Download Do Now

+
Do Now

Put your homework on your desk ready to be stamped.

Take a protractor from the front.
1.
What is the relationship between the measure of an angle
and the reflex measure of an angle?
2.
Using a protractor, draw a clock face with hands to show
8:00.
3.
Measure the angle formed by your clock hands in #2 and
use correct notation to write the measure of the angle.
4.
Draw a figure that contains at least three angles and
includes an angle bisector. Mark any congruent angles.
+
Angle Relationships
+
Today’s Objectives

Discover relationships between pairs of angles

Perform investigations and write conjectures

Use conjectures to find missing angles measures.

Use problem solving skills
+
Using Inductive Reasoning to
Discover Geometry

“Discovery consists of looking at the same thing as everyone
else and thinking something different.” –Albert SzentGyorgyi
+
Investigation:
The Linear Pair Conjecture

Step 1: Draw EH and place a point F between E and H.
Choose another point G not on EH and draw FG. You have just
created a linear pair of angles. Place the zero edge of your
protractor along EH. 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?
+
Linear Pair Conjecture

If two angles form a linear pair, then ______________________.

the measures of the angles add up to 180o.
+
Definitions

Supplementary Angles: A pair of angles whose angle
measures add up to 180o.

Complementary Angles: A pair of angles whose angle
measures add up to 90o.
+
Investigation:
Vertical Angles Conjecture

Step 1: Draw two intersecting lines onto tracing 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 that 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.
+
Vertical Angles Conjecture

If two angles are vertical angles, then ____________________.

they have equal measures (they are congruent).
+
Inductive or Deductive?

To find The Linear Pair Conjecture?

Inductive Reasoning

To find the Vertical Angles Conjecture?

Inductive Reasoning
+
Using Deductive Reasoning

If we know that The Linear Pair Conjecture is true, can
we use deductive reasoning to show that the Vertical
Angles Conjecture must be true?

What is the sum of the measures of angles 1 and 2?

180o

What is the sum of the measures of angles 2 and 3?

180o

So what can you conclude about the measures of
angles 1 and 3. Explain your reasoning.
+
Is the converse true?

Vertical Angles Conjecture: If two angles are vertical angles,
then they have equal measures.

Converse: If two angles have equal measures, then they are
vertical angles.

Is this valid?

No.

Counterexample?
+
Practice

Find the measure of each lettered angle.
+
Practice

If a=51 and e=52, find the measure of b, c, d, and f.
+
Practice
+
Today’s Objectives

Discover relationships between pairs of angles

Perform investigations and write conjectures

Use conjectures to find missing angles measures.

Use problem solving skills
+
Exit Slip

Use your two conjectures to find the missing angle measures.

Explain how you found each angle measure.

Include which conjecture(s) you used.