Slope of Perpendicular Lines Theorem

If you have already learned about systems of linear equations, then you have probably discussed that the product of the slope of perpendicular lines is -1. The proof of this theorem comes from the fact that any point (x,y) rotated 90 degrees at about the origin becomes (-y,x). One example of this is shown below. The point (3, 4), when rotated 90 degrees counterclockwise becomes (-4,3).

With this fact, we prove this theorem.

Slope of Perpendicular Lines Theorem

If two lines with slopes m_1 and m_2 are perpendicular, then m_1 m_2 = -1Continue reading…


The Sum and Product of Roots Theorem

From the quadratic formula, we know that the numbers r_1 and r_2 are the roots of the quadratic equation ax^2 + bx +c =0 where a \neq 0 if and only if

r_1 + r_2 = -\frac{b}{a}


r_1r_2 = \frac{c}{a}Continue reading…


Sample Proof on Triangle Congruence Part 3

Recall that for real numbers a, b, and c are real numbers, if a = b and b = c, then a = c. This is called Transitive Property of Equality. This is also the same with congruence. If A, B, and C are polygons, and if A is congruent to B, and B is congruent C, then A is congruent to C. This is called the Transitive Property of Congruence. We will use this to prove the following problem.

Given: \overline{BE} \cong \overline{DC} and \overline{BD} \cong \overline{CA}.

Prove: \triangle DBE \cong \triangle CAB.


It is given that \overline{BE} \cong \overline{DC}.

triangle congruence

Now, by reflexive property, that is a segment is congruent to itself, \overline{BD} \cong \overline{BD}Continue reading…

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