Download 4.1 Triangle Sum Conjecture Guided Notes DISCOVERING

4.1 Triangle Sum Conjecture
Guided Notes
DISCOVERING GEOMETRY
Name_______________________________ Block_______
LEQ: How is the sum of interior angles of a triangle calculated?
1. Complete the VOCABULARY chart below. Begin on page 200.
Term
Definition
Picture/Symbol
1. Triangulation
2. paragraph proof
3. auxiliary line
2. Complete INVESTIGATION “The Triangle Sum” below.
There are an endless variety of triangles that you can draw, with different shapes and angle measures.
Do their angle measures have anything in common? Start by drawing different kinds of triangles. Make
sure your group has at least one acute and one obtuse triangle.
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Step 1 Measure the three angles of each triangle as accurately as possible with your protractor.
Step 2 Find the sum of the measures of the three angles in each triangle. Compare results with others in
your group. Does everyone get about the same result? What is it?
Step 3 Check the sum another way. Write the letters a, b, and c in the interiors of the three angles of
one of the triangles, and carefully cut out the triangle.
Step 4 Tear off the three angles. Arrange them so that their vertices meet at a point. How does this
arrangement show the sum of the angle measures?
Step 5 Compare results with others in your group. State your observations as a conjecture.
State the Triangle Sum Conjecture:
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Developing Proof: The investigation may have convinced you that the Triangle Sum
Conjecture is true, but can you explain why it is true for every triangle?
As a group, explain why the Triangle Sum Conjecture is true by writing a paragraph proof, a deductive
argument that uses written sentences to support its claims with reasons.
Another reasoning strategy you might use is to add an auxiliary line, an extra line or segment that helps
with a proof. Your group may have formed an auxiliary line by rearranging the angles in the
investigation. If you rotated ∠A and ∠B and left ∠C pointing up, then how is the resulting line related to
the original triangle?
⃡ , an auxiliary line parallel to side 𝐴𝐵
̅̅̅̅. Use this diagram to discuss these
The figure below includes 𝐸𝐶
questions with your group.
● What are you trying to prove?
● What is the relationship among ∠1,∠2, and ∠3?
● Why was the auxiliary line drawn to be parallel to one of the sides?
● What other congruencies can you determine from the diagram?
Use your responses to these questions to mark your diagram. Discuss how you can use the information
you have to prove that the Triangle Sum Conjecture is true for every triangle. As a group, write a
paragraph proof.
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3. Complete EXERCISES on pages 203-205 # ____________________, using separate paper.
4.2 Properties of Isosceles Triangles
LEQ: How do we solve problems using properties of isosceles triangles?
1. Complete the VOCABULARY chart below. See page 206.
Term
Definition
Picture/Symbol
1. Legs
2. Complete INVESTIGATION 1 “Base Angles in an Isosceles Triangle” below.
You will need: patty paper, a straightedge, a protractor
Let’s examine the angles of an isosceles triangle. Each person in your group should draw a different
angle for this investigation. Your group should have at least one acute angle and one obtuse angle.
Step 1 Draw an angle on patty paper. Label it ∠C. This angle will be the vertex angle of your isosceles
triangle.
Step 2 Place a point A on one ray. Fold your patty paper so that the two rays match up. Trace point A
onto the other ray.
Step 3 Label the point on the other ray point B. Draw ̅̅̅̅
𝐴𝐵 . You have constructed an isosceles triangle.
Explain how you know it is isosceles. Name the base and the base angles.
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Step 4 Use your protractor to compare the measures of the base angles. What relationship do you
notice? How can you fold the paper to confirm your conclusion?
Step 5 Compare results in your group. Was the relationship you noticed the same for each isosceles
triangle? State your observations as your next conjecture.
State the Isosceles Triangle Conjecture:
3. Complete INVESTIGATION 2 “Is the Converse (to the Isosceles Triangle Conjecture) True?” below.
You will need: a compass, a straightedge
Suppose a triangle has two congruent angles. Must the triangle be isosceles?
̅̅̅̅. Draw an acute angle at point A. This angle will be a base angle.
Step 1 Draw a segment and label it 𝐴𝐵
(Why can’t you draw an obtuse angle as a base angle?)
Step 2 Copy ∠A at point B on the same side of ̅̅̅̅
𝐴𝐵. Label the intersection of the two rays point C.
Step 3 Use your compass to compare the lengths of sides ̅̅̅̅
𝐴𝐶 and ̅̅̅̅
𝐵𝐶 .
What relationship do you notice?
How can you use patty paper to confirm your conclusion?
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Step 4 Compare results in your group. State your observation as your next conjecture.
State the Converse of the Isosceles Triangle Conjecture:
4. Complete EXERCISES on pages 208-211 # ___________________, using separate paper.
4.2 Alg Skills: Solving Equations
LEQ: How do we use the properties of real numbers and equality to solve geometric problems?
1. State the Distributive property, using a variable example and a numerical example. See page 212.
What does this property allow us to do?
2. Give an example of Combining like terms, using a variable example and a numerical example. See
page 212. What does this process allow us to do?
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3. Using the Properties of Equality, give an example of each in the chart below. See page 212.
Property
Example (using variables)
Addition property
Subtraction property
Multiplication property
Division property
Substitution property
4. What do the first four properties in the chart above allow us to do? (Addition, subtraction,
multiplication and division properties)
5. What does the substitution property allow us to do?
6. Complete EXERCISES on page 214 # _________________, using separate paper.
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4.3 Triangle Inequalities
LEQ’s: How does the relationship between two sides of a triangle compare to the third side of a
triangle? How do the measures of the interior angles of a triangle relate to the lengths of their opposite
sides? How does the measure of the exterior angle of a triangle relate to the sum of its two remote
interior angles?
1. Complete the VOCABULARY chart below. See page 217.
Term
1. exterior angle
Definition
Picture/Symbol
2. adjacent interior angle
3. remote interior angles
2. Complete INVESTIGATION 1 “What is the Shortest Path from A to B?” below.
You will need: a compass, a straightedge
Each person in your group should do each construction. Compare results when you finish.
Step 1 Construct a triangle with each set of segments as sides.
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Step 2 You should have been able to construct ∆CAT, but not ∆FSH. Why? Discuss your results with
others.
State the Triangle Inequality Conjecture:
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3. Complete INVESTIGATION 2 “Where are the Largest and Smallest Angles?” below.
You will need: a ruler, a protractor
Each person should draw a different scalene triangle for this investigation. Some group members should
draw acute triangles, and some should draw obtuse triangles.
Step 1 Measure the angles in your triangle. Label the angle with greatest measure ∠L, the angle with
second greatest measure ∠M, and the smallest angle ∠S.
Step 2 Measure the three sides. Label the longest side l, the second longest side m, and the shortest side
s.
Step 3 Which side is opposite ∠L? ∠M? ∠S?
Discuss your results with others. Finish the conjecture below.
Side-Angle Inequality Conjecture: In a triangle, if one side is longer than another side, then the angle
opposite the longer side is
4. Complete INVESTIGATION 3 “Exterior Angles of a Triangle” below.
You will need: a straightedge, patty paper
Each person should draw a different scalene triangle for this investigation. Some group members should
draw acute triangles, and some should draw obtuse triangles.
̅̅̅̅ beyond point B and label a point D outside the triangle
Step 1 Draw a scalene triangle, ∆ABC. Extend 𝐴𝐵
on 𝐴𝐵 Label the angles as shown.
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Step 2 Copy the two remote interior angles, ∠A and ∠C, onto patty paper to show their sum.
Step 3 How does the sum of a and c compare with x? Use your patty paper from Step 2 to compare.
Step 4 Discuss your results with others. Finish the conjecture below.
Triangle Exterior Angle Conjecture: The measure of an exterior angle of a triangle
5. Complete EXERCISES on pages 218-220 # ___________________, using separate paper.
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4.4 Are There Congruence Shortcuts (SSS, SAS)?
LEQ: How does the relationship between two sides of a triangle compare to the third side of a triangle?
1. Complete INVESTIGATION 1 “Is SSS a Congruence Shortcut?” below.
First you will investigate the Side- Side-Side (SSS) case. If the three sides of one triangle are congruent to
the three sides of another, must the two triangles be congruent?
Step 1 Construct a triangle from the three parts shown. Be sure you match up the endpoints labeled
with the same letter.
Step 2 Compare your triangle with the triangles made by others in your group. (One way to compare
them is to place the triangles on top of each other and see if they coincide.) Is it possible to construct
different triangles from the same three parts, or will all the triangles be congruent?
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Step 3 You are now ready to complete the conjecture for the SSS case.
SSS Congruence Conjecture: If the three sides of one triangle are congruent to the three sides of
another triangle, then
2. Complete INVESTIGATION 2 “Is SAS a Congruence Shortcut?” below.
Next you will consider the Side-Angle- Side (SAS) case. If two sides and the included angle of one triangle
are congruent to two sides and the included angle of another, must the triangles be congruent?
Step 1 Construct a triangle from the three parts shown. Be sure you match up the endpoints labeled
with the same letter.
Step 2 Compare your triangle with the triangles made by others in your group. Is it possible to construct
different triangles from the same three parts, or will all the triangles be congruent?
Step 3 You are now ready to complete the conjecture for the SAS case.
SAS Congruence Conjecture: If two sides and the included angle of one triangle are congruent to two
sides and the included angle of another triangle, then
3. Complete INVESTIGATION 3 “Is SSA a Congruence Shortcut?” below.
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Finally you will consider the Side- Side-Angle (SSA) case. If two sides and a non-included angle of one
triangle are congruent to the corresponding two sides and non-included angle of another, must the
triangles be congruent?
Step 1 Construct a triangle from the three parts shown. Be sure you match up the endpoints labeled
with the same letter.
Step 2 Compare your triangle with the triangles made by others in your group. Is it possible to construct
different triangles from the same three parts, or will all the triangles be congruent?
Step 3 If two sides and a non-included angle of one triangle are congruent to the corresponding two
sides and non-included angle of another triangle, do the two triangles have to be congruent? Explain
why or show a counterexample.
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4. Complete EXERCISES on pages 224-225 # __________________, using separate paper.
4.5 Are There Other Congruence Shortcuts? ASA, SAA
LEQ: How does the relationship between two sides of a triangle compare to the third side of a triangle?
1. Complete INVESTIGATION 1 “Is ASA a Congruence Shortcut?” below.
First you will consider the Angle-Side- Angle (ASA) case. If two angles and the included side of one
triangle are congruent to two angles and the included side of another, must the triangles be congruent?
Step 1 Construct a triangle from the three parts shown. Be sure you match up the angles with the
endpoints labeled with the same letter.
Step 2 Compare your triangle with the triangles made by others in your group. Is it possible to construct
different triangles from the same three parts, or will all the triangles be congruent?
Step 3 You are now ready to complete the conjecture for the ASA case.
ASA Congruence Conjecture: If two angles and the included side of one triangle are congruent to two
angles and the included side of another triangle, then
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2. Complete INVESTIGTION 2 “Is SAA a Congruence Shortcut?” below.
Next you will consider the Side-Angle-Angle (SAA) case. If two angles and a non-included side of one
triangle are congruent to the corresponding two angles and non-included side of another, must the
triangles be congruent?
Step 1 Construct a triangle from the three parts shown. Be sure you match up the angles with the
endpoints labeled with the same letter.
Step 2 Compare your triangle with the triangles made by others in your group. Is it possible to construct
different triangles from the same three parts, or will all the triangles be congruent?
Step 3 You are now ready to complete the conjecture for the SAA case.
SAA Congruence Conjecture: If two angles and a non-included side of one triangle are congruent to the
corresponding two angles and non-included side of another triangle, then
3. Complete INVESTIGATION 3 “Is AAA a Congruence Shortcut?” below.
Finally you will investigate the Angle- Angle-Angle (AAA) case. If the three angles of one triangle are
congruent to the three angles of another, must the triangles be congruent?
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Step 1 Construct a triangle from the three parts shown.
Step 2 Compare your triangle with the triangles made by others in your group. Is it possible to construct
different triangles from the same three parts, or will all the triangles be congruent?
Step 3 If the three angles of one triangle are congruent to the three angles of another triangle, do the
two triangles have to be congruent? Explain why or show a counterexample.
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4. Complete EXERCISES on pages 229-231 # ___________________, using separate paper.
4.6 CPCTC
LEQ: How is it possible to determine if two triangles are congruent? How does CPCTC relate to the
concept of congruency?
1. Write the definition for CPCTC (see page 232).
2. What are the letters CPCTC used for?
3. Review/Study Examples A and B below.
In Lessons 4.4 and 4.5, you discovered four shortcuts for showing that two
triangles are congruent—SSS, SAS, ASA, and SAA. The definition of congruent
triangles states that if two triangles are congruent, then the corresponding parts of
those congruent triangles are congruent.We’ll use the letters CPCTC to refer to the
definition. Let’s see how you can use congruent triangles and CPCTC.
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̅̅̅̅ ≅ 𝐵𝐶
̅̅̅̅ in the figure above? Use a deductive argument to explain why they
Example A: Is 𝐴𝐷
must be congruent.
Here is one possible explanation: ∠1 ≅ ∠2 because they are vertical angles.
̅̅̅̅̅ ≅ ̅̅̅̅̅
And it is given that 𝐴𝑀
𝐵𝑀_ and ∠A ≅ ∠B. So, by ASA, ∆AMD ≅∆BMC.
Because the triangles are congruent, ̅̅̅̅
𝐴𝐷 ≅ ̅̅̅̅
𝐵𝐶 by CPCTC.
If you use a congruence shortcut to show that two triangles are congruent, then
you can use CPCTC to show that any of their corresponding parts are congruent.
When you are trying to prove that triangles are congruent, it can be hard to keep
track of what you know. Mark all the information on the figure. If the triangles are
hard to see, you can use different colors or redraw them separately.
̅̅̅̅? Write a paragraph proof explaining why.
Example B: Is ̅̅̅̅̅
𝐴𝐸 ≅ 𝐵𝐷
The triangles you can use to show congruence are ∆ABD and ∆BAE. You can separate or color them to
see them more clearly.
You can see that the two triangles have two pairs of congruent angles and they share a side.
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̅̅̅̅.
Paragraph Proof: Show that ̅̅̅̅
𝐴𝐸 ≅ 𝐵𝐷
In ∆ABD and ∆BAE, ∠D ≅ ∠E and ∠B ≅ ∠A. Also, ̅̅̅̅
𝐴𝐵 ≅ ̅̅̅̅
𝐵𝐴 because they are the same segment. So
̅̅̅̅
̅̅̅̅
∆ABD≅ ∆BAE by SAA. By CPCTC, 𝐴𝐸 ≅ 𝐵𝐷 .
4. Complete EXERCISES on pages 233-235 # _____________________, using separate paper.
4.7 Flowchart Thinking
LEQ: How are flowcharts used to introduce logical thinking?
Study Example A below.
̅̅̅̅̅ ≅ 𝐴𝐶
̅̅̅̅ and 𝐸𝑅
̅̅̅̅. Is ∠𝐴 ≅ ∠𝐸? If so, give a logical argument to explain
̅̅̅̅ ≅ 𝐴𝑅
1. In the figure above, 𝐸𝐶
why they are congruent. First, mark the given information on the figure. Then consider whether ∠A is
congruent to ∠E. It looks as if there are two congruent triangles, so use the reasoning strategy of
applying previous conjectures to explain why.
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̅̅̅̅ ≅ 𝐴𝐶
̅̅̅̅ and 𝐸𝑅
̅̅̅̅ because that information is given. 𝑅𝐶
̅̅̅̅ ≅
̅̅̅̅ ≅ 𝐴𝑅
Paragraph Proof: Show that ∠A≅ ∠E. 𝐸𝐶
̅̅̅̅ because it is the same segment, and any segment is congruent to itself. So, ∆CRE≅∆CRA by the SSS
𝑅𝐶
Congruence Conjecture. If ∆CRE≅ ∆CRA, then ∠A ≅ ∠E by CPCTC.
2. Study Example B below.
In the figure above, ̅̅̅̅̅
𝐸𝐶 ≅ ̅̅̅̅
𝐴𝐶 and ̅̅̅̅
𝐸𝑅 ≅ ̅̅̅̅
𝐴𝑅. Is ∠𝐴 ≅ ∠𝐸?
First, restate the given information clearly. It helps to mark the given information on the figure. Then
state what you are trying to show.
̅̅̅̅̅ ≅ 𝐴𝐶
̅̅̅̅ and 𝐸𝑅
̅̅̅̅
̅̅̅̅ ≅ 𝐴𝑅
Given: 𝐸𝐶
Show: ∠𝐴 ≅ ∠𝐸
3. Name two ways that flowcharts can be drawn.
4. What are the arrows in a flowchart proof saying?
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5. What other similarities or differences do you see between Example A and B?
6. Complete EXERCISES on pages 239-242 # ___________________, using separate paper.
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4.8 Proving Special Triangle Conjectures
LEQ: How can we use vertex angle bisectors to investigate properties of isosceles and equilateral
triangles?
1. Complete the VOCABULARY chart below. Begin on page 245.
Term
Definition
Picture/Symbol
1. Biconditional
2. Complete the INVESTIGATION “The Symmetry Line in an Isosceles Triangle” below.
Each person in your group should draw a different isosceles triangle for this investigation.
Step 1 Construct a large isosceles triangle on a sheet of paper. Label it ARK, with K the vertex angle.
Step 2 Construct angle bisector ̅̅̅̅
𝐾𝐷with point D on ̅̅̅̅
𝐴𝑅. Do ∆ADK and ∆RDK look congruent? If they are
̅̅̅̅
congruent, then 𝐾𝐷is a line of symmetry.
Step 3 With your compass, compare ̅̅̅̅
𝐴𝐷 and̅̅̅̅̅
𝑅𝐷. Is D the midpoint of ̅̅̅̅̅
𝐴𝑅 ? If D is the midpoint, then
̅̅̅̅?
what type of special segment is 𝐾𝐷
Step 4 Compare ∠ADK and ∠RDK. Do they have equal measures? Are they supplementary? What
conclusion can you make?
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Step 5 Compare your conjectures with the results of other students. Now combine the two conjectures
from Steps 3 and 4 into one.
Vertex Angle Bisector Conjecture: In an isosceles triangle, the bisector of the vertex angle is also
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3. Paragraph Proof: The Equilateral Triangle Conjecture
We need to show that if AB= AC = BC, then ∆ABC is equiangular. By the Isosceles Triangle Conjecture,
The converse of the Equilateral Triangle Conjecture is called the Equiangular Triangle Conjecture, and it
states: If a triangle is equiangular, then it is equilateral. Is this true? Yes, and the proof is almost identical
to the proof above, except that you use the converse of the Isosceles Triangle Conjecture. So, if the
Equilateral Triangle Conjecture and the Equiangular Triangle Conjecture are both true, then we can
combine them. Complete the conjecture below and add it to your conjecture list.
The Equilateral/Equiangular Triangle Conjecture is a biconditional conjecture: Both the statement and
its converse are true. A triangle is equilateral if and only if it is equiangular. One condition cannot be
true unless the other is also true.
Equilateral/Equiangular Triangle Conjecture: Every equilateral triangle is _________________ , and,
conversely, every equiangular triangle is__________________
4. Complete EXERCISES on pages 245-248 # __________________, using separate paper.
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