Bell Work
• 1) Solve for each variable 2) Solve for each variable
• 3 and 4)
AB  JK , JK  ST
AB  ST
Given
Definition of Congruence
Transitive Property of equality
Definition of Congruence
Outcomes
• I will be able to:
• 1) Use angle congruence properties
• 2) Prove properties about special angle
relationships
Properties of Angle Congruence
• Reflexive Property
• For any Angle A,
• Symmetric Property
• If, A  B
then B  A
• Transitive Property
• If, A  B and B  C
• then
A  C
A  A
Proof Practice
Prove the Transitive Property of Congruence for angles
Given: A  B and B  C
Prove: A  C
• Statements
• Reasons
1. ∠A ≅ ∠B and ∠B ≅∠C
1. Given
2. m∠A = m∠B
2. Definition of Congruent Angles
3. m∠B = m∠C
3. Definition of Congruent Angles
4. m∠A = m∠C
4. Transitive/Substitution
5. ∠A ≅ ∠C
5. Definition of Congruent Angles
Use the Transitive Property
Given: m∠3 = 40°, ∠1 ≅ ∠2, ∠2 ≅ ∠3
Prove: Angle 1 = 40°
Statements
1. m∠3 = 40°, ∠1 ≅ ∠2,
∠2 ≅ ∠3
2. ∠1 ≅ ∠3
Reasons
1. Given
2. Transitive Property of
Congruence.
3. m∠1 = m∠3
3. Def. of Congruence
4. m∠1 = 40°
4. Substitution
Angle Investigation
1) Fold your paper in half.
2) Place the corner of your second piece of paper
at the vertex of right angles and trace it
3) Label the four angles from left to right 1, 2, 3, 4
as shown in the picture
4) Answer the following questions on the sheet of
paper.
Complementary Questions
Angle Relationship Theorems
• Right Angle Congruence Theorem: All right angles are
congruent
• Congruent Supplements Theorem: If two angles are
supplementary to the same angle (or to congruent angles),
then they are congruent.
Ex: Angle 1 is supplementary to Angle 2
and Angle 3 is supplementary to Angle 2
Therefore, Angle 1 is congruent to Angle 3
• Congruent Complements Theorem: If two angles are
complementary to the same angle (or to congruent angles),
then they are congruent.
Ex: Angle 1 and Angle 3 are both
complementary to Angle 2.
Therefore, Angle 1 is congruent to Angle 3
Proof Practice
Prove the Congruent Complements Theorem
Statements
Reasons
1.
1. Given
2.
Angle 1 and Angle 2 are
complements; Angle 3 and
Angle 4 are complements;
Angle 2 is congruent to Angle 4
m1  m2  90 
m3  m4  90 
3.m1  m2  m3  m4
m2  m4
5. m1  m2  m3  m2
4.
6.
7.
m1  m3
1  3
2. Definition of Complementary
Angles
3. Transitive Property of Equality
4. Def. of Congruent Angles
5. Substitution Prop. of Equality
6. Subtraction Prop. of Equality
7. Def. of Congruent Angles
Angle Relationship Theorems
• Linear Pair Postulate: If two angles form a
linear pair, then they are supplementary.
• Meaning: Angle 1 + Angle 2 = 180°
• Vertical Angles Theorem:
• Vertical angles are congruent
• Meaning: 2  4 and 1  3
Example Using Angle Relationships
In the diagram, 3 is a right angle and m5  57
Find the measures of 1, 2, 3, and 4.
• By definition of a right angle, Angle 3 = 90°.
• Angle 2 and Angle 5 are vertical angles and Angle 5 =
57°, so Angle 2 = 57°.
• Angle 1 and Angle 5 form a linear pair, so Angle 1 +
Angle 5 = 180°. When you substitute 57° for angle 5,
and solve for Angle 1, the result is Angle 1 = 123°.
• Angle 4 and Angle 5 are complementary, so
Angle 4 + Angle 5 = 90°. When you substitute 57° for
Angle 5 and solve for Angle 4, the result is Angle 4 = 33°
Check Point
• ON YOUR OWN
• Using your knowledge of angle pair
relationships, solve for each angle in the
diagram.
• Then, match the diagram in column A with the
appropriate theorem/postulate in column B
Exit Quiz
• Given the following information, complete the
two-column proof below