warm up determine whether each statement is true or false. if false, give a counterexample
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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. 2. If two angles are congruent to the same angle, then they are congruent to each other. 3. Supplementary angles are congruent. - PowerPoint PPT PresentationTRANSCRIPT
Holt McDougal Geometry
2-6 Geometric Proof
Warm UpDetermine whether each statement is true or false. If false, give a counterexample.
1. It two angles are complementary, then they are not congruent.
2. If two angles are congruent to the same angle, then they are congruent to each other.
3. Supplementary angles are congruent.
false; 45° and 45°
true
false; 60° and 120°
Holt McDougal Geometry
2-6 Geometric Proof
Write two-column proofs.Prove geometric theorems by using deductive reasoning.
Objectives
Holt McDougal Geometry
2-6 Geometric Proof
theoremtwo-column proof
Vocabulary
Holt McDougal Geometry
2-6 Geometric Proof
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.
Hypothesis Conclusion
• Definitions• Postulates• Properties• Theorems
Holt McDougal Geometry
2-6 Geometric Proof
Write a justification for each step, given that A and B are supplementary and mA = 45°.
Example 1: Writing Justifications
1. A and B are supplementary.mA = 45°
Given information
2. mA + mB = 180° Def. of supp s
3. 45° + mB = 180° Subst. Prop of = Steps 1, 2
4. mB = 135° Subtr. Prop of =
Holt McDougal Geometry
2-6 Geometric Proof
Check It Out! Example 1
Write a justification for each step, given that B is the midpoint of AC and AB EF.
1. B is the midpoint of AC. Given information
2. AB BC
3. AB EF
4. BC EF
Def. of mdpt.
Given information
Trans. Prop. of
Holt McDougal Geometry
2-6 Geometric Proof
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.
Holt McDougal Geometry
2-6 Geometric Proof
POSTULATE AND THEOREMTurn to:
Holt McDougal Geometry
2-6 Geometric Proof
Linear Pair TheoremIf two angles form a linear pair; then they are supplementary
Holt McDougal Geometry
2-6 Geometric Proof
Congruent SupplementsTheorem
If two angles are supplementary to the same angle, then the two angles are congruent
Holt McDougal Geometry
2-6 Geometric Proof
Fill in the blanks to complete the two-column proof.
Given: XY
Prove: XY XY
Example 2: Completing a Two-Column Proof
Statements Reasons
1. 1. Given
2. XY = XY 2. .
3. . 3. Def. of segs.
XY
Reflex. Prop. of =
XY XY
Holt McDougal Geometry
2-6 Geometric Proof
CHECK UP!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
Proof:
.a 1 and 2 are supp., and 2 and 3 are supp.
b. m1 + m2 = m2 + m3
c. Subtr. Prop. of =
d. 1 3
Holt McDougal Geometry
2-6 Geometric Proof
TURN TO:POSTULATES AND THEOREMS
Holt McDougal Geometry
2-6 Geometric Proof
Right Angle Congruence Theorem
All right angles are congruent.
Holt McDougal Geometry
2-6 Geometric Proof
Congruent ComplementsTheorem
If two angles are complementary to the same angle, then the two angles are congruent.
Holt McDougal Geometry
2-6 Geometric Proof
Use the given plan to write a two-column proof.
Example 3: Writing a Two-Column Proof
Given: 1 and 2 are supplementary, and
1 3
Prove: 3 and 2 are supplementary.
Holt McDougal Geometry
2-6 Geometric Proof
Example 3 Continued
Statements Reasons
1. 1.
2. 2. .
3. . 3.
4. 4.
5. 5.
1 and 2 are supplementary.
1 3
Given
m1 + m2 = 180° Def. of supp. s
m1 = m3
m3 + m2 = 180°
3 and 2 are supplementary
Def. of s
Subst.
Def. of supp. s
Holt McDougal Geometry
2-6 Geometric Proof
Use the given plan to write a two-column proof if one case of Congruent Complements Theorem.
Check Up
Given: 1 and 2 are complementary, and
2 and 3 are complementary.
Prove: 1 3
Holt McDougal Geometry
2-6 Geometric ProofCheck It Out! Example 3 Continued
Statements Reasons
1. 1.
2. 2. .
3. . 3.
4. 4.
5. 5.
6. 6.
1 and 2 are complementary.
2 and 3 are complementary.
Given
m1 + m2 = 90° m2 + m3 = 90°
Def. of comp. s
m1 + m2 = m2 + m3
m2 = m2
m1 = m3
Subst.
Reflex. Prop. of =
Subtr. Prop. of =
1 3 Def. of s
Holt McDougal Geometry
2-6 Geometric Proof
Warm Up
Write each definition as a biconditional.
A.) A pentagon is a five – sided polygon.
B.) A right angle measures 90°.
Holt McDougal Geometry
2-6 Geometric Proof
Write a justification for each step, given that mABC = 90° and m1 = 4m2.
1. mABC = 90° and m1 = 4m2
2. m1 + m2 = mABC
3. 4m2 + m2 = 90°
4. 5m2 = 90°
5. m2 = 18°
Check for Understanding!!
Given
Add. Post.
Subst.
Simplify
Div. Prop. of =.
Holt McDougal Geometry
2-6 Geometric Proof
2. Use the given plan to write a two-column proof.
Given: 1, 2 , 3, 4
Prove: m1 + m2 = m1 + m4
Check for Understanding
1. 1 and 2 are supp.
1 and 4 are supp.
1. Linear Pair Thm.
2. m1 + m2 = 180°, m1 + m4 = 180°
2. Def. of supp. s
3. m1 + m2 = m1 + m4 3. Subst.
Holt McDougal Geometry
2-6 Geometric Proof
Assignment
Pg. 114 6 – 8 all
Holt McDougal Geometry
2-6 Geometric Proof
Warm Up
Complete each sentence.
1. If the measures of two angles are ____________, then the angles are congruent.
2. If two angles form a ___________, then they are supplementary.
3. If two angles are complementary to the same angle, then the two angles are _____________.