Proving Angles Congruent Geometry Honors

Vocabulary Vertical Angles – two angles whose sides form two pairs of opposite rays. 1 2

Vocabulary Adjacent Angles – two coplanar angles with a common side, a common vertex, and no common interior points. 1 2 1 2

Vocabulary Complementary Angles – two angles whose measures have a sum of 90. 2 2 1 1

Vocabulary Supplementary Angles – two angles whose measures have a sum of 180. 2 1 1 2

A picture is worth a 1000 words…. Conclusions can be drawn from pictures if you see… Adjacent Angles Adjacent Supplementary angles Vertical Angles

Conclusions that CANNOT be drawn from pictures Congruent Angles Congruent Segments Right Angles Parallel Lines Perpendicular Lines All of these must be marked to declare.

What conclusions can you draw from this picture? 3 4 2 5 1

Vocabulary Postulate– an accepted statement of fact. Theorem– a proven statement. Proof– the steps taken to show that a conjecture is true.

Theorems 1 2 Vertical Angles Theorem - Vertical angles are congruent. 1  2

Theorems Congruent Supplements Theorem - If two angles are supplements of the same angle (or of congruent angles), then the two angles are congruent. 2 1 3 If 1 is supplementary to 2, and 3 is supplementary to 2, then 1  3.

Theorems Congruent Complements Theorem - If two angles are complements of the same angle (or of congruent angles), then the two angles are congruent. 2 1 3 If 1 is complementary to 2, and 3 is complementary to 2, then 1  3.

Theorems If two angles are congruent and supplementary, then each is a right angle. 2 3

Theorems All right angles are congruent. 2 3 2  3

What theorem did you use? Applying Theorems Find x. 4x Answer x = 5 3x + 5 What theorem did you use? Theorem Vertical Angle Theorem

What theorem did you use? Applying Theorems Find x. 3x Answer 80 - x x = 20 What theorem did you use? Theorem Vertical Angle Theorem

What reasons did you use? Applying Theorems Find x and y. 3x + 8 5x - 20 5x + 4y Answer x = 14 y = 15 What reasons did you use? Reasons Vertical Angle Theorem & Definition of Supplementary

Applying Theorems Find x. x = 9 54 4x What reason did you use? Answer Definition of Complementary

Applying Theorems Find x. x = 10 12x - 15 3x + 45 Answer What reason did you use? Reason Definition of Supplementary

What reasons did you use? Applying Theorems Find x. 3x 2x Answer x = 18 What reasons did you use? Reasons Vertical Angles Theorem & Definition of Complementary

Applying Theorems Find x. 4x + 5 x = 11 3x + 8 Answer x = 11 What reason did you use? Reason Definition of Complementary

What reasons did you use? Applying Theorems x + y + 5 2x Find x and y. y x = 35 y = 70 Answer What reasons did you use? Reason Vertical Angles Theorem, Substitution, & Def. of Supplementary

A and B are supplementary. Applying Theorems A and B are supplementary. mA = 3x + 12 and mB = 2x – 22. What is the measure of each angle? Answer x = 38 mA =126 mB = 54

Applying Theorems A is half as large as its complement, B . Find the measure of each angle? mA = 30 mB = 60 Answer