Math 310Math 310

Section 10.1Section 10.1

Congruence and Congruence and ConstructionsConstructions

Congruence vs. Congruence vs. SimilaritySimilarity

DefDef

SimilarSimilar means that two objects have means that two objects have the same shape, but not necessarily the same shape, but not necessarily the same size. the same size. CongruenceCongruence means means that two objects have both the same that two objects have both the same size and the same shape.size and the same shape.

ExEx

Similar shapes:Similar shapes:

ExEx

Congruent shapes:Congruent shapes:

Congruent Segments & Congruent Segments & AnglesAngles

DefDef

Segments AB is congruent to segment Segments AB is congruent to segment CD iff mAB = mCD (their measures)CD iff mAB = mCD (their measures)

<ABC is congruent to <DEF iff <ABC is congruent to <DEF iff

m(<ABC) = m(<DEF)m(<ABC) = m(<DEF)

CirclesCircles

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

A CircleA Circle

DefDef

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

Parts of a CircleParts of a Circle

major arc

minor arc

radius

diameter

Parts of a Circle (cont)Parts of a Circle (cont)

O

semicircle

semicircle

center

Why Circles?Why Circles?

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

Construction: Congruent Construction: Congruent SegmentsSegments

Before AnglesBefore Angles

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

Congruent TrianglesCongruent Triangles

DefDef

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

Note: Order here is Note: Order here is veryvery important. important.

ExEx

A

C B

GH

I

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.

SSSSSS

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

SSS Congruence SSS Congruence PostulatePostulate

ThrmThrm

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

ExEx

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 TrianglesCopying Triangles

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

Construction: Congruent Construction: Congruent TrianglesTriangles

I will demonstrate two possibilities.I will demonstrate two possibilities. You are given a triangle to copyYou are given a triangle to copy You are given the measurements of a You are given the measurements of a

triangle to Constructtriangle to Construct

How Bout Angles?How Bout Angles?

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

Construction: Congruent Construction: Congruent AnglesAngles

Triangle InequalityTriangle Inequality

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

SAS Congruence SAS Congruence PostulatePostulate

ThrmThrm

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

ExEx

60O

C

W

60D

B

R

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 What do you do with these?these?

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

Perpendicular BisectorPerpendicular Bisector

DefDef

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

Construction: Construction: Perpendicular BisectorPerpendicular Bisector

Perpendicular Bisector Perpendicular Bisector TheoremsTheorems

ThrmThrm

Any point equidistant from the Any point equidistant from the endpoints of a segment is on the endpoints of a segment is on the perpendicular bisector of the perpendicular bisector of the segment.segment.

Any point on the perpendicular bisector Any point on the perpendicular bisector of a segment is equidistant from the of a segment is equidistant from the endpoints of the segment.endpoints of the segment.

Altitude of a TriangleAltitude of a Triangle

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

altitude

altitude

QuestionQuestion

How many altitudes does a triangle How many altitudes does a triangle have?have?

33

Isosceles Triangle Isosceles Triangle TheoremsTheorems

ThrmThrmThe angles opposite the congruent The angles opposite the congruent

sides are congruent. (Base angles of sides are congruent. (Base angles of an isosceles triangle are congruent.)an isosceles triangle are congruent.)

The angle bisector of an angle formed The angle bisector of an angle formed by two congruent sides contains the by two congruent sides contains the altitude of the triangle and is the altitude of the triangle and is the perpendicular bisector of the third perpendicular bisector of the third side of the triangle.side of the triangle.

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If these sides are congruent…

…then these angles are congruent.

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.

Circumscribe & Circumscribe & CircumcenterCircumcenter

DefDef

To circumscribe a circle about some To circumscribe a circle about some polygon is to construct the circle so polygon is to construct the circle so that each vertex of the polygon lies on that each vertex of the polygon lies on the circle, and the polygon is the circle, and the polygon is contained by the circle.contained by the circle.

The circumcenter of a triangle is the The circumcenter of a triangle is the point that is equidistant from all three point that is equidistant from all three vertices of a triangle. (i.e. a circle can vertices of a triangle. (i.e. a circle can be circumscribed about the triangle)be circumscribed about the triangle)

Construction: Construction: CircumcenterCircumcenter