This Is to Give Students More Practice with Proofs

Chapter 4 Proof Practice

This is to give students more practice with proofs.

1)Print out the proofs on heavy paper or cardstock and put in a paper protector

2)Print out the justifications and statements on cardstock. (I have done statements on one color, justifications on another, or make it a little more challenging and have them all the same color – students should be able to figure out which is which).

3)Have students work on the proofs in groups or pairs. (I have several of each proof printed so there is enough for the whole class to work in pairs).

4)Either have students write down the proof once they are done or just check it off in some way so you know which ones they have completed.

5)Occasionally I had students waiting over another students shoulder for a particular number towards the end. But generally students were engaged and on task.

6)I ended up taking out the triangle congruence postulates as I expected students to know which one it was, you can do it either way. Also, you could give them just the statements or just the justifications to make it more challenging.
Chapter 4 Proof #1

Given: F is the midpoint of

Prove:

Proof 1 StatementsProof 1 Justifications

/ Given
/ Given
F is the midpoint of / Given
/ Defintion of a midpoint
A point is the midpoint if and only if it divides the segment into 2 equal parts.
/ SSS Postulate
If the three sides of one triangle are congruent to the three sides of another triangle, then the triangles are congruent.

Proof 2 statementsproof 2 justifications

/ Given
/ Given
/ Reflexive Property of Congruence
/ SSS Postulate
If the three sides of one triangle are congruent to the three sides of another triangle, then the triangles are congruent.

Chapter 4 Proof #2

Given:

Prove:
Chapter 4 Proof #3

Given: ,

Prove:
Proof 3 StatementsProof 3 Justifications

/ Given
/ Given
/ Reflexive Property of Congruence
/ SAS Postulate
If the two sides and the included angle of one triangle are congruent to the two sides and the included angle of another triangle, then the triangles are congruent.

Proof 4 StatementsProof 4 justifications

X is the midpoint of and / Given
/ Defintion of a midpoint
A point is the midpoint if and only if it divides the segment into 2 equal parts.
/ Defintion of a midpoint
A point is the midpoint if and only if it divides the segment into 2 equal parts.
/ Vertical Angles Theorem
Vertical Angles are congruent
/ SAS Postulate
If the two sides and the included angle of one triangle are congruent to the two sides and the included angle of another triangle, then the triangles are congruent.

Chapter 4 Proof #4

Given: X is the midpoint of and

Prove:
Chapter 4 Proof #5

Given: bisects ,

Prove:

Proof #5 StatementsProof #5 Justifications

/ Given
bisects / Given
/ Reflexive Property of Congruence
/ SAS Postulate
If the two sides and the included angle of one triangle are congruent to the two sides and the included angle of another triangle, then the triangles are congruent.
/ Definition of an Angle Bisector
A segment is the angle bisector if and only if it divides an angle into 2 congruent angles.

Proof #6 StatementsProof #6 Justifications

/ Definition of an Segment Bisector
A segment is the segment bisector if and only if it divides a segment into 2 segments.
and bisect each other / Given
/ Definition of an Segment Bisector
A segment is the segment bisector if and only if it divides a segment into 2 segments.
/ SAS Postulate
If the two sides and the included angle of one triangle are congruent to the two sides and the included angle of another triangle, then the triangles are congruent.
/ Vertical Angles Theorem
Vertical Angles are congruent

Chapter 4 Proof #6

Given: and bisect each other

Prove:
Chapter 4 Proof #7

Given: ;

Prove:

Proof #7 StatementsProof #7 Justifications

/ Given
/ Given
/ Reflexive Property of Congruence
/ SAS Postulate
If the two sides and the included angle of one triangle are congruent to the two sides and the included angle of another triangle, then the triangles are congruent.
/ Alternate Interior Angles Theorem
If two parallel lines are intersected by a transversal then alternate interior angles are congruent.

Proof #8 StatementsProof #8 Justifications

/ Given
/ Given
/ Reflexive Property of Congruence
/ ASA Postulate
If the two angles and the included side of one triangle are congruent to the two angles and the included side of another triangle, then the triangles are congruent.
/ All Right Angles are Congruent
and are right angles / Definition of Perpendicular Lines
Two segments are perpendicular if and only if they intersect to form 4 right angles.

Chapter 4 Proof #8

Given:

Prove:
Chapter 4 Proof #9

Given:

Prove:

Proof #9 StatementsProof #9 Justifications

/ Given
/ Given
/ Reflexive Property of Congruence
/ AAS Theorem
If the two angles and a non-included side of one triangle are congruent to the two angles and a non-included side of another triangle, then the triangles are congruent.
/ Definition of an Angle Bisector
A segment is the angle bisector if and only if it divides an angle into 2 congruent angles.

Proof #10 StatementsProof #10 Justifications

/ Given
/ Given
T is the midpoint of / Given
/ AAS Theorem
If the two angles and a non-included side of one triangle are congruent to the two angles and a non-included side of another triangle, then the triangles are congruent.
/ All Right Angles are Congruent
is a right angle / Definition of Perpendicular Lines
Two segments are perpendicular if and only if they intersect to form 4 right angles.
is a right angle / Definition of Perpendicular Lines
Two segments are perpendicular if and only if they intersect to form 4 right angles.
/ Defintion of a midpoint
A point is the midpoint if and only if it divides the segment into 2 equal parts.
/ Vertical Angles Theorem
Vertical angles are congruent

Chapter 4 Proof #10

Given:

Prove:

Chapter 4 Proof #11

Given: ;

Prove:

Proof #11

/ Given
/ Given
/ Vertical Angles are congruent
/ AAS Theorem
If the two angles and a non-included side of one triangle are congruent to the two angles and a non-included side of another triangle, then the triangles are congruent

Proof #12

/ Given
/ Given
/ Given
/ ASA Postulate
If the two angles and the included side of one triangle are congruent to the two angles and the included side of another triangle, then the triangles are congruent.
/ Corresponding Angles Postulate
If two parallel lines are intersected by a transversal then corresponding angles are congruent.

Proof #13

/ Given
/ Given
/ Reflexive Property of Congruence
/ ASA Postulate
If the two angles and the included side of one triangle are congruent to the two angles and the included side of another triangle, then the triangles are congruent.
/ Definition of an Angle Bisector
A segment is the angle bisector if and only if it divides the angle into two congruent angles.

Proof #12

Given:

Prove:
Proof #13

Given:

Prove: