2.3 Angle Relationships

2.3 Angle Relationships

Angle Relationships

In this section, we will be looking at a number of theorems regarding the relationship between angles. While some of these theorems MAY seem a bit simple, remember that Geometry is all built out from the three basic units (point, line, and plane), everything else needs to be defined and proven from there.

Similar to the ruler postulate, the protractor postulate simply states that we can measure angles (and that for our purposes, those angle measures are positive - no matter which direction we measure from - which is why there are two set of numbers on a protractor!).

Multiple Choice

What is the measure of

the given angle?

• 21°

• 159°

• 19°

• 161°

Multiple Choice

What is the measure of

the given angle?

• 82°

• 102°

• 100°

• 88°

Also like its counterpart, the segment addition postulate from yesterday, the angle addition postulate is really telling us that the sum of the measure of two angles make the measure of the larger angle.

The angle addition postulate can be combined with the ideas of complementary (add to 90 deg.) and supplementary (add to 180 deg.) to give us the supplement theorem and complement theorem. These are just special cases of the angle addition postulate where the sum happens to be 90 or 180. Because these have been proven to always be true, they are considered theorems and not just the angle addition postulate.

Multiple Choice

• 50°

• 58°

• 115°

• 226°

Multiple Choice

• x = 6°

• x = 7°

• x = 12°

• x = 144°

Multiple Choice

Calculate m∠DEF.

• 180°

• 60°

• 120°

• 80°

Multiple Choice

Calculate m∠ABC.

• 45°

• 60°

• 105°

• 15°

Multiple Choice

Find the value of x.

• 18°

• 28°

• 38°

• 118°

The first words (reflexive, symmetric, and transitive) should be a bit familiar!

These two theorems are really just extensions of the Transitive property of congruence....but someone felt that it was necessary to call your attention to them specifically.

Remember when we talked about the type of angles in the last unit, we mentioned that vertical angles are always congruent. **This is a theorem that you MIGHT be asked to prove on the summative :-)

These 5 theorems might all seem like common sense, but remember that they have been proven so that we can build on only thoughts that are supported by proof or postulate.

Multiple Choice

Find the missing angle.

• 60°
• 70°
• 50°
• 140

Multiple Choice

What is m∠ABD

• 80
• 120
• 20
• 100

Multiple Choice

Find the value of ∠A.

• 79 degrees

• 87 degrees

• 90 degrees

• 97 degrees

Multiple Choice

Find the value of ∠A.

• 25 degrees
• 30  degrees
• 35 degrees
• 90 degrees

Multiple Choice

What is the justification?

• Vertical Angles Theorem

• Transitive Property of Equality

• Compliment Theorem

• Reflexive Property of Congruence

Multiple Choice

Multiple Choice

What is the reasoning?

• Symmetric Property

• Angle addition postulate

• Reflexive Property

• Transitive Property

Multiple Choice

Given: <1 and <3 are vertical angles. What should you conclude by the vertical angles theorem?

• <1 and <2 are a linear pair

• <1 and <3 are right angles

• <2 and <4 are vertical angles

•  $<1\cong<3$  

Multiple Choice

Give the reason for statement #3

• Definition of supplementary angles
• Definition of congruence
• Vertical angles theorem
• Congruent supplements theorem 

Multiple Choice

Match the correct words into the blanks for statement 5 and statement 7.

• Substitution; Reflexive Property

• Definition of Supplementary Angles; Definition of Congruence

• Transitive Property; Definition of Congruence

• Definition of Complementary Angles; Reflexive Property

Multiple Choice

Give the reason for statement #6

• Definition of supplementary angles

• Substitution Property of Equality

• Vertical angles theorem

• Congruent supplements theorem