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Unit 1 Logical Arguments and Constructions; Proof and Congruence > Topic 2 Reasoning and Proof > 2-6 Proving Angles
Congruent
2-6 Proving Angles Congruent
Teks Focus
TEKS (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.
TEKS (1)(D) Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and
language as appropriate.
Additional TEKS (1)(E), (1)(G), (4)(A)
Vocabulary
• Paragraph proof – A paragraph proof gives statements and reasons or justifications of a proof, written as sentences in a paragraph.
• Theorem – a conjecture or statement that you prove true
• Implication – a conclusion that follows from previously stated ideas or reasoning without being explicitly stated.
• Representation – a way to display or describe information. You can use a representation to present mathematical ideas and data.
ESSENTIAL UNDERSTANDING
You can use given information, definitions, properties, postulates, and previously proven theorems as reasons in a proof.
take note
Key Concept Paragraph Proof
The proof in Problem 3 is two-column, but there are many ways to display a proof. A paragraph proof is written as sentences in a paragraph. Below is a paragraph
proof from Problem 3. Each statement in the Problem 3 proof is red in the paragraph proof.
Proof Given: ∠1 ≅ ∠4
Prove: ∠2 ≅ ∠3
Proof: ∠1 ≅ ∠4 is given. ∠4 ≅ ∠2 because vertical angles are congruent. By the Transitive Property of Congruence, ∠1 ≅ ∠2. ∠1 ≅ ∠3 because vertical angles are
congruent. By the Transitive Property of Congruence, ∠2 ≅ ∠3.
take note
Theorem 2-1 Vertical Angles Theorem
Vertical angles are congruent.
∠1 ≅ ∠3 and ∠2 ≅ ∠4
d
For a proof of Theorem 2-1, see Problem 1.
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Unit 1 Logical Arguments and Constructions; Proof and Congruence > Topic 2 Reasoning and Proof > 2-6 Proving Angles
Congruent
take note
Theorem 2-2 Congruent Supplements Theorem
Theorem
If …
∠1 and ∠3 are supplements and ∠2 and ∠3 are
supplements
If two angles are supplements of the same angle (or of congruent angles), then the two angles
are congruent.
Then …
∠1 ≅
∠2
For a proof of Theorem 2-2, see Problem 5.
Theorem 2-3 Congruent Complements Theorem
Theorem
If …
∠1 and ∠2 are complements and ∠3 and ∠2 are
complements
Then …
∠1 ≅
∠3
If two angles are complements of the same angle (or of congruent angles), then the two angles
are congruent.
You will prove Theorem 2-3 in Exercise 4.
Theorem 2-4
Theorem
If …
∠1 and ∠2 are right angles
Then …
∠1 ≅ ∠2
All right angles are congruent.
You will prove Theorem 2-4 in Exercise 23.
Theorem 2-5
Theorem
If …
∠1 ≅ ∠2, and ∠1 and ∠2 are supplements
If two angles are congruent and supplementary, then each is a right angle.
Then …
m∠1 = m∠2 = 90
You will prove Theorem 2-5 in Exercise 11.
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Unit 1 Logical Arguments and Constructions; Proof and Congruence > Topic 2 Reasoning and Proof > 2-6 Proving Angles
Congruent
Problem 1
TEKS Process Standards (1)(D)
Proof Proving the Vertical Angles Theorem
Given: ∠1 and ∠3 are vertical angles.
Prove: ∠1 ≅ ∠3
Statements
1) ∠1 and ∠3 are vertical angles.
Reasons
1) Given
2) ∠
s that form a linear pair are supplementary.
2) ∠1 and ∠2 are supplementary. ∠2 and ∠3 are supplementary. ∠1 and ∠2 form a linear pair.
∠2 and ∠3 form a linear pair.
3) m∠1 + m∠2 = 180
3) The sum of the measures of supplementary ∠
s is 180.
m∠2 + m∠3 = 180
4) m∠1 + m∠2 = m∠2 + m∠3
4) Transitive Property of Equality
5) m∠1 = m∠3
5) Subtraction Property of Equality
6) ∠1 ≅ ∠3
s with the same measure are ≅.
6) ∠
Proof Verifying the Vertical Angles Theorem Using Line Segments
Plan
How can you use the
relationship between ∠1, ∠2,
and ∠3 in your proof?
∠1 and ∠3 both form linear
pairs with ∠2. Since angles that
form a linear pair are
supplementary, you can show
that ∠1 and ∠2 are
supplements, and ∠2 and ∠3
are supplements.
The Vertical Angles Theorem works for vertical angles formed by line segments as well as lines. Suppose line
¯¯¯¯
¯¯¯¯
segments AB and CD intersect at point E. Write a paragraph proof to show that ∠AEC ≅ ∠BED.
Both ∠AEC and ∠BED are supplementary to ∠CEB, because ∠AEC and ∠CEB form a linear pair, and ∠CEB and ∠BED form
a linear pair. By the definition of supplementary angles, m∠AEC + m∠CEB = 180 and m∠CEB + m∠BED =180. Then m∠AEC +
m∠CEB = m∠CEB + m∠BED by the Transitive Property of Equality. Subtract m∠CEB from each side. By the Subtraction
Property of Equality, m∠AEC = m∠BED. Angles with the same measure are congruent, so ∠AEC = ∠BED.
Problem 2
Applying the Vertical Angles Theorem
What is the value of x?
Plan
How do you get started?
Think
The two labeled angles are vertical angles, so set them equal.
Write
2x + 21 = 4x
21 = 2x
Solve for x by subtracting 2x from each side and then dividing by 2. 21
=x
2
Look for a relationship in the
diagram that allows you to write
an equation with the variable.
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Unit 1 Logical Arguments and Constructions; Proof and Congruence > Topic 2 Reasoning and Proof > 2-6 Proving Angles
Congruent
Problem 3
Proof Writing a Proof Using the Vertical Angles Theorem
Given: ∠1 ≅ ∠4
Prove: ∠2 ≅ ∠3
Statements
Reasons
1) ∠1 ≅ ∠4 1) Given
2) ∠4 ≅ ∠2 2) Vertical angles are ≅.
3) ∠1 ≅ ∠2 3) Transitive Property of Congruence
4) ∠1 ≅ ∠3 4) Vertical angles are ≅.
5) ∠2 ≅ ∠3 5) Transitive Property of Congruence
Think
Why does the Transitive
Property work for statements
3 and 5?
In each case, an angle is
congruent to two other angles,
so the two angles are
congruent to each other.
Problem 4
Distinguishing Between Mathematical Concepts
For each statement, determine whether it is an undefined term, a definition, a postulate, a conjecture, or a theorem.
I. A segment is a part of a line that consists of two endpoints and all points between them.
II. If ∠1 and ∠3 are supplements and ∠2 and ∠3 are supplements, then ∠1 ≅ ∠2.
III. A plane contains infinitely many lines.
Think
IV. If ∠1 and ∠2 are supplementary, then one of the angles is obtuse.
Why are some terms
undefined?
V. If two distinct planes R and W intersect, they intersect in exactly one line.
Statement I is the definition of the term segment. Statement II is a theorem, specifically the Congruent Supplements Theorem.
Statement III describes a plane, but plane is an undefined term. A counterexample can show that Statement IV is incorrect, so
it is a conjecture. It is an accepted fact that two planes intersect in exactly one line, so Statement V is a postulate.
The terms point, line, and
plane are not defined because
their definitions would require
terms that also need defining.
Problem 5
TEKS Process Standard (1)(G)
Proof Writing a Paragraph Proof
Plan
How can you use the given
information?
∠1 and ∠3 are supplementary.
∠2 and ∠3 are supplementary.
Prove: ∠1 ≅ ∠2
∠1 and ∠3 are supplementary because it is given. So m∠1 + m∠3 = 180 by the definition of supplementary angles.
∠2 and ∠3 are supplementary because it is given, so m∠2 + m∠3 = 180 by the same definition. By the Transitive
Proof:
Property of Equality, m∠1 + m∠3 = m∠2 + m∠3. Subtract m∠3 from each side. By the Subtraction Property of
Equality, m∠1 = m∠2. Angles with the same measure are congruent, so ∠1 ≅ ∠2.
Given:
Both ∠1 and ∠2 are
supplementary to ∠3. Use their
relationship with ∠3 to relate
∠1 and ∠2 to each other.
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Unit 1 Logical Arguments and Constructions; Proof and Congruence > Topic 2 Reasoning and Proof > 2-6 Proving Angles
Congruent
PRACTICE and APPLICATION EXERCISES
Find the value of each variable and the measure of each labeled angle.
Scan page for a Virtual
Nerd™ tutorial video.
1.
2.
For additional
Proof 3. Justify Mathematical Arguments (1)(G) Complete the following proof by filling in the blanks.
support when completing your
homework, go to
PearsonTEXAS.com .
Given: ∠1 ≅ ∠3
Prove: ∠6 ≅ ∠4
Statements
Reasons
1) ∠1 ≅ ∠3 1) Given
2) ∠3 ≅ ∠6 2) a. _?_
3) b. _?_
3) Transitive Property of Congruence
4) ∠1 ≅ ∠4 4) c. _?_
5) ∠6 ≅ ∠4 5) d. _?_
Proof 4. Fill in the blanks to complete this proof of the Congruent Complements Theorem (Theorem 2−3).
If two angles are complements of the same angle, then the two angles are congruent.
∠1 and ∠2 are complementary.
∠3 and ∠2 are complementary.
Prove: ∠1 ≅ ∠3
∠1 and ∠2 are complementary and ∠3 and ∠2 are complementary because it is given. By the definition of
complementary angles, m∠1 + m∠2 = a. _?_ and m∠3 + m∠2 = b. _?_. Then m∠1 + m∠2 = m∠3 + m∠2 by the
Proof:
Transitive Property of Equality. Subtract m∠2 from each side. By the Subtraction Property of Equality, you get m∠1=
c. _?_. Angles with the same measure are d. _?_, so∠1 ≅ ∠3.
5. Apply Mathematics (1)(A) What is the measure of the angle formed by Park St. and 116th St.?
Given:
6. Apply Mathematics (1)(A) Give an example of vertical angles in your home or classroom.
d
Proof 7. Use Multiple Representations to Communicate Mathematical Proof Ideas (1)(D) Write a paragraph proof for the
Vertical Angles Theorem (Theorem 2−1). Include a sketch of intersecting lines and label each angle.
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Unit 1 Logical Arguments and Constructions; Proof and Congruence > Topic 2 Reasoning and Proof > 2-6 Proving Angles
Congruent
8. Explain Mathematical Ideas (1)(G) In the figure at the right, m∠2 = 21 and m∠5 = 138. Find m∠1. Show your work.
9. Two lines that intersect form four angles. If one of the angles has a measure of 55, what are the measures of the remaining angles?
10. Apply Mathematics (1)(A) In the game of miniature golf, the ball bounces off the wall at the same angle it hits the wall. (This is the angle formed by the path of
the ball and the line perpendicular to the wall at the point of contact.) In the diagram, the ball hits the wall at a 40° angle. Using Theorem 2−3, what are the values of x
and y?
d
Proof 11. Justify Mathematical Arguments (1)(G) Fill in the blanks to complete this proof of Theorem 2−5.
If two angles are congruent and supplementary, then each is a right angle.
Given: ∠W and ∠V are congruent and supplementary.
Prove: ∠W and ∠V are right angles.
∠W and ∠V are congruent because a. _?_. Because congruent angles have the same measure, m∠W = b. _?_. ∠W and ∠V are supplementary because it
is given. By the definition of supplementary angles, m∠W + m∠V = c. _?_. Substituting m∠W for m∠V, you get m∠W +m∠W = 180, or 2m∠W = 180. By the
Proof:
d. _?_ Property of Equality, m∠W = 90. Since m∠W = m∠V, m∠V = 90 by the Transitive Property of Equality. Both angles are e. _?_ angles by the definition
of right angle.
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Unit 1 Logical Arguments and Constructions; Proof and Congruence > Topic 2 Reasoning and Proof > 2-6 Proving Angles
Congruent
12. Apply Mathematics (1)(A) In the photograph below, the legs of the table are constructed so that ∠1 ≅ ∠2. What theorem can you use to justify the statement that
∠3 ≅ ∠4?
d
13. Explain Mathematical Ideas (1)(G) Explain why this statement is true: If m∠ABC + m∠XYZ = 180 and ∠ABC = ∠XYZ, then ∠ABC and ∠XYZ are right angles.
Find the measure of each angle.
14. ∠A is twice as large as its complement, ∠B.
15. ∠A is half as large as its complement, ∠B.
16. ∠A is twice as large as its supplement, ∠B.
17. ∠A is half as large as twice its supplement, ∠B.
Proof 18. Write a proof for this form of Theorem 2-2.
If two angles are supplements of congruent angles, then the two angles are congruent.
∠1 and ∠2 are supplementary.
Given: ∠3 and ∠4 are supplementary.
∠2 ≅ ∠4
Prove: ∠1 ≅ ∠3
Proof 19. Justify Mathematical Arguments (1)(G) Two lines intersect and one of the angles formed is a right angle. Prove that all four angles are right angles using
the Vertical Angles Theorem.
Determine whether the statement is an undefined term, a definition, a postulate, a conjecture, or a theorem. Explain your reasoning.
20. If an angle is obtuse, then it measures 100°.
21. If ∠1 and ∠2 are complements and ∠3 and ∠2 are complements, then ∠1 ≅ ∠3.
¯¯¯¯
¯¯¯¯
22. If two distinct lines AB and CD intersect, then they intersect in exactly one point.
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Unit 1 Logical Arguments and Constructions; Proof and Congruence > Topic 2 Reasoning and Proof > 2-6 Proving Angles
Congruent
Proof 23. Justify Mathematical Arguments (1)(G) Fill in the blanks to complete this proof of Theorem 2-4.
All right angles are congruent.
Given: ∠X and ∠Y are right angles.
Prove: ∠X ≅ ∠Y
∠X and a. _?_ are right angles because it is given. By the definition of b. _?_, m∠X = 90 and m∠Y = 90. By the Transitive Property of Equality, m∠X = c. _?
Proof:
_. Because angles of equal measure are congruent, d. _?_.
24. Analyze Mathematical Relationships (1)(F) Explain the relationship between undefined terms and terms with definitions. Include an example to illustrate your
explanation.
Find the value of each variable and the measure of each angle.
25.
26.
27.
28. Justify Mathematical Arguments (1)(G) Sketch a pair of intersecting line segments. Label each of the four resulting angles. Write a paragraph proof to show that
one pair of vertical angles in your sketch have equal measures.
29. Given: ∠7 ≅ ∠8
Prove: ∠5 ≅ ∠6
TEXAS Test Practice
30. ∠1 and ∠2 are vertical angles. If m∠1 = 63 and m∠2 = 4x − 9, what is the value of x?
31. What is the area in square centimeters of a triangle with a base of 5 cm and a height of 8 cm?
32. What is the measure of an angle with a supplement that is four times its complement?
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