1. Topic 2: Reasoning and Proof
Pearson Unit 1
Topic 2: Reasoning and Proof
2-6: Proving Angles Congruent Pearson Texas Geometry ©2016 Holt Geometry Texas ©2007

2. TEKS Focus:
Foundation to TEKS (6) Use the process skills with deductive reasoning to prove and apply theorems by using a variety of methods such as coordinate, transformational, and axiomatic and formats such as two-column, paragraph, and flow chart.
(1)(G) Display, explain, or justify mathematical ideas and arguments using precise mathematical language in written or oral communication.
(1)(D) Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate.
(1)(E) Create and use representations to organize, record, and communicate mathematical ideas.
(1)(F) Analyze mathematical relationships to connect and communicate mathematical ideas.

3. Vocabulary: Define each term by using the pictures to write a good definition.
Adjacent Angles:
Two angles that share a vertex and common side.

4. Vocabulary: Define each term by using the pictures to write a good definition.
Linear Pair of Angles:
Adjacent angles that add up to 180.

5. Vocabulary: Define each term by using the pictures to write a good definition.
Verical Angles: 1 2 4 3
Angles that are formed by two intersecting lines and are opposite of each other so they do NOT share a side.

6. Vocabulary: Define each term by using the pictures to write a good definition.
Complementary Angles:
120
60
Two angles that add up to 90.

7. Vocabulary: Define each term by using the pictures to write a good definition.
Supplementary Angles:
145
35
Two angles that add up to 180.

8. #2 #3 #4 #5 #1

9.  1 = 30
 2 = 90
 3 = 60
 4 = 30

12. Example 1: Write a paragraph proof to show
1  3 2 4
Both 1 and 3 are supplementary to 2 because 1 and 2 form a linear pair and 3 and 2 form a linear pair.
By definition of supplementary angles, m1 + m 2 = 180° and m3 + m 2 = 180°. So, by the Transitive Property of Equality (or Substitution Property), m1 + m 2 = m3 + m 2.
If we subtract m2 from each side of the equation by the
Subtraction Property of Equality, then m1 = m3.
Therefore, by definition of congruent angles, 1  3.

13. Theorem 2-1: Vertical Angles Theorem
Vertical angles are congruent.
1  3 and 2  4
Given: 1 and 3 are vertical
Prove: 1  3 2 4

14. EXAMPLE 2: 2x + 21 = 4x 21 = 2x 10.5 = x Check: 2(10.5) + 21 = 42
4(10.5) = 42

15. EXAMPLE 3:
3x= 2x + 40
x= 40
Check:
3(40) = 120
2(40) + 40 = 120 

16. Example: 4 Use the given plan to write a two-column proof.
Given: 1 and 2 are complementary, and
2 and 3 are complementary.
Prove: 1  3
Plan: The measures of complementary angles add to 90° by definition. Use substitution to show that the sums of both pairs are equal. Use the Subtraction Property and the definition of congruent angles to conclude that 1  3.

17. Example: 4 cont. Statements Reasons 1. 2. 2. . 3. . 3. 4. 5. 6. Given
2. .
3. . 3. 4. 5. 6.
1 and 2 are complementary.
2 and 3 are complementary.
Given
m1 + m2 = 90° m2 + m3 = 90°
Def. of comp. s
m1 + m2 = m2 + m3
Substitution Prop
m2 = m2
Reflexive Prop. of =
m1 = m3
Subtraction Prop. of =
1  3
Def. of  s

18. Example: 5
Fill in the blanks to complete a two-column proof of one case of the Congruent Supplements Theorem.
Given: 1 and 2 are supplementary, and
2 and 3 are supplementary.
Prove: 1  3
Statements
Reasons
1. 1 and 2 are supp., and 2 and 3 are supp.
1. Given
2. 1 + 2 = 180⁰
2 + 3 = 180⁰
2. Def. of Supp.  ‘s
3. m1 + m2 = m2 + m3
3. Substitution Property
4. m2 = m2
4. Reflexive Property of =
5. m1 = m3
5. Subtraction Prop. of =
6. 1  3
6. Def. of   ‘s

19. Example 6:
m1 = 90°
m2 = 50°
m3 = 40°