Proofs from The Book

Proof That The Opposite Sides of a Parallelogram Are Congruent

Posted on 23 May, 2013 by Guillermo Bautista

A parallelogram is a quadrilateral whose opposite sides are parallel. In the figure below, $PQRS$ is a parallelogram. $PQ$ is parallel to $RS$ and $PS$ is parallel to $QR$ .

In this post, aside from being parallel, we will also prove that the opposite sides of a parallelogram are congruent.

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Posted in Geometry | Tagged opposite sides are congruent, opposite sides of a parallelogram, parallelogram | Leave a comment

The Angle Secant Theorem

Posted on 19 May, 2013 by Guillermo Bautista

A secant is a line that intersects a circle at two points. In the figure below, $\angle E$ is formed by two secants. The angle intercepts two arcs $\overline{AB}$ and $\overline{CD}$ . In this post, we will prove that the measure of the angle formed by two secants intersecting outside a circle is half the difference of the arcs intercepted by it.

To prove this theorem we will connect $\overline{BC}$ and use the Inscribed Angle Theorem and Exterior Angle Theorem.

Angle Secant Theorem

The angle measure formed by two secants intersecting outside a circle is half the difference of the arcs intercepted by it Continue reading →

Posted in Geometry | Tagged angle secant theorem, exterior angle theorem, intercepted arc, secant of a circle | Leave a comment

Math Proof Problem of the Week 1

Posted on 17 May, 2013 by Guillermo Bautista

It’s Friday and as I have mentioned in the previous post, I will be posting proof problems that are accessible to advanced high school students every Friday. Here is the Mathematics Proof Problem of the Week.

If $a > b > 0$ , and $a^2 + b^2 = 6ab$ , prove that