Chapter 2 Similarity and Congruence

Definitions

Definition

AB ∼= CD if and only if AB = CD

Remember, mAB = AB.

Definition∠ABC ∼= ∠DEF if and only if m∠ABC = m∠DEF

Definitions

Definition

AB ∼= CD if and only if AB = CD

Remember, mAB = AB.

Definition∠ABC ∼= ∠DEF if and only if m∠ABC = m∠DEF

Congruence Postulates

PostulateSide-Side-SideIf three sides of a triangle are congruent to the corresponding sides inanother triangle, then the triangles are congruent.

Congruence Postulates

PostulateSide-Angle-SideIf two sides and the included angle of a triangle are congruent to thecorresponding sides and angle in another triangle, then the trianglesare congruent.

Congruence Postulates

PostulateHypotenuse-LegIf the hypotenuse and leg of one right triangle are congruent to thehypotenuse and corresponding leg of another triangle, then the twotriangles are congruent.

Euclidean Tools

A compass looks like our compass, but it has no markings on it.So we can’t set it to draw circles with predetermined radii. Also,when we pick up the compass, it collapses, so we cannot copy acircle by picking up the compass and drawing another. We canonly draw circles when given a center and a point on thecircumference.

The straightedge is like a ruler with no markings. We can makestraight lines as long as we choose using any two points, or wecan extend an existing line segment as long as we want.

Euclidean Tools

A compass looks like our compass, but it has no markings on it.So we can’t set it to draw circles with predetermined radii. Also,when we pick up the compass, it collapses, so we cannot copy acircle by picking up the compass and drawing another. We canonly draw circles when given a center and a point on thecircumference.

The straightedge is like a ruler with no markings. We can makestraight lines as long as we choose using any two points, or wecan extend an existing line segment as long as we want.

What We Can Construct

Create a line through two points

Create a point at the intersection of lines

QuestionWhat is the difference between intersecting and concurrent lines?

Create a circle with two points where one is the center and theother is any point on the circumference

Create point(s) of intersection of lines and circles

Create point(s) of intersection of two circles For these two, howmany points could there be?

QuestionFor the intersection of lines, circles or one of each, how many pointsof intersection could there be?

What We Can Construct

Create a line through two points

Create a point at the intersection of lines

QuestionWhat is the difference between intersecting and concurrent lines?

Create a circle with two points where one is the center and theother is any point on the circumference

Create point(s) of intersection of lines and circles

Create point(s) of intersection of two circles For these two, howmany points could there be?

QuestionFor the intersection of lines, circles or one of each, how many pointsof intersection could there be?

What We Can Construct

Create a line through two points

Create a point at the intersection of lines

QuestionWhat is the difference between intersecting and concurrent lines?

Create a circle with two points where one is the center and theother is any point on the circumference

Create point(s) of intersection of lines and circles

Create point(s) of intersection of two circles For these two, howmany points could there be?

QuestionFor the intersection of lines, circles or one of each, how many pointsof intersection could there be?

What We Can Construct

Create a line through two points

Create a point at the intersection of lines

QuestionWhat is the difference between intersecting and concurrent lines?

Create a circle with two points where one is the center and theother is any point on the circumference

Create point(s) of intersection of lines and circles

Create point(s) of intersection of two circles For these two, howmany points could there be?

QuestionFor the intersection of lines, circles or one of each, how many pointsof intersection could there be?

What We Can Construct

Create a line through two points

Create a point at the intersection of lines

QuestionWhat is the difference between intersecting and concurrent lines?

Create a circle with two points where one is the center and theother is any point on the circumference

Create point(s) of intersection of lines and circles

Create point(s) of intersection of two circles

For these two, howmany points could there be?

QuestionFor the intersection of lines, circles or one of each, how many pointsof intersection could there be?

What We Can Construct

Create a line through two points

Create a point at the intersection of lines

QuestionWhat is the difference between intersecting and concurrent lines?

Create a circle with two points where one is the center and theother is any point on the circumference

Create point(s) of intersection of lines and circles

Create point(s) of intersection of two circles For these two, howmany points could there be?

QuestionFor the intersection of lines, circles or one of each, how many pointsof intersection could there be?

What We Can Construct

Bisect an angle

Find the perpendicular bisector of a segment

Construct regular polygons

Circumscribe regular polygons

Circumscribe some other polygons

What We Can Construct

Bisect an angle

Find the perpendicular bisector of a segment

Construct regular polygons

Circumscribe regular polygons

Circumscribe some other polygons

What We Can Construct

Bisect an angle

Find the perpendicular bisector of a segment

Construct regular polygons

Circumscribe regular polygons

Circumscribe some other polygons

What We Can Construct

Bisect an angle

Find the perpendicular bisector of a segment

Construct regular polygons

Circumscribe regular polygons

Circumscribe some other polygons

What We Can Construct

Bisect an angle

Find the perpendicular bisector of a segment

Construct regular polygons

Circumscribe regular polygons

Circumscribe some other polygons

Terms We’ll Need

DefinitionA perpendicular bisector is a line that passes through the midpoint ofanother line segment and the intersection forms four right angles.

DefinitionThe altitude of a triangle is a line segment that begins at a vertex of atriangle and is perpendicular to the opposite side.

DefinitionAn angle bisector is a line that passes through the vertex of an angleand divides the angle into two equal angles.

Terms We’ll Need

DefinitionA perpendicular bisector is a line that passes through the midpoint ofanother line segment and the intersection forms four right angles.

DefinitionThe altitude of a triangle is a line segment that begins at a vertex of atriangle and is perpendicular to the opposite side.

DefinitionAn angle bisector is a line that passes through the vertex of an angleand divides the angle into two equal angles.

Terms We’ll Need

DefinitionA perpendicular bisector is a line that passes through the midpoint ofanother line segment and the intersection forms four right angles.

DefinitionThe altitude of a triangle is a line segment that begins at a vertex of atriangle and is perpendicular to the opposite side.

DefinitionAn angle bisector is a line that passes through the vertex of an angleand divides the angle into two equal angles.

Circumscribing Polygons

DefinitionCircumscribing a polygon means we draw a circle that passes throughall of the vertices of the polygon.

DefinitionThe point at which the perpendicular bisectors of the sides of atriangle meet is the circumcenter. The circle we draw that passesthrough each vertex is called the circumcircle.

Circumscribing Polygons

DefinitionCircumscribing a polygon means we draw a circle that passes throughall of the vertices of the polygon.

DefinitionThe point at which the perpendicular bisectors of the sides of atriangle meet is the circumcenter. The circle we draw that passesthrough each vertex is called the circumcircle.

When Can We Circumscribe a Quadrilateral?

Theorema. If a circle can be circumscribed about a convex quadrilateral, then

the opposite angles are supplementary.

b. If the opposite angles of a quadrilateral are supplementary, then acircle can be circumscribed about the quadrilateral.

ASA Postulate

Angle-Side-AngleIf two angles and the included side of one triangle are congruent tothe two angles and the included side in another triangle, respectively,then the triangles are congruent.

AAS Postulate

Angle-Angle-SideIf two angles and a side opposite one of these angles of a triangle arecongruent to the two angles and the corresponding side in anothertriangle, then the triangles are congruent.

SSA Postulate

Side-Side-Angle PostulateThis postulate doesn’t exist The question is, why?

The Postulates at Work

Example

Given that PQ||RS and ∠PRQ ∼= ∠SQR, prove that ∆PQR ∼= ∆SRQ.

But first, why did I have to give you that ∠PRQ ∼= ∠SQR instead ofjust telling you PQ||RS?

The Postulates at Work

Example

Given that RN bisects ∠ERV and ∠NER ∼= ∠NVR, prove that∆ENR ∼= ∆VNR.

The Postulates at Work

Example

If AM and BN bisect the base angles of the given isosceles triangle,prove AM ∼= BN.

���������������

LLLLLLLLLLLLLLL

����������

HHHH

HHH

HHH

BA

C

N M

What We Want To Construct

1 Parallel lines2 Parallelograms3 Perpendicular line to a given point4 Angles other than 90◦

5 Incenter

Definition

DefinitionThe altitude of a triangle is the perpendicular from the base to theopposite vertex.

Angle Bisectors

Theorema. Any point P on an angle bisector is equidistant from the sides of

the angle.

b. Any point in the interior of an angle that is equidistant from thesides of the angle is on the angle bisector of the angle.

Incenter

DefinitionThe incenter of a triangle is the point of concurrency for the anglebisectors of a triangle.

TheoremThe incenter of a triangle is equidistant from the three sides of thetriangle.

Incenter

DefinitionThe incenter of a triangle is the point of concurrency for the anglebisectors of a triangle.

TheoremThe incenter of a triangle is equidistant from the three sides of thetriangle.

Definition of Similarity

Definition∆ABC is similar to ∆DEF, denoted as ∆ABC ∼ ∆DEF, if and onlyif the corresponding angles are congruent and the corresponding sidesare proportional.

A•

46

D•

23

E•2

•F

B•4

•C

ABDE

=BCEF

=ACDF

Question

Explain the following:Are all isosceles triangles similar?

Ways to Prove Similarity of Triangles

TheoremSSS Similarity for TrianglesIf the lengths of corresponding sides of two triangles areproportional, then the triangles are similar.

Ways to Prove Similarity of Triangles

TheoremSAS Similarity for TrianglesIf two sides are proportional to the corresponding sides and theincluded angles are congruent, then the triangles are similar.

Ways to Prove Similarity of Triangles

TheoremAA Triangle SimilarityIf two angles in one triangle are congruent to the correspondingangles in another triangle, then the triangles are similar.

Example

Triangle SimilarityExplain why ∆DBE ∼ ∆ABC.

What is the length of BE?

412

=x

x + 98x = 36

x =92

Example

Triangle SimilarityExplain why ∆DBE ∼ ∆ABC.

What is the length of BE?

412

=x

x + 98x = 36

x =92

Example

Triangle SimilarityExplain why ∆ABC ∼ ∆ADE.

Find the value of x.

Solution

x3

=x + 4

66x = 3(x + 4)

6x = 3x + 12

3x = 12

x = 4

Notice now that the length of the side AD is twice the length of AB,giving us a ratio of 1

2 for the measures of the sides in ∆ABCcompared to the corresponding sides of ∆ADE.

Solution

x3

=x + 4

66x = 3(x + 4)

6x = 3x + 12

3x = 12

x = 4

Notice now that the length of the side AD is twice the length of AB,giving us a ratio of 1

2 for the measures of the sides in ∆ABCcompared to the corresponding sides of ∆ADE.

Theorem

TheoremIf a line parallel to one side of a triangle intersects the other sidesthen it divides those sides into proportional segments.

TheoremIf a line divides two sides of a triangle into proportional segments,then the line is parallel to the third side.

TheoremIf a parallel line cuts off congruent segments on one transversal, thenthey cut off congruent segments on any transversal.

Theorem

TheoremIf a line parallel to one side of a triangle intersects the other sidesthen it divides those sides into proportional segments.

TheoremIf a line divides two sides of a triangle into proportional segments,then the line is parallel to the third side.

TheoremIf a parallel line cuts off congruent segments on one transversal, thenthey cut off congruent segments on any transversal.

Theorem

TheoremIf a line parallel to one side of a triangle intersects the other sidesthen it divides those sides into proportional segments.

TheoremIf a line divides two sides of a triangle into proportional segments,then the line is parallel to the third side.

TheoremIf a parallel line cuts off congruent segments on one transversal, thenthey cut off congruent segments on any transversal.

Midpoints

DefinitionThe midsegment is the segment connecting the midpoint of adjacentsides of a triangle or quadrilateral.

TheoremThe Midpoint TheoremThe midsegment joining the midpoint of two sides of a triangle isparallel to and is half as long as the third side.

TheoremIf a line bisects one side of a triangle and is parallel to a second sidethen it bisects the third side and therefore is a midsegment.

Midpoints

DefinitionThe midsegment is the segment connecting the midpoint of adjacentsides of a triangle or quadrilateral.

TheoremThe Midpoint TheoremThe midsegment joining the midpoint of two sides of a triangle isparallel to and is half as long as the third side.

TheoremIf a line bisects one side of a triangle and is parallel to a second sidethen it bisects the third side and therefore is a midsegment.

Midpoints

DefinitionThe midsegment is the segment connecting the midpoint of adjacentsides of a triangle or quadrilateral.

TheoremThe Midpoint TheoremThe midsegment joining the midpoint of two sides of a triangle isparallel to and is half as long as the third side.

TheoremIf a line bisects one side of a triangle and is parallel to a second sidethen it bisects the third side and therefore is a midsegment.

Centroid

DefinitionThe median of a triangle is the segment joining a vertex and themidpoint of the opposite side.

DefinitionThe centroid is the point of concurrency of the three medians of atriangle.

Centroid

DefinitionThe median of a triangle is the segment joining a vertex and themidpoint of the opposite side.

DefinitionThe centroid is the point of concurrency of the three medians of atriangle.

Example

Triangle SimilarityExplain why ∆abc ∼ ∆fde

Example

SimilarityFind the value of z.

610

=4z

z =203

Example

SimilarityFind the value of z.

610

=4z

z =203

Example

More SimilarityJustify why these triangles are similar and then find the value of x andy.

12x

=1620

=20y

So, x = 15 and y = 25.

Example

More SimilarityJustify why these triangles are similar and then find the value of x andy.

12x

=1620

=20y

So, x = 15 and y = 25.

Example

More SimilarityJustify why these triangles are similar and then find the value of x andy.

12x

=1620

=20y

So, x = 15 and y = 25.

Similarity and Other Polygons

DefinitionAny two polygons with the same number of sides are similar if andonly if the corresponding angles are congruent and the correspondingsides are proportional.

Same idea without the ‘named’ theorems and postulates.

Similarity and Other Polygons

DefinitionAny two polygons with the same number of sides are similar if andonly if the corresponding angles are congruent and the correspondingsides are proportional.

Same idea without the ‘named’ theorems and postulates.

Example

Similarity

Suppose you wanted to make a copy of a document at 18 of the original

size, but you made a mistake and made a copy of the original at 25 of

the original size. You are stubborn, so instead of starting at over, youwant to use the copy you made and reduce it to make the final productbe 1

8 of the original size. What ratio should you use to do this?

We think of this as 18 is the part we want and 2

5 is the whole, since thatis what we are working with now. But we want to know what part ofthe original whole this corresponds to. This gives

1825

=x

100516

=x

10016x = 500

x = 31.25

Example

Similarity

Suppose you wanted to make a copy of a document at 18 of the original

size, but you made a mistake and made a copy of the original at 25 of

the original size. You are stubborn, so instead of starting at over, youwant to use the copy you made and reduce it to make the final productbe 1

8 of the original size. What ratio should you use to do this?

We think of this as 18 is the part we want and 2

5 is the whole, since thatis what we are working with now. But we want to know what part ofthe original whole this corresponds to. This gives

1825

=x

100

516

=x

10016x = 500

x = 31.25

Example

Similarity

Suppose you wanted to make a copy of a document at 18 of the original

size, but you made a mistake and made a copy of the original at 25 of

the original size. You are stubborn, so instead of starting at over, youwant to use the copy you made and reduce it to make the final productbe 1

8 of the original size. What ratio should you use to do this?

We think of this as 18 is the part we want and 2

5 is the whole, since thatis what we are working with now. But we want to know what part ofthe original whole this corresponds to. This gives

1825

=x

100516

=x

10016x = 500

x = 31.25

Similarity

ExampleSuppose we have ∆ABC, ∆DEF, and ∆GHI such that ∆ABC is 70%of ∆DEF and ∆GHI is 30% of ∆DEF. What is the ratio between∆ABC and ∆GHI?

We haveABDE

=70100

,GHDE

=30

100

ABDEGHDE

=70

10030

100

abgh

=7030

So, the ratio between the first and third triangles is 73 .

Similarity

ExampleSuppose we have ∆ABC, ∆DEF, and ∆GHI such that ∆ABC is 70%of ∆DEF and ∆GHI is 30% of ∆DEF. What is the ratio between∆ABC and ∆GHI?

We haveABDE

=70100

,GHDE

=30

100

ABDEGHDE

=70

10030

100

abgh

=7030

So, the ratio between the first and third triangles is 73 .

Similarity

ExampleSuppose we have ∆ABC, ∆DEF, and ∆GHI such that ∆ABC is 70%of ∆DEF and ∆GHI is 30% of ∆DEF. What is the ratio between∆ABC and ∆GHI?

We haveABDE

=70100

,GHDE

=30

100

ABDEGHDE

=70

10030

100

abgh

=7030

So, the ratio between the first and third triangles is 73 .

Similarity

ExampleSuppose we have ∆ABC, ∆DEF, and ∆GHI such that ∆ABC is 70%of ∆DEF and ∆GHI is 30% of ∆DEF. What is the ratio between∆ABC and ∆GHI?

We haveABDE

=70100

,GHDE

=30

100

ABDEGHDE

=70

10030

100

abgh

=7030

So, the ratio between the first and third triangles is 73 .

Similarity and Slope

Do you remember the formula for slope? How about the phrase weuse when working with slope?

Slope

m =riserun

=y2 − y1

x2 − x1

How does this relate to similar triangles?

Similarity and Slope

Do you remember the formula for slope? How about the phrase weuse when working with slope?

Slope

m =riserun

=y2 − y1

x2 − x1

How does this relate to similar triangles?

Similarity and Slope

Do you remember the formula for slope? How about the phrase weuse when working with slope?

Slope

m =riserun

=y2 − y1

x2 − x1

How does this relate to similar triangles?