2.8 Proving Angle Relationships

Objectives

Write proofs involving supplementary and complementary angles

Write proofs involving congruent and right angles

Postulates

Postulates 2.10 (Protractor Rule)All angles have measures between 0° and 180°.

Postulate 2.11 (Angle Addition Postulate) R is in the interior of PQS iff mPQR + mRQS = mPQS.

S

P

Q

R

Postulates and Theorems

Copy all of the Theorems from Section 2.8.

Theorems 2.3 thru 2.13

If the second hand stops where the angle is bisected,

then the angle between the minute and second hands is

one-half the measure of the angle formed by the hour

and minute hands, or .

TIME At 4 o’clock, the angle between the hour and minute hands of a clock is 120º. If the second hand stops where it bisects the angle between the hour and minute hands, what are the measures of the angles between the minute and second hands and between the second and hour hands?

Example 1:

By the Angle Addition Postulate, the sum of the two angles is 120, so the angle between the second and hour hands is also 60º.

Answer: They are both 60º by the definition of angle bisector and the Angle Addition Postulate.

Example 1:

Answer: 50

QUILTING The diagram below shows one square for a particular quilt pattern. If and is a right angle, find

Your Turn:

Supplement Theorem

Subtraction Property

Answer: 14

form a linear pair and find If and

Example 2:

Answer: 28

are complementary angles and .

andIffind

Your Turn:

Given: form a linear pair.

Prove:

In the figure, form a linear pair, and Prove that are congruent.

andand

Example 3:

1. Given1.

3. Definition of supplementary angles

3.

4. Subtraction Property4.

5. Substitution5.

6. Definition of congruent angles

6.

Proof:

Statements Reasons

2. Linear pairs are supplementary.

2.

Example 3:

1 & 4 linear pair;

In the figure, NYR and RYA form a linear pair,AXY and AXZ form a linear pair, and RYA and AXZ are congruent. Prove that RYN and AXY are congruent.

Your Turn:

Proof:

Statements Reasons

1. Given

2. If two s form a linear pair, then they are suppl. s.

3. Given

4.

1.

2.

3.

4.

linear pairs.

Your Turn:

Substitution

Add 2d to each side.

Add 32 to each side.

Divide each side by 3.

If 1 and 2 are vertical angles and m1 and m2 find m1 and m2.

Vertical Angles Theorem1 2

Definition of congruent anglesm1 m2

Example 4:

Answer: m1 = 37 and m2 = 37

Example 4:

Answer: mA = 52; mZ = 52

find andIf and are vertical angles and and

Your Turn:

Assignment

Geometry: Pg. 111 – 113 #6, 16 – 24, 27 - 32

Pre-AP Geometry:Pg. 111 – 113 #6, 16 – 24, 27 - 32 , 34,

36, 38