Geometry Theorems

Exterior Angles of a TriangleThe exterior angle has two interesting properties that follow from one another. 1) The exterior angle at a given vertex is equal in measure to the sum of the two remote interior angles. These remote interior angles are those at the other two vertices of the triangle. 2) Knowing this, it follows that the measure of any exterior angle is always greater than the measure of either remote interior angle. The first fact (1), the equality, is useful for proving congruence; the second fact (2), the inequality, is useful for disproving congruence.

There are Two Inequalities involving TrianglesThe first, the triangle inequality, states that the length of a side of a triangle is always less than the sum of the lengths of the other two sides.The second states that when a pair of angles are unequal, the sides opposite them are also unequal. The converse is also true: when a pair of sides are unequal, so are their opposite angles.

In this image: Angle 4 is greater than angle 2 and angle 3; angle 4 = angle 2 + angle 3

Altitudes of a Triangle The lines containing the altitudes of a triangle meet at one point called the orthocenter of the triangle. Because the orthocenter lies on the lines containing all three altitudes of a triangle, the segments joining the orthocenter to each side are perpendicular to the side. Keep in mind that the altitudes themselves aren't necessarily concurrent; the lines that contain the altitudes are the only guarantee. This means that the orthocenter isn't necessarily in the interior of the triangle. Figure %: The lines containing the altitudes of a triangle and the orthocenter There are two other common theorems concerning altitudes of a triangle. Both concern the concept of similarity. The first states that the lengths of the altitudes of similar triangles follow the same proportions as the corresponding sides of the similar triangles.The second states that the altitude of a right triangle drawn from the right angle to the hypotenuse divides the triangle into two similar triangles. These two triangles are also similar to the original triangle. The figure below illustrates this concept.

The lines containing the altitudes of a triangle and the orthocenter

There are two other common theorems concerning altitudes of a triangle. Both concern the concept of similarity. The first states that the lengths of the altitudes of similar triangles follow the same proportions as the corresponding sides of the similar triangles.The second states that the altitude of a right triangle drawn from the right angle to the hypotenuse divides the triangle into two similar triangles. These two triangles are also similar to the original triangle. The figure below illustrates this concept

Triangles ABC, DAC, and DBA are similar to one another

Medians of a Triangle Every triangle has three medians, just like it has three altitudes, angle bisectors, and perpendicular bisectors. The medians of a triangle are the segments drawn from the vertices to the midpoints of the opposite sides. The point of intersection of all three medians is called the centroid of the triangle. The centroid of a triangle is twice as far from a given vertex than it is from the midpoint to which the median from that vertex goes. For example, if a median is drawn from vertex A to midpoint M through centroid C, the length of AC is twice the length of CM. The centroid is 2/3 of the way from a given vertex to the opposite midpoint. The centroid is always on the interior of the triangle. The lengths of the medians of similar triangles are of the same proportion as the lengths of corresponding sides.  The median of a right triangle from the right angle to the hypotenuse is half the length of the hypotenuse.

A triangle's medians and centroid Two more interesting things are true of medians. 

Angle Bisectors of Triangles The angle bisectors of a triangle intersect each other at a point called the incircle of the triangle. The incircle of a triangle is the same as the center of a circle inscribed in a triangle. Every triangle can have exactly one inscribed circle, whose center is the incircle of the triangle, which is the point at which the angle bisectors of the triangle intersect. The incircle, then, is equidistant from the three sides of the triangle--a property that results from the inherent congruency of the radii of a circle.Another property of angle bisectors has to do with the side opposite the bisected angle. An angle bisector divides the side opposite the bisected angle into two segments that are of the same proportion as the other two sides. For example, in triangle ABC above, let the angle at vertex A be bisected, and let the bisector intersect BC at point D. BD/DC = BA/CA.

A triangle's angle bisectors and incircle

Perpendicular Bisectors of Triangles The three perpendicular bisectors of a triangle intersect at one point called the circumcenter of a triangle. The circumcenter is the center of the circle circumscribed about the triangle and is equidistant from all the vertices of the triangle. In this case the perpendicular bisectors of the sides of the triangles are lines, not segments. Therefore, the circumcenter of a triangle does not necessarily exist in the interior of the triangle. Often the perpendicular bisectors of a triangle intersect outside the triangle.

A triangle with its perpendicular bisectors and circumcenter

Theorem for Regular Polygons One additional theorem applicable to all regular polygons must be mentioned. You have probably already assumed as much from drawings, but to make it official, we'll state it as a theorem: the radii of a regular polygon bisect the interior angles. Theorem for Perpendicular Bisectors A final handy theorem with polygons has to do with perpendicular bisectors. The points on a perpendicular bisector are equidistant from the endpoints of the segment that they bisect.

The diagonals of a rhombus have useful properties

An isosceles trapezoid

Theorems for Quadrilaterals Rhombi The diagonals of a rhombus have three special properties. They are perpendicular to each other. They bisect each other. They bisect the interior angles of the rhombus. Thus, when you draw a rhombus, you can also draw this:

Squares Now recall that a square is both a rhombus and a rectangle. Its sides and angles are all congruent. From this fact, it follows: The diagonals of a square are perpendicular to each other. They bisect each other. They bisect the angles of the square. They are equal. Isosceles Trapezoids An isosceles trapezoid is the name given to a trapezoid with equal legs. The angles whose vertices are the vertices of the longer base are called the lower base angles, and the other two angles are called the upper base angles.For every isosceles trapezoid, the following is true: The legs are equal. The lower base angles are equal. The upper base angles are equal. The diagonals are equal.

The radii of a regular polygon bisect its internal angles

A perpendicular bisector

An angle whose vertex lies on a circle and whose sides intercept the circle (the sides contain chords of the circle) is called an inscribed angle. The measure of an inscribed angle is half the measure of the arc it intercepts.

The inscribed angle measures half of the arc it intercepts

If the vertex of an angle is on a circle, but one of the sides of the angle is contained in a line tangent to the circle, the angle is no longer an inscribed angle. The measure of such an angle, however, is equal to the measure of an inscribed angle. It is equal to one-half the measure of the arc it intercepts.

 An angle whose sides are a chord and a tangent segment

The angle ABC is equal to half the measure of arc AB (the minor arc defined by points A and B, of course).An angle whose vertex lies in the interior of a circle, but not at its center, has rays, or sides, that can be extended to form two secant lines. These secant lines intersect each other at the vertex of the angle. The measure of such an angle is half the sum of the measures of the arcs it intercepts.

An angle whose vertex is in the interior of a circle

The measure of angle 1 is equal to half the sum of the measures of arcs AB and DE.When an angle's vertex lies outside of a circle, and its sides don't intersect with the circle, we don't necessarily know anything about the angle. The angle's sides, however, can intersect with the circle in three different ways. Its sides can be contained in two secant lines, one secant line and one tangent line, or two tangent lines. In any case, the measure of the angle is one-half the difference between the measures of the arcs it intercepts. Each case is pictured below.In part (A) of the figure above, the measure of angle 1 is equal to one-half the difference between the measures of arcs JK and LM. In part (B), the measure of angle 2 is equal to one-half the difference between the measures of arcs QR and SR. In part (C), the measure of angle 3 is equal to one-half the difference between the measures of arcs BH and BJH. In this case, J is a point labeled just to make it easier to understand that when an angle's sides are parts of lines tangent to a circle, the arcs they intercept are the major and minor arc defined by the points of tangency. Here, arc BJH is the major arc.

An angle whose vertex lies outside of a circle

Tangent Segments Given a point outside a circle, two lines can be drawn through that point that are tangent to the circle. The tangent segments whose endpoints are the points of tangency and the fixed point outside the circle are equal. In other words, tangent segments drawn to the same circle from the same point (there are two for every circle) are equal.

Tangent segments that share an endpoint not on the circle are equal

Chords Chords within a circle can be related many ways. Parallel chords in the same circle always cut congruent arcs. That is, the arcs whose endpoints include one endpoint from each chord have equal measures.When congruent chords are in the same circle, they are equidistant from the center.In the figure above, chords WX and YZ are congruent. Therefore, their distances from the center, the lengths of segments LC and MC, are equal.

Arcs AC and BD have equal measures

Congruent chords in the same circle are equidistant from the center

Basic Theorems for Triangles

Theorems for Segments within Triangles

Theorems for Other Polygons

Theorems for Angles and Circles

Theorems for Segments and Circles