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Math 070 – Chapter 4: Discovering and Proving Triangle Properties
4.1: Triangle Sum Conjecture
Ex. Draw a triangle, measure the three angles with a protractor, then find the sum
of the angles.
Conjecture C-17: Triangle Sum Conjecture
The sum of the measures of the angles in every triangle is ___________________.
A paragraph proof is a deductive argument that uses written sentences to support
its claims with reasons.
An auxiliary line is an extra line or segment drawn on a diagram to help with a
proof.
Ex. Draw a diagram of a triangle with an auxiliary line drawn through one vertex
parallel to the opposite side, then write a paragraph proof for the Triangle Sum
Conjecture.
Ex. Do Example A from Condensed Lesson 4.1.
Ex. Do Example B from Condensed Lesson 4.2.
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Math 070 – Chapter 4: Discovering and Proving Triangle Properties
4.2: Properties of Isosceles Triangles
Ex. Construct two different isosceles triangles, then use a protractor to measure the
base angles of both triangles.
Conjecture C-18: Isosceles Triangle Conjecture
If a triangle is isosceles, then __________________________________________.
Is the converse of the Isosceles Triangle Conjecture true?
Ex. Draw an acute angle, then construct a triangle ABC in which  A and B are
both congruent to the given angle. Measure the sides AC and BC . What do
you notice about them?
Conjecture C-19: Converse of the Isosceles Triangle Conjecture
If a triangle has two congruent angles, then _______________________________.
Ex. Do example from Condensed Lesson 4.2.
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Math 070 – Chapter 4: Discovering and Proving Triangle Properties
4.3: Triangle Inequalities
Ex. Do problems #11 and 12 from Practice Your Skills Worksheet 4.3.
Conjecture C-20: Triangle Inequality Conjecture
The sum of the lengths of any two sides of a triangle is ______________________
the length of the third side.
Another explanation: The shortest distance between two points is a straight line, so
the distance from A to C plus the distance from C to B cannot be shorter than the
path from A to B.
Ex. Draw a sketch to illustrate the Triangle Inequality Conjecture.
Ex. Perform the following steps.
1. Draw a triangle.
2. Use a protractor to measure each angle in the triangle.
3. Label the angle with the greatest measure  L , the angle with the second
greatest measure M , and the angle with the smallest measure S .
4. Use a ruler to measure each side of the triangle.
5. Label the longest side l, the second longest side m, and the shortest side s.
What do you notice?
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Math 070 – Chapter 4: Discovering and Proving Triangle Properties
Conjecture C-21: Side-Angle Inequality Conjecture
In a triangle, if one side is longer than another side, then the angle opposite the
longer side is _______________________________________________________.
An exterior angle of a triangle is an angle formed by an extension of one side past
a vertex and the other side of the triangle shared by that vertex.
Ex. Draw an example of a triangle with an exterior angle.
For an exterior angle, its adjacent interior angle is the angle of the triangle that has
the same vertex; the other two angles in the triangle are its remote interior angles.
Ex. Label the adjacent interior angle and the remote interior angles in the diagram
above.
Ex. Go back to the diagram of an exterior angle. What can we say about the
measure of the exterior angle in terms of the interior angles of the triangle?
Conjecture C-22: Triangle Exterior Angle Conjecture
The measure of an exterior angle of a triangle _____________________________.
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Math 070 – Chapter 4: Discovering and Proving Triangle Properties
4.4: Are There Congruence Shortcuts?
Two triangles are congruent if all corresponding sides and corresponding angles
are congruent (six congruency statements). Sometimes, we can determine that two
triangles are congruent if we are given three congruency statements.
Ex. Draw a sketch of each possible combination of three congruency statements.
Side-Side Side (SSS)
Side-Angle-Side (SAS)
Angle-Side-Angle (ASA)
Side-Angle-Angle (SAA)
Side-Side-Angle (SSA)
Angle-Angle-Angle (AAA)
Ex. Is SSS a congruence shortcut? Draw three sides of different length and use
them to construct a triangle. Is it possible to construct another triangle that is not
congruent to the first one?
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Math 070 – Chapter 4: Discovering and Proving Triangle Properties
Conjecture C-23: SSS Congruence Conjecture
If the three sides of one triangle are congruent to the three sides of another triangle,
then _____________________________________.
Ex. Is SAS a congruence shortcut? Draw two sides and one angle, and use them to
construct a triangle (with the angle between the two sides). Is it possible to
construct another triangle that is not congruent to the first one?
Conjecture C-24: 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 _______________________________.
Ex. Is SSA a congruence shortcut? Draw two sides and one angle, and use them to
construct a triangle (with the angle not between the two sides). Is it possible to
construct another triangle that is not congruent to the first one?
Ex. Do example from Condensed Lesson 4.4.
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Math 070 – Chapter 4: Discovering and Proving Triangle Properties
4.5: Are There Other Congruence Shortcuts?
An included side is a side that is included between two angles of a triangle.
Ex. Is ASA a congruence shortcut? Draw one side and two angles of different
measure, then use them to construct a triangle with the side included between the
two angles. Is it possible to construct another triangle that is not congruent to the
first one?
Conjecture C-25: ASA Congruence Conjecture
If two sides and the included side of one triangle are congruent to two angles and
the included side of another triangle, then ________________________________.
Ex. Is SAA a congruence shortcut? Draw one side and two angles of different
measure, then use them to construct a triangle with the side not included between
the two angles. Is it possible to construct another triangle that is not congruent to
the first one?
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Math 070 – Chapter 4: Discovering and Proving Triangle Properties
Conjecture C-26: 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
___________________________________________________________.
Ex. Is AAA a congruence shortcut? Draw three angles of different measure
(measure them with a protractor to make sure they add to 180 o ), then use them to
construct a triangle. Is it possible to construct another triangle that is not congruent
to the first one?
Ex. Do the example from Condensed Lesson 4.5.
4.6: Corresponding Parts of Congruent Triangles
According to the definition of congruent triangles, corresponding parts of
congruent triangles are congruent. We will use the letters CPCTC to refer to this
definition.
Ex. Do Example A from Condensed Lesson 4.6.
Ex. Do Example B from Condensed Lesson 4.6.
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Math 070 – Chapter 4: Discovering and Proving Triangle Properties
4.7: Flowchart Thinking
A flowchart proof is a visual way of organizing a proof in which statements are
written in boxes with the reasons for them written underneath; arrows then connect
the boxes to show how facts lead to conclusions.
A column proof is a proof written in a two-column table in which the statements
are listed in the left column, while the corresponding reasons for them are written
in the right column.
Ex. Do Example A from Condensed Lesson 4.7.
Ex. Do Example B from Condensed Lesson 4.7.
Ex. Create a proof to explain why the Angle Bisector Construction works.
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