2.6 Proving Statements about Angles

Report
2.6 Proving Statements about
Angles
Properties of Angle Congruence
Reflexive
For any angle, A <A  <A.
Symmetric
If <A  <B, then <B  <A.
Transitive
If <A  <B and <B  <C,
then <A  <C.
Right Angle Congruence Theorem
• All right angles are congruent.
A
.
X
B
.C
Y
.
.Z
Congruent Supplements Theorem
•
If two angles are supplementary to the same angle,
then they are congruent
– If m<1 + m<2 = 180° and m<2 + m<3 = 180°,
then m<1 = m<3 or  1   3
Congruent Complements Theorem
•
If two angles are complementary to the same
angle, then the two angles are congruent.
– If m<4 + m<5 = 90° and m<5 + m<6 = 90°,
then m<4 = m<6 or  4   6
Linear Pair Postulate
• If two angles form a linear pair, then they are
supplementary.
1
2
m<1 + m<2 = 180°
Example:
• < 1 and < 2 are a linear pair.
If m<1 = 78°, then find m<2.
Vertical Angles Theorem
•
Vertical angles are congruent.
1
4
2
3
1   3 ,  2   4
Example
<1 and <2 are complementary angles.
<1 and <3 are vertical angles.
If m<3 = 49°, find m<2.
Proving the Right Angle
Congruence Theorem
Given: Angle 1 and angle 2 are right angles
1   2
Prove:
Statements
Reasons
1 . 1 and  2 right  ' s
1. Given
2 . 1  90 and  2  90
2. Def. of right ’s


3 . m 1  m  2
3. Trans. POE
4. 1   2
4. Def. of  ’s
Proving the Vertical Angles Theorem
5
6
7
Given: 5 and 6 are a linear pair. 6 and 7 are a
linear pair.
Prove: 5  7
Statements
Reasons
1. 5 and 6 are a linear pair. 6
and 7 are a linear pair.
1. Given
2. 5 and 6 are supplementary.
6 and 7 are supplementary.
2. Linear Pair Postulate
3.  5   7
3.  Supplements
Theorem
Solve for x.
Give a reason for each step of the proof. Choose from the list of reasons
given.
Given: 6  7
Prove: 5  8
Plan for Proof:
First show that 5  6 and 7  8.
Then use transitivity to show that 5  8.)
Statements
Reasons
1. 6  7
1. Given
2. 7  8
2. Vertical ’s Theorem
3. 6  8
3. Trans. POC
4. 5  6
4. Vertical ’s Theorem
5. Trans. POC
5. 5  8

similar documents