1. Proving Angle Relationships
LESSON 2–8
Proving Angle Relationships

2. Five-Minute Check (over Lesson 2–7) TEKS Then/Now
Postulate 2.10: Protractor Postulate
Postulate 2.11: Angle Addition Postulate
Example 1: Use the Angle Addition Postulate
Theorems 2.3 and 2.4
Example 2: Real-World Example: Use Supplement or Complement
Theorem 2.5: Properties of Angle Congruence
Proof: Symmetric Property of Congruence
Theorems 2.6 and 2.7
Proof: One Case of the Congruent Supplements Theorem
Example 3: Proofs Using Congruent Comp. or Suppl. Theorems
Theorem 2.8: Vertical Angles Theorem
Example 4: Use Vertical Angles
Theorems 2.9–2.13: Right Angle Theorems
Lesson Menu

3. D. Segment Addition Postulate
Justify the statement with a property of equality or a property of congruence.
A. Transitive Property
B. Symmetric Property
C. Reflexive Property
D. Segment Addition Postulate
5-Minute Check 1

4. D. Segment Addition Postulate
Justify the statement with a property of equality or a property of congruence.
A. Transitive Property
B. Symmetric Property
C. Reflexive Property
D. Segment Addition Postulate
5-Minute Check 2

5. D. Segment Addition Postulate
Justify the statement with a property of equality or a property of congruence. If H is between G and I, then GH + HI = GI.
A. Transitive Property
B. Symmetric Property
C. Reflexive Property
D. Segment Addition Postulate
5-Minute Check 3

6. State a conclusion that can be drawn from the statement given using the property indicated. W is between X and Z; Segment Addition Postulate.
A. WX > WZ
B. XW + WZ = XZ
C. XW + XZ = WZ
D. WZ – XZ = XW
5-Minute Check 4

7. State a conclusion that can be drawn from the statements given using the property indicated.
LM  NO
___ A. B. C. D.
5-Minute Check 5

8. Given B is the midpoint of AC, which of the following is true?
___
A. AB + BC = AC
B. AB + AC = BC
C. AB = 2AC
D. BC = 2AB
5-Minute Check 6

9. Mathematical Processes G.1(E), G.1(G)
Targeted TEKS
G.6(A) Verify theorems about angles formed by the
intersection of lines and line segments, including
vertical angles, and angles formed by parallel lines cut by a transversal and prove equidistance between the endpoints of a segment and points on its perpendicular bisector and apply these relationships to solve problems.
Mathematical Processes
G.1(E), G.1(G)
TEKS

10. You identified and used special pairs of angles.
Write proofs involving supplementary and complementary angles.
Write proofs involving congruent and right angles.
Then/Now

11. Concept

12. Concept

13. m1 + m2 = 90 Angle Addition Postulate 42 + m2 = 90 m1 = 42
Use the Angle Addition Postulate
CONSTRUCTION Using a protractor, a construction worker measures that the angle a beam makes with a ceiling is 42°. What is the measure of the angle the beam makes with the wall?
The ceiling and the wall make a 90 angle. Let 1 be the angle between the beam and the ceiling. Let 2 be the angle between the beam and the wall.
m1 + m2 = 90 Angle Addition Postulate
42 + m2 = 90 m1 = 42
42 – 42 + m2 = 90 – 42 Subtraction Property of Equality
m2 = 48 Substitution
Example 1

14. Answer: The beam makes a 48° angle with the wall.
Use the Angle Addition Postulate
Answer: The beam makes a 48° angle with the wall.
Example 1

15. Find m1 if m2 = 58 and mJKL = 162.
B. 94
C. 104
D. 116
Example 1

16. Concept

17. Use Supplement or Complement
TIME At 4 o’clock, the angle between the hour and minute hands of a clock is 120º. When the second hand bisects the angle between the hour and minute hands, what are the measures of the angles between the minute and second hands and between the second and hour hands?
Analyze Make a sketch of the situation. The time is 4 o’clock and the second hand bisects the angle between the hour and minute hands.
Example 2

18. Answer: Both angles are 60°.
Use Supplement or Complement
Formulate Use the Angle Addition Postulate and the definition of angle bisector.
Determine Since the angles are congruent by the definition of angle bisector, each angle is 60°.
Answer: Both angles are 60°.
Justify Use the Angle Addition Postulate to check your answer.
m1 + m2 = 120
= 120
120 = 120 
Example 2

19. Use Supplement or Complement
Evaluate The sketch we drew helps us determine an appropriate solution method. Our answer is reasonable.
Example 2

20. QUILTING The diagram shows one square for a particular quilt pattern
QUILTING The diagram shows one square for a particular quilt pattern. If mBAC = mDAE = 20, and BAE is a right angle, find mCAD.
A. 20
B. 30
C. 40
D. 50
Example 2

21. Concept

22. Concept

23. Concept

24. Concept

25. Given: Prove: Proofs Using Congruent Comp. or Suppl. Theorems
Example 3

26. 1. m3 + m1 = 180; 1 and 4 form a linear pair.
Proofs Using Congruent Comp. or Suppl. Theorems
Proof:
Statements Reasons
1. Given
1. m3 + m1 = 180; 1 and 4 form a linear pair.
2. Linear pairs are supplementary.
2. 1 and 4 are supplementary.
3. Definition of supplementary angles
3. 3 and 1 are supplementary.
4. s suppl. to same  are .
4. 3  4
Example 3

27. In the figure, NYR and RYA form a linear pair, AXY and AXZ form a linear pair, and RYA and AXZ are congruent. Prove that NYR and AXY are congruent.
Example 3

28. Which choice correctly completes the proof? Proof:
Statements Reasons
1. Given
1. NYR and RYA, AXY and AXZ form linear pairs.
2. If two s form a linear pair, then they are suppl. s.
2. NYR and RYA are supplementary. AXY and AXZ are supplementary.
3. Given
3. RYA  AXZ
4. NYR  AXY
4. ____________ ?
Example 3

29. B. Definition of linear pair
A. Substitution
B. Definition of linear pair
C. s supp. to the same  or to  s are .
D. Definition of supplementary s
Example 3

30. Concept

31. 2. Vertical Angles Theorem
Use Vertical Angles
If 1 and 2 are vertical angles and m1 = d – 32 and m2 = 175 – 2d, find m1 and m2. Justify each step.
Statements Reasons
Proof:
1. 1 and 2 are vertical s.
1. Given
2. 1  2
2. Vertical Angles Theorem
3. m1 = m2
3. Definition of congruent angles
4. d – 32 = 175 – 2d
4. Substitution
Example 4

32. Statements Reasons 5. Addition Property 6. Addition Property
Use Vertical Angles
Statements Reasons
5. 3d – 32 = 175
5. Addition Property
6. 3d = 207
6. Addition Property
7. d = 69
7. Division Property
m1 = d – 32 m2 = 175 – 2d
= 69 – 32 or 37 = 175 – 2(69) or 37
Answer: m1 = 37 and m2 = 37
Example 4

33. A. B. C. D.
Example 4

34. Concept

35. Proving Angle Relationships
LESSON 2–8
Proving Angle Relationships