The Diagonals of an Isosceles Trapezoid Are Congruent

An isosceles trapezoid is a trapezoid whose sides are congruent.  An example of an isosceles trapezoid is shown below. The trapezoid $latex ABCD$ is isosceles with $latex AB$ parallel to $latex CD$ and $latex AD$ congruent to $latex BC$.

In this post, we are going to show that the diagonals of an isosceles trapezoid are congruent. In the figure below, we will show that $latex AC$ is congruent to $latex BD$. Continue reading

The Proof of the Second Case of the Inscribed Angle Theorem

In December last year, we have proved one case of the Inscribed Angle Theorem. In this post, we prove the second case. Note that the order of the cases does not matter; we just placed ordinal numbers to distinguish one from the other.

Recall that the first case of the theorem involved drawing an auxiliary line. We drew a line segment passing through the center. In this case, one of the sides of the triangle is passing through the center, so it is not possible to repeat this strategy. However, we will use another line to prove the theorem. Continue reading

Quadrilaterals with Congruent Opposite Angles are Parallelograms

In last week’s post, we have learned that quadrilaterals with congruent opposite sides are parallelograms. In this post, we show a related theorem. That is, quadrilaterals whose opposite angles are congruent are parallelograms.

In the figure above, $latex ABCD$ is a quadrilateral where $latex \angle A$ and $latex \angle C$ are congruent and $latex \angle D$ and $latex \angle B$ are congruent. In proving the theorem, we need to show that the opposite sides of $latex ABCD$ are parallel. That is $latex \overline {AB}$ is parallel to $latex \overline{CD}$ and $latex \overline {AD}$ is parallel to $latex \overline {BC}$. Continue reading