Homework #1 on Proofs

Geometry 1Name: ______

Homework #1 on Proofs

The following Definitions, Postulates and Theorems can be used in this assignment:

Definition of Midpoint: M is the midpoint of if and only if M is between A and B and .

Definition of Bisects: A line bisects a segment if it intersects the segment at its midpoint, and a ray bisects an angle if it is interior to the angle, its vertex coincides with the vertex of the angle, and it forms two congruent angles.

Definition of Complementary: Two angles are complementary if their measures add to 90o.

Definition of Supplementary: Two angles are supplementary if their measures add to 180o.

Definition of Adjacent Angles: Two angles are adjacent if they have a common side.

Definition of Linear Pair: Two angles form a linear pair if they are adjacent and their non-common sides are opposite rays (form a line).

Definition of Parallelogram: A quadrilateral is a parallelogram if opposite sides are parallel.

Transitive: If two segments or angles are congruent to a third segment or angle then they are congruent to each other.

Vertical Angle Theorem: Vertical angles are congruent.

Linear Pair Theorem: The angles in a linear pair are supplementary..

Parallel Postulate: Given a point and a line not containing that point, there is one and only one line through the given point that is parallel to the given line.

Parallel Lines:

Two lines cut by a transversal are parallel if and only if corresponding anglesare congruent.

Two lines cut by a transversal are parallel if and only if alternate interior anglesare

congruent.

Two lines cut by a transversal are parallel if and only if alternate exterior anglesare

congruent.

Two lines cut by a transversal are parallel if and only if same-side interior anglesare

supplementary.

Two lines cut by a transversal are parallel if and only if same-side exterior anglesare

supplementary.

Problems:Mark the diagrams according to the given information, and fill in the missing statements or reasons for each of the following proofs.

1.Given: and are supplementary.

Prove: and are supplementary.

Proof:

Statement: / Reason:
1. / and are supplementary. / Given
2. /
3. / and are supplementary.

2.Given: and

Prove: bisects

Proof:

Statement: / Reason:
1. / / Given
2. /
3. /
4. / bisects

3.Given: and

Prove:

Proof:

Statement: / Reason:
1. / / Given
2. /
3. /
4. /

4.Given: , , and

Prove:

Proof:

Statement: / Reason:
1. /
2. /
3. /
4. /
5. /

5.Given: and

Prove: ACDE is a parallelogram.

Proof:

Statement: / Reason:
1. /
2. / Two lines cut by a transversal are parallel if and only if alternate interior anglesare congruent.
3. / Given
4. /
5. / ACDE is a parallelogram.

6.Given: and

Prove:

Proof:

Statement: / Reason:
1. /
2. /
3. /
4. /
5. /
6. / Two lines cut by a transversal are parallel if and only if alternate interior anglesare congruent.

7.Given: bisects and

Prove:

Proof:

Statement: / Reason:
1. / bisects
2. / Definition of “bisects.”
3. /
4. / / Transitive Property
5. /

8.Prove that the angles of a triangle add to 180o:

Given: with

Prove: a + b + c = 180o

Proof:

Statement: / Reason:
1. / Let be the line containing point B that is parallel to .
2. / Let and / Definition of d and e.
3. / d = a
4. / These are alternate interior angles for parallel lines and with transversal .
5. / and are adjacent angles. / Definition of adjacent angles
6. / / Angle Addition Postulate
7. / and are a linear pair. / Definition of linear pair
8. / and are supplementary.
9. /
10. / d + b + e= 180o / Substitution using steps 6 and 9.
11. / a + b + c= 180o / Substitution using steps 3, 4 and 10.

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