2.8 Notes: Proving Angle Relationships How can you prove a mathematical statement?
Vocab! Angle Addition Postulate Supplement Theorem Complement Theorem Reflexive Property of Angle Congruence D is the interior of ∠𝐴𝐵𝐶 if and only if 𝑚∠𝐴𝐵𝐷+𝑚∠𝐷𝐵𝐷=𝑚∠𝐴𝐵𝐶 If two angles form a linear pair, then they are supplementary angles If the noncommon sides of two adjacent angles form a right angle, then the angles are complementary angles ∠1≅∠1
Vocab! If ∠1≅∠2 then ∠2≅∠1 If ∠1≅∠2 and ∠2≅∠3 then ∠1≅∠3 Symmetric Property of Angle Congruence Transitive Property of Angle Congruence Congruent Supplement Theorem Congruent Complement Theorem If ∠1≅∠2 then ∠2≅∠1 If ∠1≅∠2 and ∠2≅∠3 then ∠1≅∠3 Angles supplementary to the same angle or to congruent angles are congruent Angles complementary to the same angle or to congruent angles are congruent
Example 1 ∠𝟏 & ∠𝟑 are complementary ∠𝟐 & ∠𝟑 are complementary Definition of complementary angles Definition of complementary angles Substitution Reflexive Property 𝒎∠𝟏=𝒎∠𝟐 ∠𝟏≅∠𝟐
Example 2 In the figure, ∠1 and ∠ 4 form a linear pair, and m ∠ 3 + m ∠ 1 = 180°. Prove that ∠ 3 and ∠ 4 are congruent. 𝒎∠𝟑+𝒎∠𝟏=𝟏𝟖𝟎 Given Supplement Theorem 𝒎∠𝟏+𝒎∠𝟒=𝟏𝟖𝟎 ∠𝟑≅∠𝟒 Congruent Supplement Theorem
Vertical Angles Theorem Vocab! Vertical Angles Theorem If two angles are vertical angles, then they are congruent.
Example 3 1 and 2 are vertical angles and m1 = (d – 32)° and m2 = (175 – 2d)°, find m1 and m2. Justify each step. ∠𝟏 & ∠𝟐 are vertical angles Vertical Angles Theorem Definition of Congruency 𝒅−𝟑𝟐=𝟏𝟕𝟓−𝟐𝒅 𝟑𝒅−𝟑𝟐=𝟏𝟕𝟓 𝟑𝒅=𝟐𝟎𝟕 Addition Property Division Property 𝒎∠𝟏=𝟑𝟕 Substitution
Right Angles Congruent Theorem Vocab! Right Angles Congruent Theorem Right angles are always congruent to one another 2.9 Perpendicular lines intersect to form four right angles 2.10 All right angles are congruent
Vocab! 2.11 Perpendicular lines form congruent adjacent angles 2.12 If two angles are congruent and supplementary, then each angle is a right angle 2.13 If two congruent angles form a linear pair, then they are right angles.