Making Conclusions
Mathematics and other fields allow us to make conjectures (observations) and conclusions based on the evidence that we have.
Example 1:
How would you order the following movie titles from Marvel Comics?
Mathematics and other fields allow us to make conjectures (observations) and conclusions based on the evidence that we have.
Example 1:
How would you order the following movie titles from Marvel Comics?
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How did you come to a conclusion about your arrangement?
Example 2:
What conclusions can you make about the angles? |
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Solution:
Move the blue and purple rectangles to reveal the answers. Were you surprised? What led to your conclusions?
Just using the diagram, it is very easy to make assumptions. In math we use mathematical reasoning to make conclusions based on what we know using properties, postulates, definitions and theorems. Properties, postulates, and definitions are accepted as true and do not have to be proven. Below are some properties that will be useful from Algebra 1.
Move the blue and purple rectangles to reveal the answers. Were you surprised? What led to your conclusions?
Just using the diagram, it is very easy to make assumptions. In math we use mathematical reasoning to make conclusions based on what we know using properties, postulates, definitions and theorems. Properties, postulates, and definitions are accepted as true and do not have to be proven. Below are some properties that will be useful from Algebra 1.
Property |
Let \(a, b\) and \(c\) be real numbers |
Addition Property |
If \(a = b\), then \( a + c = b + c\) |
Subtraction Property |
If \(a = b\), then \( a - c = b - c\) |
Multiplication Property |
If \(a = b\), then \(ac = bc\) |
Division Property |
If \(a = b\) and \(c \neq 0\), then \(\Large\frac{a}{c} = \frac{b}{c}\) |
Substitution Property |
If \(a = b\), then \(a\) can be substituted in for \(b\) in any expression or equation. |
Distributive Property |
\(a(b + c) = ab + ac\) |
If you were solving the equation \(2x - 5 = \dfrac{1}{3}x + 1\), what Algebraic properties would you use?
Here are some important properties that we will use from Geometry.
Here are some important properties that we will use from Geometry.
Properties |
Congruence of Segments |
Congruence of Angles |
Reflexive |
\(\overline{AB}\cong\overline{AB}\) or \(\overline{AB}\cong\overline{BA}\) |
\(\angle A \cong \angle A\) or \(\angle PAT \cong \angle PAT\) or \(\angle PAT \cong \angle TAP\) |
Symmetric |
If \(\overline{AB} \cong \overline{CD}\), then \(\overline{CD}\cong\overline{AB}\) |
If \(\angle{A} \cong \angle{B}\), then \(\angle{B}\cong\angle{A}\) |
Transitive |
If \( \overline{AB} \cong \overline{CD} \) and \( \overline{CD} \cong \overline{EF} \), then \( \overline{AB} \cong \overline{EF} \) |
If \( \angle{A} \cong \angle{B} \) and \( \angle{B} \cong \angle{C} \), then \( \angle{A} \cong \angle{C} \) |
Example 3:
Name the property that is used. We are given that \(HU = TA\)
a) \(HU = TA\)
b) \(LK = LK\)
c) \(HU + LK = TA + LK\)
Solution:
a) This is given
b) Reflexive property
c) Addition property
Definitions are also statements that are true and can be used for mathematical reasoning. Here is the definition of perpendicular lines.
Name the property that is used. We are given that \(HU = TA\)
a) \(HU = TA\)
b) \(LK = LK\)
c) \(HU + LK = TA + LK\)
Solution:
a) This is given
b) Reflexive property
c) Addition property
Definitions are also statements that are true and can be used for mathematical reasoning. Here is the definition of perpendicular lines.
When we write conclusions, we will be using many theorems. A theorem is a statement that can be proven. Here is a very important theorem about right angles.
Quick Check
1) Name the property that is used.
a) \(\overline{NC} \cong \overline{CN}\)
b) \(\angle R\) is a right angle. \(m\angle R = 90^{\circ}\)
2) What conclusion(s) can be made from the diagram?
1) Name the property that is used.
a) \(\overline{NC} \cong \overline{CN}\)
b) \(\angle R\) is a right angle. \(m\angle R = 90^{\circ}\)
2) What conclusion(s) can be made from the diagram?