Section 10.1 Congruence and Constructions Math 310 Section 10.1 Congruence and Constructions

Congruence vs. Similarity Def Similar means that two objects have the same shape, but not necessarily the same size. Congruence means that two objects have both the same size and the same shape.

Ex Similar shapes:

Ex Congruent shapes:

Congruent Segments & Angles Def Segments AB is congruent to segment CD iff mAB = mCD (their measures) <ABC is congruent to <DEF iff m(<ABC) = m(<DEF)

Circles Circles are the backbone of constructions, therefore, we begin constructions with some info about circles.

A Circle Def A circle is the loci (think “set of”) all points equidistant from a given center.

Parts of a Circle radius minor arc major arc diameter

Parts of a Circle (cont) semicircle center semicircle

Why Circles? So why are circles so important to constructions? By the very definition of circles, they allow us to copy distances, ie create congruent segments. And, as we will see, the use of two circles, one can copy an angle. Therefore, it is the use of circles that allow us to create congruent geometric objects. (restricted primarily of course to segments and angles)

Construction: Congruent Segments

Before Angles Before we can copy angles however, we need one more definition and one more postulate, both regarding triangles.

Congruent Triangles Def Two triangles are congruent if all of their parts are congruent. That is to say, triangle ABC is congruent to triangle EFG iff m<A = m<E, m<B = m<F, m<C = m<G, and sides mAB = mEF, mBC = mFG, and mCA = mGE. Note: Order here is very important.

Ex Let us suppose the following two figures are congruent and that they are drawn to scale. It would then be inappropriate to say that side AB was congruent to side HI. Clearly side AB is congruent to side IG and side CA is congruent to side HI. Thus is we call the upper triangle ABC, the lower triangle must then be called IGH.

SSS It would be highly complicated and tiring to prove for every set of triangles (or other figures) that all their corresponding sides and angles were congruent every time. Therefore, mathematicians have discovered certain conditions which guarantee that all parts are congruent. The first of these is called the Side-Side-Side Congruence Postulate, or SSS for short.

SSS Congruence Postulate Thrm If the three sides of one triangle are congruent, respectively, to the three sides of a second triangle, then the triangles are congruent.

Ex Use the SSS congruence postulate and your compass to demonstrate that these two triangles are congruent. Then name the triangles so that corresponding parts match up. Assume the triangles are drawn to scale.

Copying Triangles Now, since we have SSS, to copy a triangle we simply need to copy all three lengths of the triangle and we are guaranteed that the angles will be copied also.

Construction: Congruent Triangles I will demonstrate two possibilities. You are given a triangle to copy You are given the measurements of a triangle to Construct

How Bout Angles? If you are given an angle, simply attaching another side yields a triangle, and thus by copying the triangle we can also copy the angle!

Construction: Congruent Angles

Triangle Inequality The sum of the measures of any two sides of a triangle must be greater than the measure of the third side.

SAS Congruence Postulate Thrm If two sides and the included angle of one triangle are congruent to the two sides and the included angle of another triangle, respectively, then the two triangles are congruent.

Ex Given the following diagrams, state why the two triangles are congruent, and then, taking the name of the triangle at left to be WOC, what is the name of the triangle below?

What do you do with these? Constructions allow us to see properties of the geometric objects we are constructing. By constructing a geometric figure to given specification, we are, in essence, proving that what we have constructed satisfies those conditions. Alternately, to construct an object, we can follow a proof of its properties.

Perpendicular Bisector Def The perpendicular bisector of a segment is a line passing through the midpoint of the segment, perpendicular to the segment.

Construction: Perpendicular Bisector

Perpendicular Bisector Theorems Thrm Any point equidistant from the endpoints of a segment is on the perpendicular bisector of the segment. Any point on the perpendicular bisector of a segment is equidistant from the endpoints of the segment.

Altitude of a Triangle Def The altitude of a triangle is a segment drawn perpendicularly from one side of a triangle through the vertex opposite it. altitude altitude

Question How many altitudes does a triangle have? 3

Isosceles Triangle Theorems Thrm The angles opposite the congruent sides are congruent. (Base angles of an isosceles triangle are congruent.) The angle bisector of an angle formed by two congruent sides contains the altitude of the triangle and is the perpendicular bisector of the third side of the triangle.

Ex And if this angle is bisected by this segment… …then this segment is the altitude of the triangle and the perpendicular bisector of this side. If these sides are congruent… …then these angles are congruent.

Circumscribe & Circumcenter Def To circumscribe a circle about some polygon is to construct the circle so that each vertex of the polygon lies on the circle, and the polygon is contained by the circle. The circumcenter of a triangle is the point that is equidistant from all three vertices of a triangle. (i.e. a circle can be circumscribed about the triangle)

Construction: Circumcenter