Page 202 - Math Course 2 (Book 2)

Proving Angle Relationships

12.7 Angle complementary to the same angle
or the congruent angles are congruent. Let’s Begin
Abbreviation ⦞ compl. to same ∠ or ≅ ⦞ are ≅.

Example: If m∠1 + m∠2 = 90 and
m∠2 + m∠3 = 90, then Use Supplementary Angles
∠1 ≅ ∠3.
Example

1 2
1 4
3
2 3

Vertical Angles Theorem ∠1 and ∠4 form a linear pair.
Given:
12.8 If two angle are vertical angle, then they m∠3 + m∠1 = 180
are congruent. Prove: ∠3 ≅ ∠4
Proof:
Abbreviation Vert. ⦞ are ≅. Statements Reasons
1. m∠3 + m∠1 = 180
∠1 ≅ ∠3 and ∠2 ≅ ∠4 ∠1 and ∠4 form a 1. Given
linear pair
2.∠1 and ∠4 are 2. Linear pairs are
2 supplementary. supplementary.
1 3
4
3. ∠3 and ∠1 are 3. Definition of
supplementary supplementary angles
4. ∠’s suppl. to same ∠
Vertical Angles Theorem 4. ∠3 ≅ ∠4 are ≅.

12.9 Perpendicular lines intersect to from Vertical Angles
four right angles.

12.10 All right angles are congruent Example

12.11 Perpendicular line form congruent If ∠1 and ∠2 are vertical angles and m∠1 = d – 32
adjacent angles. and m∠2 = 175 – 2d, find m∠1 and m∠2.

12.12 If two angles are congruent and ∠1 ≅ ∠2 Vertical Angles Theorem
supplementary, then each angle is a m∠1 = m∠2 Definition of congruent angles
right angle.
d – 32 = 175 – 2d Substitution
12.13 If two congruent angle form a linear pair, 3d – 32 = 175 Add 2d to each side.
then they are right angles. 3d = 207 Add 32 to each side.
d = 69 Divide each side by 3.
m∠1 = d – 32 m∠2 = 175 – 2d
= 69 – 32 or 37 = 175 – 2(69) or 37

Answer m∠1 = 37 and m∠2 = 37

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