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Geometry, 1.4 Notes – Simple Proofs, Exams of Geometry

Geometry

We prove mathematical statements using Two-column proofs: Examples: Theorem 1: If two angles are right angles, then they are congruent. Statement.

Typology: Exams

2022/2023

Uploaded on 02/28/2023

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Geometry, 1.4 Notes – Simple Proofs
What is a proof? It is a step-by-step argument to convince someone that something is true. It gives reasons for
each step in the argument.
Example: You want to convince your mom to let you go to the mall.
Statement Reason
1. I cleaned my room. 1. Fact
2. You gave me $10. 2. I get $10 if I clean my room.
3. I get to go to the mall. 3. You said if I had money, I could go to the mall.
A Theorem is: _________________________________________
We prove mathematical statements using Two-column proofs:
Examples:
Theorem 1: If two angles are right angles, then they are congruent.
Statement | Reason
1. | 1.
2. | 2.
3. | 3.
4. | 4.
5. | 5.
pf3
pf4
pf5
pf8

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Geometry, 1.4 Notes – Simple Proofs

What is a proof? It is a step-by-step argument to convince someone that something is true. It gives reasons for each step in the argument.

Example: You want to convince your mom to let you go to the mall. Statement Reason

  1. I cleaned my room. 1. Fact
  2. You gave me $10. 2. I get $10 if I clean my room.
  3. I get to go to the mall. 3. You said if I had money, I could go to the mall.

A Theorem is: _________________________________________

We prove mathematical statements using Two-column proofs: Examples: Theorem 1: If two angles are right angles, then they are congruent.

Statement | Reason

  1. | 1.
  2. | 2.
  3. | 3.
  4. | 4.
  5. | 5.

Theorem 2: If two angles are straight angles, then they are congruent.

Statement | Reason

  1. | 1.
  2. | 2.
  3. | 3.
  4. | 4.
  5. | 5.

Geometry, 1.5 Notes – Division of segments and angles

Bisect means _________________________________________________ Trisect You can bisect a line segment. means _________________________________________________ The point that divides the segment in half is called the _________________

Line segments can have a midpoint, but rays and lines cannot:

Midpoints have to be collinear with line segment endpoints:

Geometry, 1.7 Notes – Deductive Structure

Deductive Structure is a ___________________ in which ___________________ are justified

by _______________________________________ statements.

A deductive structure contains:

1) Undefined terms – e.g.: lines, points.

2) Postulates , which are _____________________.

e.g. Two points make a straight line.

3) Theorems , which are _____________________________.

e.g. If two angles are right angles, then they are congruent.

4) Definitions , which state the ________________________________.

e.g. If points lie on the same line, then they are collinear.

Definitions are always reversible: If points lie on the same line, then they are collinear.

If points are collinear, then they lie on the same line.

Postulates and Theorems are not always reversible:

If two angles are right angles, then they are congruent.

If two angles are congruent, then they are right angles.

Many postulates, theorems, and definitions are in the form of a conditional statement:

If p , then q symbolized by:

Examples of conditional statements:

If it is raining, then it is cloudy.

If two angles have the same measure, then they are congruent.

Converse: A statement with the 'if' and 'then' reversed

Examples:

Statement: If 2 angles are right angles, then they are congruent.

Converse: If 2 angles are congruent, then they are right angles.

Statement: If a person is female, then the person is a girl.

Converse: If a person is a girl, then the person is female.

Try it...

Write the converse of each statement. Is the converse true?

#1. If a person is a boy, then the person is male.

#2. If a person was born 75 years ago, then the person is old.

#3. If an angle is a 45 degree angle, then it is acute.

#4. If a point is the midpoint of a segment, it divides the segment into two congruent segments.