENTS THEOREM
(2-2)

If two angles are supplements of the same angle
(or of congruent angles), then the two angles are
congruent.
2

∠1 and ∠2 are supplementary
(∠2 is a supplement of ∠1)
3

∠1 and ∠3 are supplementary
(∠3 is a supplement of ∠1)

So ∠2 ≅ ∠3
1
PROVING THEOREM 2-2

Given: ∠1 and ∠2 are supplementary
∠1 and ∠3 are supplementary

Prove: ∠2 ≅ ∠3

Proof:
2
3
1.
By the definition of ___________________________
m∠1 + m∠2 = 180° and m∠1 + m∠3 = 180°.
2.
By substitution, m∠1 + m∠2 _________ m∠1 + m∠3
3.
_____________ m∠1 from each side.
4.
You get ________________________________ or ________________
1
CONGRUENT COMPLEMENTS THEOREM



(2-3) If two angles are complements of the same
angle (or of congruent angles), then the two
angles are congruent.
Theorem 2-4: All right angles are congruent.
Theorem 2-5: If two angles are congruent and
supplementary, then each is a right angle.
TICKET OUT THE DOOR
PRACTICE!!
Pg.
100-101
1-9, 10, 13, 15, 19, 20-25, 31, and
39-42