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Proving Angles Congruent

Learn to prove angles congruent using vertical, adjacent, complementary, and supplementary angles, identify angle pairs, make conclusions from diagrams, and understand key angle theorems through paragraph proofs.

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Proving Angles Congruent

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  1. Proving Angles Congruent

  2. Proving Angles Congruent • Vertical Angles: Two angles whose sides form two pairs of opposite rays; form two pairs of congruent angles 1 2 4 <1 and <3 are Vertical angles <2 and <4 are Vertical angles 3

  3. Proving Angles Congruent • Adjacent Angles: Two coplanar angles that share a side and a vertex 1 2 1 <1 and <2 are Adjacent Angles 2

  4. 2.5 Proving Angles Congruent • Complementary Angles: Two angles whose measures have a sum of 90° • Supplementary Angles: Two angles whose measures have a sum of 180° 50° 2 40° 1 105° 75° 3 4

  5. Identifying Angle Pairs In the diagram identify pairs of numbered angles that are related as follows: • Complementary • Supplementary • Vertical • Adjacent 2 1 3 5 4

  6. 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

  7. 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

  8. Theorems About Angles Theorem 2-1 Vertical Angles Theorem Vertical Angles are Congruent Theorem 2-2 Congruent Supplements If two angles are supplements of the same angle or congruent angles, then the two angles are congruent

  9. Theorems About Angles Theorem 2-3 Congruent Complements If two angles are complements of the same angle or congruent angles, then the two angles are congruent Theorem 2-4 All right angles are congruent Theorem 2-5 If two angles are congruent and supplementary, each is a right angle

  10. 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. 1 3 2

  11. 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 __________.

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