16. 5x – 3 = 4(x + 2) Given
5x – 3 = 4x + 8
Dist. Prop
x–3=8
Subt.
x = 11
Subt.
17. 1.6 = 3.2n
0.5 = n
Given
Division
23.  Add Post; Subst.; Simplify; Subtr.; Add; Division
24.  Add Post; Subst.; Dist. Prop; Simplify; Subtr; Division
25. Sym Prop
26. Reflex. Prop
34. 169.50 = 35 + 21(3) + 1.10x Given
169.50 = 98 + 1.10x
Simplify
27. Trans. Prop
71.50 = 1.10x
Subtr.
28. Reflex. Prop
65 = x
Division
30. 3x – 1
39. B
31. A  T
40. H
32. NP  BC
41. D
33. x = 5, y = 9
42. 90
Warm Up
Determine whether each statement is true or
false. If false, give a counterexample.
1. It two angles are complementary, then they are
not congruent. false; 45° and 45°
2. If two angles are congruent to the same angle,
then they are congruent to each other. true
3. Supplementary angles are congruent.
false; 60° and 120°
When writing a proof, it is important to justify each logical step
with a reason. You can use symbols and abbreviations, but they
must be clear enough so that anyone who reads your proof will
understand them.
• Definitions
Hypothesis
Example 1:
• Postulates
• Properties
• Theorems
Conclusion
Write a justification for each step,
given that A and B are
supplementary and mA = 45°.
1. A and B are supplementary.
mA = 45°
2. mA + mB = 180°
Given information
3. 45° + mB = 180°
Subst. Prop of =
Def. of supp s
Steps 1, 2
4. mB = 135°
Subtr. Prop of =
A theorem is any statement that you can prove. Once
you have proven a theorem, you can use it as a
reason in later proofs.
A geometric proof begins with Given and Prove statements,
which restate the hypothesis and conclusion of the
conjecture. In a two-column proof, you list the steps of
the proof in the left column. You write the matching reason
for each step in the right column.
Example 2:
Fill in the blanks to complete the two-column
proof.
Given: XY
Prove: XY  XY
Statements
1.
Reasons
1. Given
3. XY
.
2. Reflex.
.
Prop. of =
3. Def. of  segs.
XY
2. XY = XY

XY
Before you start writing a proof, you should plan out your
logic. Sometimes you will be given a plan for a more
challenging proof. This plan will detail the major steps of the
proof for you.
Helpful Hint
If a diagram for a proof is not provided, draw your own and
mark the given information on it. But do not mark the
information in the Prove statement on it.
Example 3: Use the given plan to write a two-column proof.
Given: 1 and 2 are supplementary, and
1  3
Prove: 3 and 2 are supplementary.
Plan: Use the definitions of supplementary and congruent angles and
substitution to show that m3 + m2 = 180°. By the definition of
supplementary angles, 3 and 2 are supplementary.
Statements
1. 1 and 2 are supplementary. 1  3
2.
m1 + m2 = 180°
3. .m1 = m3
Reasons
1. Given
of supp. s
2. Def.
.
3. Def. of  s
Subst.
4.
m3 + m2 = 180°
4.
5.
3 and 2 are supplementary
5. Def. of supp. s
Example 3 Use the given plan to write a two-column proof
if one case of Congruent Complements
Theorem.
Given: 1 and 2 are complementary, and
2 and 3 are complementary.
Prove: 1  3
Statements
1. 1 and 2 are complementary.
Reasons
1.
Given
2 and 3 are complementary.
2. m1 + m2 = 90°
m2 + m3 = 90°
of comp. s
2. Def.
.
3. .m1 + m2 = m2 + m3
3. Subst.
4.
m2 = m2
4. Reflex.
5.
m1 = m3
5. Subtr.
6.
1  3
6. Def. of  s