2.6 Proven Angles Congruent. Objective: To prove and apply theorems about angles. 2.6 Proven Angles Congruent.

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Presentation transcript:

2.6 Proven Angles Congruent

Objective: To prove and apply theorems about angles. 2.6 Proven Angles Congruent

Vocabulary: Theorem Vertical Prove Apply Paragraph Proof Congruent 2.6 Proven Angles Congruent

Solve it! 2.6 Proven Angles Congruent

Vertical Angles:  Theorem 2.1: Two angles whose sides form two pairs of opposite rays; form two pairs of congruent angles <1 and <3 are Vertical angles <2 and <4 are Vertical angles and 2.6 Proven Angles Congruent

Vertical Angles: 2.6 Proven Angles Congruent

Vertical Angles: 2.6 Proven Angles Congruent

Adjacent Angles:  Two coplanar angles that share a side and a vertex <1 and <2 are Adjacent Angles 2.6 Proven Angles Congruent

Complementary Angles:  Two angles whose measures have a sum of 90°  Two angles whose measures have a sum of 180° ° 40° 34 75° 105° Supplementary Angles: 2.6 Proven Angles Congruent

Identifying Angle Pairs: In the diagram identify pairs of numbered angles that are related as follows: a. Complementary b. Supplementary c. Vertical d. Adjacent Proven Angles Congruent <2 and <3 <4 and <5 <3 and <5 <1 and <2 <2 and <3 <3 and <4 <4 and <5

Making Conclusions: Whether you draw a diagram or use a given diagram, you can make some conclusions directly from the diagrams. You CAN conclude that angles are  Adjacent angles  Adjacent supplementary angles  Vertical angles 2.6 Proven Angles Congruent

Making Conclusions: Unless there are markings that give this information, you CANNOT assume  Angles or segments are congruent  An angle is a right angle  Lines are parallel or perpendicular 2.6 Proven Angles Congruent

Theorems About Angles 2.6 Proven Angles Congruent

Theorems About Angles 2.6 Proven Angles Congruent

Theorems About Angles All these theorems can be proven using Deductive Reasoning. 2.6 Proven Angles Congruent

Proving Theorems Paragraph Proof: Written as sentences in a paragraph Given: <1 and <2 are vertical angles Prove: <1 = <2 Paragraph Proof: By the Angle Addition Postulate, m<1 + m<3 = 180 and m<2 + m<3 = 180. By substitution, m<1 + m<3 = m<2 + m<3. Subtract m<3 from each side. You get m<1 = m<2, which is what you are trying to prove Proven Angles Congruent

Proving Theorems Given: <1 and <2 are supplementary <3 and <2 are supplementary Prove:<1 = <3 Proof: By the definition of supplementary angles, m<___ + m<____ = _____ and m<___ + m<___ = ____. By substitution, m<___ + m<___ = m<___ + m<___. Subtract m<2 from each side. You get __________ <1 = < 3 You might want to draw a picture first! 2.6 Proven Angles Congruent

Lesson Check 2.6 Proven Angles Congruent

Lesson Check 2.6 Proven Angles Congruent