GeometryLesson Notes2.8Date ______
Objective: Solve problems and write geometric proofs involving angles.
In chapter 1.4, we measured the angles with a protractor.
Protractor Postulate: Given a ray and a number r between 0 and 180, there is exactly one ray with endpoint A, extending on either side of, such that the measure of the angle formed is r.
Remember, angles can be acute, obtuse, right, or straight.
Angle Addition Postulate:
If R is in the interior of PQS, then mPQR + mRQS = mPQS.
Likewise,If mPQR + mRQS = mPQS, then R is in the interior of PQS.
This theorem, and the two that follow, justify what we did in chapter 1 when we assumed that
the whole is equal to the sum of its parts. Geometry is based on rules. We are exploring and developing more of the rules.
Example 1 (p 107): Angle Addition
Given: mRXQ = x + 8, mRXT = 7x − 6, mQXT = 10x − 30;
Find x and the measure of mQXT.
Theorem 2.3 Supplement Theorem: If two angles form a linear pair, then they are
supplementary angles.
Abbrev: Linear pairs of s are suppl.
Prove theorem 2.3.
Given: 1 and2 form a linear pair
Prove: 1 and 2 are supplementary s
ABC is straight ; defn of linear pair
mABC = 180; defn of straight
m1 + m2 = mABC: angle addition postulate
m1 + m2 = 180; substitution
1 and 2 are supplementary s; defn of supplementary s
Example 2 (p 108): Supplementary Angles
If 1 and 2 form a linear pair, and m1 = 2x − 7, m2 = 25 + x,find x and the
measure of each angle.
Theorem 2.4 Complement Theorem: If the noncommon sides of two adjacent angles form
a right angle, then the angles are complementary angles.
Draw a picture.
Theorem 2.5: Properties of Congruence for Angles
Congruence of angles is reflexive, symmetric, and transitive.
Reflexive Property of Congruence:1 1
Symmetric Property of Congruence:If1 2, then 2 1
Transitive Property of Congruence:If 1 2 and2 3, then1 3
Theorem 2.6: Angles supplementary to the same angle or to congruent anglesare congruent.
Abbreviation: s suppl.to same or s are
Prove theorem 2.6.
Theorem 2.7: Angles complementary to the same angle or to congruent anglesare congruent.
Abbreviation: s compl. to same or s are
Practice: Identify congruent angles.
Example 3 (p 109): Use Supplementary Angles
In the figure,1 and 2 form a linear pair and
2 and 3 form a linear pair. Prove that
1 and 3 are congruent.
Proof:
Given: 1 and 2 form a linear pair
2 and 3 form a linear pair
Prove: 1 3
Think about it:
Implication! the angles of a linear pair are supplementary (by theorem).
Looking at the picture, I clearly see that 2 is part of each linear pair so it’s
supplementary to 1 and to 3. I have a plan!
StatementsReasons
1.1 and 2 form a linear pair 1. Given
2 and 3 form a linear pair
2.1 and 2 are supplementary 2. Supplement Theorem
2 and 3 are supplementary(Linear pairs of s are suppl.)
3.1 33. s suppl. to same or s are
We’ve proved a theorem which justifies what we said in chapter 1.
Theorem 2.8 Vertical Angles Theorem: If two angles are vertical angles, then they are
congruent.
Abbreviation: Vert. s are
Example 4 (p 110): Vertical Angles
If 1 and 2 are vertical angles and m1 = d – 32d and m2 = 175 – 2d,find d and
the measure of 1 and 2.
Remember: Draw a picture!
Theorem 2.9: Perpendicular lines intersect to form four right angles.
Theorem 2.10: All right angles are congruent.
Theorem 2.11: Perpendicular lines form congruent adjacent angles.
Theorem 2.12: If two angles are congruent and supplementary, then eachangle is aright angle.
Theorem 2.13: If two congruent angles form a linear pair, then they are rightangles.
HW: A9a pp 112-113 #16-24, 27-32
A9b pp 112-114 #7, 38-39*, 44-47 * number the angles on your paper!
A9c fms.Geometry Worksheet 2.8 - Proofs
fms-Geometry Lesson Notes 2.8Page 1 of 5