Triangles

What is a transversal and how does it help us in the case of parallel lines?

A transversal to two lines in the plane is a line that intersects them in two different points.

The lines cut by the transversal are not necessarily parallel, as shown in the first picture above. In both cases, the angles labeled by 1 and 2 are referred to as alternate interior angles. Note that there are two pairs of alternate interior angles.

The most important theorems involving transversals are in the case of parallel lines:

• • If two lines are cut by a transversal and any pair of alternate interior angles are congruent, then the lines are parallel.

• If two parallel lines are cut by a transversal, then any pair of alternate interior angles is congruent.

In fact, the example above is a special case of a corollary of the two theorems on transversals.

• If two lines are cut by a transversal, and one pair of alternate interior angles is congruent, then the other pair of alternate interior angles is also congruent.

The proof of this is sketched in the explanation of the correct answer to the example question above. If we replace the number 140 with an arbitrary angle measure of x and follow the same reasoning, then we have a proof.

When do parallel lines exist?

The answer to this question is actually a postulate created by Euclid, the famous geometer.

Existence and Uniqueness of Parallel Lines: Let L be any line and P be a point not on L. Then there is only one line containing P parallel to L.

What is the angle sum of a triangle?

We can label any triangle by labeling its vertices. The triangle ABC is shown below.

The most basic property concerning the angles of the triangle is given by the following theorem.

The Triangle Angle Sum Theorem:The sum of the interior angles of a triangle is 180°.

How can the Triangle Angle Sum Theorem be proven using these theorems about transversals and parallel lines?

As is often the case in geometric proofs, it boils down to thinking about the right picture. If we consider the first picture below, then we are in an ideal position to use the theorem on transversals. But, as in the case with any mathematical proof, we need to justify each step and our picture.

Step 1 For any triangle span >ABC, we can label the measure of the three interior angles by the variables x, y, and z.

Step 2 By the theorem on the existence of parallel lines in the previous subsection, we know that there exists a line parallel to  containing C. This justifies the picture drawn below.

Step 3 Apply the transversal theorem involving alternate interior angles to the angles yspan > and z.

Step 4 Thus, the angles x, y, and z are supplementary, .

Just by drawing the right picture and using one other theorem, we were able to show that the angle sum of the triangle is 180°.

Now use the Triangle Angle Sum Formula to prove the following theorem:

If two angles of one triangle are congruent to two angles of another triangle, then the remaining angles are congruent.

Suppose we denote the measure of the first pair of angles by x and the second by y. Then  and  by the Triangle Angle Sum Theorem. So w = span > z, which shows that the remaining angles are congruent.

When are two triangles congruent?

We compare triangles by comparing their sides and angles. Angles are congruent if they have the same angular measure. Sides are congruent if they have the same length. We can extend the idea of congruence to triangles as follows.

A correspondence between two triangles is a match-up between its vertices. For example, in the two triangles shown below, if the vertex A corresponds to D and the vertex B corresponds to E and the vertex C corresponds to F, then the triangle ABC corresponds to the triangle DEF. We write ABC  DEF.

The ordering is very important. Any correspondence between triangles automatically gives us a correspondence between its sides and its angles. This will allow us to define congruent triangles.

A correspondence ABC  DEF between two triangles is called congruence if every pair of corresponding sides is congruent and every pair of angles is congruent. If such congruence exists, we denote this relationship by ABC DEF. In fact, congruence between two triangles is actually six relations: three angle congruences and three side congruences.

When comparing triangles, indicate congruence of sides and angles by marking the figures as shown below.

In the example above, we showed that if two pairs of corresponding angles are congruent then the third pair of corresponding angles is congruent. If we relax some of these six conditions, however, is the congruence of the triangles still implied? There are three well-known situations in which this is the case. These are known as the SSS postulate, the SAS postulate, and the ASA postulate.

Side-Side-Side (SSS) postulate

If three sides of one triangle are congruent to three sides of another triangle, then the two triangles are congruent.

Side-Angle-Side (SAS) postulate

If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent.

Angle-Side-Angle (ASA) postulate

If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent.

Although it is not as well known as its siblings, there is one more postulate we can use to show that two triangles are congruent.

Angle -Angle-Side (AAS) postulate

If two angles and a non-included side of one triangle are congruent to two angles and the non-included side of another triangle, then the two triangles are congruent.

Review of New Vocabulary and Concepts

• A figure has symmetry if there is an isometry of the plane mapping the figure to itself.
• There are three main types of symmetry: reflectional, point, and rotational symmetry.
• A polygon is convex if no two points of it lie on opposite sides of a line containing a side of the polygon or that a line sontaining a side of the polygon does not cross the interior of the polygon.
• A polyhedron is convex if no two points of it lie on opposite sides of a line containing a side of the polygon or if a segment connecting any two points is on the polyhedron or the interior of the polyhedron.
• Any polyhedron satisfies Euler’s Formula: , where F = number of faces, E = number of edges, and V = number of vertices .
• Perspective drawing uses the notion of parallel lines and vanishing point to give the viewer a sense of three dimensions.
• A vanishing point is a point on the horizon line where parallel lines appear to meet.
• In one-point perspective drawing, parallel lines appear to meet at one point in the horizon.
• In two-point perspective drawing, parallel lines appear to meet at two distinct points in the horizon.
• If a transversal intersects two parallel lines, then any pair of alternate interior angles is congruent.
• If a transversal intersects two lines and any pair of alternate interior angles is congruent, then the other pair or alternate interior angles is congruent.
• The Triangle Angle Sum Theorem : The sum of the angles of a triangle is 180 °.
• A correspondence between two triangles is a correspondence between its vertices.
• A correspondence ABC DEF between two triangles is called a congruence if every pair of corresponding sides is congruent and every pair of angles is congruent.
• The SSS, SAS, ASA, and AAS postulates give sufficient conditions for two triangles to be congruent.