Math-2 Lesson 5-3 Two Column Proofs
Vocabulary Adjacent Angles have a common side and share a common vertex Vertex. B C D A Common Side
A Two-Column Proof is a logical argument written so that the 1st column contains a statement and the 2nd column provides a justification for the truthfulness of the statement. Statement Justification (reason) A drawing is NOT a proof!!!
The justification may be: 1. A definition (of adjacent angles, vertical angles, etc.) 2. Theorems 3. Postulates/Axioms 4. Mathematical properties 5. Mathematical operations. 6. Substitution (combining previously proven statements) 7. Information provided or given. Statement Justification (reason) A drawing is NOT a proof!!!
The Side Angle Side Triangle Congruence Theorem If two pairs of sides of two triangles are congruent and the two included angles are congruent then the two triangles are congruent. To prove triangle congruence using this theorem, we would have to show that the hypothesis is true (the statement after the if statement). 1. We have two pairs of congruent sides 2. The included angles between those two sides are congruent.
The following figure is given: We will prove that the two triangles are congruent. C B A D E Statement C F BC DF AC FE F is include between sides DF and EF Justification (reason) given in figure given in figure given in figure given in figure C is include between sides AC and BC given in figure ABC FDE SAS congruence theorem A drawing is NOT a proof!!!
The Angle Side Angle Triangle Congruence Theorem If two pairs of angles of two triangles are congruent and the included sides are congruent then the two triangles are congruent. To prove triangle congruence using this theorem, we would have to show: 1. We have two pairs of congruent angles 2. The included sides between those two sides are congruent.
The following figure is given: Fill in the blanks of the proof. C B A Z Y Statement Justification (reason) A Y given in figure B Z given in the figure AB YZ side AB is included between A and B side YZis included between Y and Z given in figure given in figure given in figure ABC YZX ASA congruence theorem A drawing is NOT a proof!!!
Angle Addition Postulate If two angles are adjacent, then their measures can be added. Therefore: Vertex. D B C m ABC m CBD m ABD A Common Side
Straight Angle is formed by two opposite rays (that are collinear) and whose measure is 180. A B C Supplementary Angles are any two angles whose measures add up to 180.
Linear Pair of Angles are adjacent angles with two sides that form a straight angle. A B C D Linear Pair Theorem: If two angles form a linear pair, then they are supplementary.
A B D C Prove the Linear Pair Theorem (that Linear Pairs of angles are supplementary)...... Statement Justification (reason) ABC is a straight angle. given in figure m ABC 180 Definition of a straight ABD DBC ABC Angle addition Postulate m m ABD m m DBC m m 180 angle. substitution (steps 2 and 3) ABD and DBC are a a linear pair definition of a linear pair Linear Pairs are supplementary. QED A drawing is NOT a proof!!!
Q.E.D. is an initialism of the Latin phrase quod erat demonstrandum, meaning "which is what had to be proven".
Vertical Angles are formed by two intersecting lines. They are the angle pairs that share a vertex but do not share any sides. A D B C ABD and CBE are vertical angles. E CBD and ABE are vertical angles. Vertical Angle Theorem: If two angles are vertical angles, then they are congruent (have the same measure).
Given the two intersection lines, Prove the Vertical Angle Theorem 3 1 2 Statement Justification (reason) Linear Pair Theorem Linear Pair Theorem m 1 m 2 180 m 2 m 3 180 m 1 m 2 m 2 m 3 substitution (steps1and 2) m 1 and 3 are vertical angles. Vertical equality m 1 3 Property of angles are congruent. Def n of vertical angles QED
Transversal line: A line that intersects two other lines (usually parallel lines). 8 5 1 4 6 7 3 2 (not counting straight angles), eight angles are formed.
Corresponding Angles: pairs of angles that are in the same relative position at the two intersections. 1 4 3 2 1, 5 8 5 7 6 Name the three other corresponding angle pairs. 2, 6 3, 7 4, 8
Alternate Interior Angles: pairs of angles that are in between the parallel lines and on alternate sides of the transversal. 1 4 3 2 4, 6 8 5 7 6 Name the one other alternate interior angle pair. 3, 5
Alternate Exterior Angles: pairs of angles that are outside the parallel lines and on alternate sides of the transversal. 1 4 3 2 1, 7 8 5 7 6 Name the one other alternate exterior angle pair. 2, 8
Consecutive Interior Angles: pairs of angles that are in between the parallel lines and are on same side of the transversal. 1 4 3 2 3, 6 8 5 7 6 Name the one other consecutive interior angle pair. 4, 5
Corresponding Angles Postulate: If two parallel lines are cut by a transversal, then Corresponding angles are congruent. 1 4 3 2 m 1 m 5 m 2 m 6 8 5 7 6 m 3 m 7 m 4 m 8
Use the Corresponding Angles Postulate, Linear Pair Theorem, and Vertical Angle Theorem to prove that Alternate Interior Angles are congruent. 1 2 4 3 5 6 8 7 Two parallel lines are cut by a transversal m m 1 3 m m 1 5 m m 3 5 3, 5 are Alt Int Angles Alternate Interior Angles are congruent. Given in the figure Vertical Angles Theorem Corresponding Angles Postulate Substitution (steps 1 and 2) Definition of Alt Int Angles QED
Use the Corresponding Angles Postulate, Linear Pair Theorem, and Vertical Angle Theorem to prove that Alternate Exterior Angles are congruent. 1 2 4 3 5 6 8 7 Two parallel lines are cut by a transversal m m 1 5 m m m m 5 7 1 7 1, 7 are Alt Ext Angles Alternate Exterior Angles are congruent. Given in the figure Corresponding Angles Postulate Vertical Angles Theorem Substitution (steps 1 and 2) Definition of Alt ext Angles QED
Use the Corresponding Angles Postulate, Linear Pair Theorem, and Vertical Angle Theorem to prove that Consecutive Interior Angles are supplementary. 1 2 4 3 5 6 8 7 Two parallel lines are cut by a transversal m 1 m 4 m 1 m 5 m 4 m 5 4, 5 are Cons. 180 180 Int Angles Consec Int Angles are supplementary Given in the figure Linear Pair Theorem Corresponding Angles Postulate Substitution (steps 1 and 2) Definition of Cons Int Angles QED
One pair of parallel lines 3 1 4 2 8 5 7 6
Two pairs of parallel lines 3 1 4 2 13 14 16 15 8 5 7 6 9 10 12 11
What sequence of angles would you link to prove m 1 m 11 3 1 4 2 13 14 16 15 Alternate Exterior Alternate Interior 5 6 9 10 Vertical 8 7 12 11
What sequence of angles would you link to prove m 1 m 11 3 1 4 2 13 14 16 15 Corresponding Corresponding 5 6 9 10 Vertical 8 7 12 11
What sequence of angles would you link to prove m 4 m 9 3 1 4 2 13 14 16 15 Alternate Interior Corresponding 8 5 7 6 9 10 12 11
What sequence of angles would you link to prove m 4 m 10 180 3 1 4 2 13 14 16 15 Consecutive Interior Corresponding 5 6 9 10 Substitution 8 7 12 11