SECTION 2-5Angle Relationships

You have now experienced using inductive reasoning. So we will begin discovering geometric relationships using inductive reasoning along with your geometric tools.

Investigation: The Linear Pair Conjecture• Step 1: On a sheet of paper, draw and place a point

R between P and Q. Choose another point S not onand draw . You have just created a linear pair of angles. Place the “zero edge” of your protractor along . What do you notice about the sum of the measures of the linear pair of angles?

• Step 2: Compare your results with those of your group. Does everyone make the same observation? Complete the conjecture.

PQsuur

PQsuur

RSsuur

PQsuur

If two angles form a linear pair, then

the measures of the angles add up to 180o.

Investigation: Vertical Angles Conjecture• Step 1: Draw two intersecting lines on a sheet of patty

paper. Label the angles as shown. Which angles are vertical angles?

• Step 2: Fold the paper so that the vertical angles lie over each other. What do you notice about their measures?

• Step 3: Fold the paper so the other pair of vertical angles lie over each other. What do you notice about their measures?

• Step 4: Compare your results with the results of others. Complete the conjecture.

If two angles are vertical, then

they have equal measure or they are congruent.

Example

• Use the Linear Pair conjecture and the diagram at the right to write a deductive argument explaining why angle 1 must be congruent to angle 3.

You discovered the Vertical Angle Conjecture: If two angles are vertical angles, then they are congruent.

Does this mean that all congruent angles are vertical angles? This is the converse of the Vertical Angle Conjecture. Is it true?

• In some of the previous lessons you have been using inductive reasoning by making conjectures based on patterns you have observed. But when you made a conjecture, the discovery process that led to the conjecture did not always help you explain why the conjecture works.

• Deductive Reasoning is the process of showing that certain statements follow logically from agreed=upon assumptions and proven facts.

• When you use deductive reasoning, you try to reason in an orderly way to convince yourself or someone else that your conclusion if valid.

• In a trial, lawyers use deductive arguments to show how the evidence that they present proves their case. A lawyer might make a good argument. But first, the court must believe the evidence and accept it as true.

• You use deductive reasoning in algebra. When you provide a reason for each step in the process of solving an equation, you are using deductive reasoning.

Solve the equation for x. Give a reason for each step in the process.

3(2x+1) +2(2x+1) + 7 = 42 – 5x

EXAMPLE

The next example show how to use both kinds of reasoning:

Inductive reasoning to discover the property

and

Deductive reasoning to explain why it works.

EXAMPLEIn each diagram, bisects obtuse angle BAD. Classify as acute, right or obtuse. Then complete the conjecture.

ACuuur

, ,BAD DAC and CAB

Conjecture: If an obtuse angle is bisected, then the two newly formed congruent angles are acute angles.

60o

60o79o

79o46o 46

o

All the newly formed angles are acute angle.

Deductive Argument

To explain why this is true, a useful reasoning strategy is to represent the situation algebraically. Let m represent the measure of any obtuse angle.

By definition, an angle measure is less than 180o.

180om

When you bisect an angle, the newly formed angles each measure half the original angle.

1 1180

2 2om 1

902

om

The new angles measure less than 90o, so they are acute angles.

Good use of deductive reasoning depends on the quality of the argument. A conclusion in a deductive argument is true only if all the statements in the argument are true and the statements in your argument clearly follow from each other.

Inductive and deductive reasoning work well together.

In the next investigation you will use inductive reasoning to form the conjecture.Then, in your groups, you will use deductive reasoning to explain why it’s true.

Overlapping Segments

AB CD• In each segment, .

• From the markings on each diagram, determine the lengths of and . What do you discover about these segments?

AC BD

• Draw a new segment. Label it . Place your own points B and C on so that .

ADAD AB CD

Measure and . How do these lengths compare?

Complete the conclusion of this conjecture:

AC BD

If has points A, B, C, and D in that order with , then AC

AB CD

In the investigation you used inductive reasoning to discover the Overlapping Segments Conjecture. In your group discussion you then used deductive reasoning to explain why this conjecture is always true. You will use a similar process to discover and prove the overlapping angles conjecture in exercises 10 and 11.