Geometry Triangles - NJCTLcontent.njctl.org/courses/math/archived-courses... · C isosceles D...

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Table of Contents

· Questions from Released PARCC Test

· Triangle Sum Theorem

· Exterior Angle Theorem

· Triangles

· Inequalities in Triangles

· Similar Triangles

Click on the topic to go to that section

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Geometry

Triangles

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Return to Tableof Contents

Triangles

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Geometric Figures

Euclid now makes the transitions to geometric figures, which are created by a boundary which separates space into that which is within the figure and that which is not.

Definition 13. A boundary is that which is an extremity of anything.

Definition 14. A figure is that which is contained by any boundary or boundaries.

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Geometric Figures

His definitions from 15 to 18 relate to circles, which we will discuss later. In this chapter, we will be discussing triangles, which are an example of a rectilinear figure: a figure bounded by straight lines.

A triangle is bounded by three lines.

Definition 19. Rectilinear figures are those which are contained by straight lines, trilateral figures being those contained by three, quadrilateral those contained by four, and multilateral those contained by more than four straight lines.

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Parts of a TriangleEach triangle has three sides and three vertices.

Each vertex is where two sides meet.

A pair of sides and the vertex define an angle, so each triangle includes three angles.

Write "side" next to each side and circle the vertices on the triangle below.

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1 The letter on this triangle that corresponds to a side is:

A

B

C

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2 The letter on this triangle that represents a vertex is:

AB

C

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Parts of a Triangle

C

A B

Each vertex is named with a letter.

The sides can then be named with the letters of the two vertices on either side of it.

The triangle is named with a triangle symbol Δ in front followed by the three letters of its vertices.

Name the 3 sides of this triangle

______ ______ ______

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3 What is the name of the side shown in red?

A ABB BCC AC

C

A B

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4 What is the name of the side shown in red?

A AB

B BC

C AC

C

A B

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5 Which of the following are names of this triangle?

A ΔABCB ΔBCAC ΔACB

C

A B

D ΔCABE all of these

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Parts of a Triangle

C

A B

In the above, the red side is ________________ A,

while the green sides are ________________ to A.

A side is opposite an angle if it does not touch it. Otherwise, it is adjacent to the angle.

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6 Which side is opposite angle B?

A AB

B CA

C BC

D None

C

A B

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7 Which side is opposite angle A?

A AB B CA

C BC D None

C

A B

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8 Which sides are adjacent to angle C?

A AB & BC

B CA & BA

C BC & CAD None

C

A B

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9 Which sides are adjacent to angle B?

A AB & BCB CA & BA C BC & CAD None

C

A B

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Types of Triangles

In general, a triangle can have sides of all different lengths and angles of all different measure.

However, there are names given to triangles which have specific or special angles or some number of equal sides or angles.

Euclid defined the names for a number of these in his definitions.

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Definition 20: Of trilateral figures, an equilateral triangle is that which has its three sides equal, an isosceles triangle is that which has two of its sides alone equal, and a scalene triangle is that which has its three sides unequal

Classifying Triangles

Triangles can be classified by their sides or by their angles.

In this definition, Euclid used the sides.

In his next definition, Euclid uses the angles.

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Definition 21: Further, of trilateral figures, a right-angled triangle is that which has a right angle, an obtuse-angled triangle is that which has an obtuse angle, and an acute-angled triangle is that which has

its three angles acute.


We will draw from both definitions, since in several cases both definitions apply to the same triangle.

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Acute Triangles

In an acute triangle, every angle of a triangle is acute.

Notice that no angle is equal to or greater than 90º in this triangle.


Definition 21: "...an acute-angled triangle is that which has its three angles acute."

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Right Triangles

A right triangle has one right angle and two acute angles.

Notice that one angle is 90º, which means that the other two sum to 90º; and they are acute.

The side opposite the right angle is called the hypotenuse and the other two sides are called the legs.


Definition 21: "...a right-angled triangle is that which has a right angle..."

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Isosceles Triangles

An isosceles triangle has two sides with equal length.

The angles opposite those equal sides are of equal measure.

x x


Definition 20: "...an isosceles triangle is that which has two of its sides alone equal..."

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Isosceles Triangles

The equal angles, of measure x in this diagram, are called the base angles. The side between them is called the base.

The other two sides, opposite the base angles and congruent to each other are called the legs.

This is a special case of an acute triangle.

x x


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Obtuse Triangles

An obtuse triangle has one angle which is greater than 90º and two acute angles.

Notice that one angle is greater than 90º, which means that the other two sum to less than 90º; and they are acute..


Definition 21: "...an obtuse-angled triangle is that which has an obtuse angle..."

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Equiangular / Equilateral Triangles

An equiangular, or equilateral, triangle has angles of equal measure and sides of equal length.

Definition 20: "...an equilateral triangle is that which has its three sides equal..."

All the angles are of equal measure and all the sides are of equal length.

Each angle measures 60º.

This is a special acute triangle.x x

x


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Scalene Triangles

None of the sides or angles of a scalene triangle are congruent with one another.

Definition 20: "...a scalene triangle is that which has its three sides unequal..."

Note that in this triangle none of the sides or angles are equal.

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10 An isosceles triangle is _______________ an equilateral triangle.

A SometimesB AlwaysC Never

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11 An obtuse triangle is _______________ an isosceles triangle.

A SometimesB AlwaysC Never

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12 A triangle can have more than one obtuse angle.

TrueFalse

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13 A triangle can have more than one right angle.

TrueFalse

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14 Each angle in an equiangular triangle measures 60°

TrueFalse

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15 An equilateral triangle is also an isosceles triangle

TrueFalse

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16 This triangle is classified as _____. (Choose all that apply.)

A acute

B right

C isosceles

D obtuse

E equilateral

F equiangular

G scalene

60º8.6

60º

60º8.68.6

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A acute

B right

C isosceles

D obtuse

E equilateral

F equiangular

G scalene

57º

79º 44º

6.1 8.7

7.4

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A acute

B right

C isosceles

D obtuse

E equilateral

F equiangular

G scalene

26°

128° 26°

2.5

2.5

4.5

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19 This triangle is classified as _____. Choose all that apply.

A acute

B right

C isosceles

D obtuse

E equilateral

F equiangular

G scalene

4.8 4.8

45° 45°

6.8

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Measure and Classify the triangle by sides and anglesExample

isosceles, acuteClick for AnswerClick for Answer

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scalene, obtuseClick for AnswerClick for Answer

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scalene, acuteClick for AnswerClick for Answer

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20 Classify the triangle with the given information: Side lengths: 3 cm, 4 cm, 5 cm

A EquilateralB IsoscelesC Scalene

D AcuteE EquiangularF RightG Obtuse

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23 Classify the triangle with the given information: Angle Measures: 25°, 120°, 35°



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24 Classify the triangle with the given information: Angle Measures: 30°, 60°, 90°



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25 Classify the triangle with the given information: Side lengths: 3 cm, 4 cm, 5 cmAngle measures: 37°, 53°, 90°



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26 Classify the triangle by sides and angles

A EquilateralB IsoscelesC ScaleneD AcuteE EquiangularF RightG Obtuse

AB

120°

C

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L

MN



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H

J

K45°

85°

50°



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Triangle Sum Theorem


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A

B C

We can use what we learned about parallel lines to determine the sum of the measures of the angles of any triangle.

First, let's draw two parallel lines. The first along the base of the triangle and the other through the opposite vertex.

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And extend AB to make it a transversal.

Then, let's label some of the angles.


A

B C

x

x

y

y

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29 What is the name for the pair of angles labeled x and what is the relationship between them?

A outside exterior, they are unequalB alternate interior, they are unequalC alternate interior, they are equalD outside exterior, they are equal

Is the same true for the pair of angles labeled y?

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A

B C

Therefore, both angles labeled x are equal and can be called x, and x has the same measure as B.

x

x

Repeat the same process with side AC and find an angle along the upper parallel line equal to angle C


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A

B C

x

x

y

y

Let's just re-label the upper angles with A, B and C.


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A

B C

The sum of those angles along that upper parallel line equals 180º, so A + B + C = 180º

B C

We made no special assumptions about this triangle, so this proof applies to all triangles: the sum of the interior angles of any triangle is 180º


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The measures of the interior angles of a triangle sum to 180°

Click here to go to the lab titled, "Triangle Sum Theorem"


A

B C

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Example: Triangle Sum Theorem

320

J

K L 200

Find the measure of the missing angle.

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30 What is m∠B?

A B

C

52°

53°

https://njctl.org/courses/math/geometry/congruent-triangles/triangle-sum-theorem-lab/

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31 What is the measurement of the missing angle?

57°L

M

N

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32 In ΔABC, if m∠B is 84° and m∠C is 36°, what is m∠A?

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33 In ΔDEF, if m∠D is 63° and m∠E is 12°, find m∠F.

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Solve for x

55°

(12x+8)°

(8x-3)°P

Q

R

Example

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Q

R

S2x° 5x°

8x°34 Solve for x.

Then find: m∠Q =

m∠ R =

m∠S =

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35 What is the measure of ∠B?

C

B

A

(3x-17)0

(x+40)0 (2x-5)0

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Corollary to Triangle Sum TheoremThe acute angles of a right triangle are complementary.

A

B

C

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Given: Triangle ABC is a right triangle

Prove: Its acute angles, Angles B and C, are complementary

A

B

C

Proof of Triangle Sum Theorem Corollary

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36 Which reason applies to step 1?A Subtraction Property of EqualityB Substitution Property of EqualityC GivenD Definition of right triangle E Definition of a right angle

A

B

C

Statement Reason1 Triangle ABC is a right triangle ?

2 Right triangles contain a right angle. ?

3 ? Interior Angles Theorem4 m∠A = 90º ?5 90º + m∠B + m∠C = 180º ?

6 m∠B + m∠C = 90º ?

7 ? Definition of complementary

Ans

wer

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A

B

C



3 ? Interior Angles Theorem4 m∠A = 90º ?5 90º + m∠B + m∠C = 180º ?6 m∠B + m∠C = 90º ?


Ans

wer

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38 Which reason applies to step 3?

A

B

C

A The measure of a straight angle is 180ºB m∠A + m∠B + m∠C = 180ºC m∠B + m∠C = 90ºD m∠B + m∠C = 180ºE ∠A is a right angle




6 m∠B + m∠C = 90º ?


Ans

wer

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A

B

CA

nsw

erStatement Reason

1 Triangle ABC is a right triangle ?




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A

B

C




6 m∠B + m∠C = 90º ?


Ans

wer

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A

B

C





Ans

wer

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42 Which reason applies to step 7?

A

B

C




6 m∠B + m∠C = 90º ?


A The measure of a straight angle is 180ºB The sum of the interior angles of a

triangle is 180ºC The acute angles are complementaryD The acute angles are supplementaryE ∠A is a right angle

Ans

wer

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A

B

C

Given: Triangle ABC is a right triangle

Prove: Its acute angles, Angles B and C, are complementary

Statement Reason1 Triangle ABC is a right triangle Given

2 Right triangles contain a right angle. Definition of right triangle

3 m∠A + m∠B + m∠C = 180º Interior Angles Theorem4 m∠A = 90º Definition of right angle

5 90º + m∠B + m∠C = 180º Substitution Property of Equality

6 m∠B + m∠C = 90º Subtraction Property of Equality

7The acute angles are complementary

Definition of complementary

Proof of Triangle Sum Theorem Corollary

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Example

The measure of one acute angle of a right triangle is five times the measure of the other acute angle.

Find the measure of each acute angle.

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43 In a right triangle, the two acute angles sum to 90°

TrueFalse

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44 What is the measurement of the missing angle?

57°L

M

N

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45 Solve for x

A

B CCB

AWhat are the measures of the three angles?

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46 Solve for x

What are the measures of the three angles?

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47 m∠1 + m∠2 =

1

23

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48 m∠1 + m∠3 =

1

23

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20°

X°

49 Find the value of x in the diagram

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Exterior Angle Theorem

Return to Table of Contents

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Exterior angles are formed by extending any side of a triangle.

The exterior angle is then the angle between that extended side and the nearest side of the triangle.

One exterior angle is shown below.

Take a moment and draw another.

Exterior Angles

A

B Cx

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Since a triangle has three vertices and two external angles can be drawn at each vertex, it is possible to draw six external angles to a triangle.

Draw the other external angle at Vertex A.

Exterior Angles

A

B C

x

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A

B C

xx

The exterior angles at each vertex are congruent, since they are vertical angles.

Exterior Angles

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The interior angles of this triangle are ∠A, ∠ABC and ∠C.

Once an exterior angle is drawn, one interior angle is adjacent, and the two others are remote.

Since you can draw exterior angles at any vertex, any interior angle can be the remote depending on at which vertex you draw the external angle.

Remote Interior Angles

A

B Cx

In this case, ∠A and ∠C are the remote interior angles and ∠ABC is the adjacent interior angle.

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50 Which are the remote interior angles in this instance?

A ∠A & ∠BB ∠A & ∠CC ∠B & ∠C

A

B C

xx

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51 If line AB is a straight line, what is the sum of ∠2 and ∠1?

1A B

2

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52 In this diagram, what is the sum of angles P, Q and R?

P

R Q

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A

B Cx

D

The measure of any exterior angle of a triangle is equal to the sum of its remote interior angles.

m∠DBA = m∠A + m∠C

or

x = m∠A + m∠C

Exterior Angles Theorem

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Given: ∠DBA is an exterior angle of ΔABC and ∠A and ∠C are remote interior angles.

Prove: m∠DBA = m∠A + m∠C

Proof of Exterior Angles Theorem

A

B Cx

D

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53 Which reason applies to step 2?A Angles that form a linear pair are supplementaryB Definition of complementaryC Interior Angles TheoremD Substitution Property of EqualityE Definition of a right angle

A

B Cx

D

Statement Reason

1∠DBA is an exterior angle of ΔABC and ∠A and ∠C are remote interior angles

Given

2 ∠DBA and ∠ABC are supplementary ?

3 ? Definition of supplementary

4 m∠A+ m∠ABC + m∠C = 180° ?

5 m∠DBA + m∠ABC = m∠A + m∠ABC + m∠C ?

6 ? Subtraction Property of Equality

Ans

wer

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54 Which statement applies to step 3?A m∠DBA + m∠ABC = 180°B m∠DBA = m∠A + m∠C C m∠A + m∠B = 180° D m∠DBA + m∠A = 90°E m∠DBA + m∠A = 180°

A

B Cx

D

Statement Reason


Given



4 m∠A+ m∠ABC + m∠C = 180° ?



Ans

wer

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A

B Cx

D

Statement Reason


Given



4 m∠A+ m∠ABC + m∠C = 180° ?



Ans

wer

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A

B Cx

D

Statement Reason


Given



4 m∠A+ m∠ABC + m∠C = 180° ?



Ans

wer

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57 Which statement applies to step 6?A m∠DBA + m∠ABC = 180°B m∠DBA = m∠A + m∠C C m∠A + m∠B = 180° D m∠DBA + m∠A = 90°E m∠DBA + m∠A = 180°

A

B Cx

D

Statement Reason


Given



4 m∠A+ m∠ABC + m∠C = 180° ?



Ans

wer

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Statement Reason


Given

2 ∠DBA and ∠ABC are supplementary Angles that form a linear pair are supplementary

3 ∠DBA + m∠ABC = 180° Definition of supplementary

4 m∠A+ m∠ABC + m∠C = 180° Interior Angles Theorem

5 m∠DBA + m∠ABC = m∠A + m∠ABC + m∠C Substitution Property of Equality

6 m∠DBA = m∠A + m∠C Subtraction Property of Equality

Proof of Exterior Angles TheoremGiven: ∠DBA is an exterior angle of ΔABC and ∠A and ∠C are remote interior angles.

Prove: m∠DBA = m∠A + m∠C

A

B Cx

D

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58 In this case, what must be the relationship between the interior angles of ΔPQR and ∠1?

A m∠Q = m∠1B m∠1 = m∠PC m∠1 = m∠Q + m∠RD m∠1 = m∠P + m∠RE m∠1 = m∠Q + m∠P

1

P

R Q

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59 In this case, what must be the relationship between the interior angles of ΔPQR and ∠2?

A m∠Q = m∠2B m∠2 = m∠PC m∠2 = m∠Q + m∠RD m∠2 = m∠P + m∠RE m∠2 = m∠Q + m∠P

2P

R Q

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Example: Using the Exterior Angle Theorem

140ºXº

Xº

P

QR

What is the value of x?

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ExampleSolve for x and y.

21°

34°x° y°

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xº yº

75º

50º

ExampleSolve for x and y.

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60 Solve for x.

xº yº

60º

55º

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61 Solve for y.

xºyº

60º

55º

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62 Find the value of x.

2xº

yº

60º

94º

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(2x+3)º

yº100º

51º

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(x+2)°

y°(3x-5)°

33°

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65 Segment PS bisects ∠RST, what is the value of w?

w

25°

P

S

TR

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Example

Find the missing angles in the diagram.

60°

7

103°

43°45°

30°

5 43

2 1

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40º

1

24 53

60º

66 Find the measure of ∠1.

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40º

1

24 53

60º

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40º

1

24 53

60º

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40º

1

24 53

60º

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40º

1

24 53

60º

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Inequalities in Triangles


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Inequalities in one Triangle

To investigate inequalities in one triangle download the sketch, "inequalities in one triangle" and the worksheet, "inequalities in one triangle"

Go to the sketch, "Inequalities in one

triangle."

Go to the worksheet,"Inequalities in one triangle."

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Angle Inequalities in a Triangle

The longest side is always opposite the largest angle.

The shortest side is always opposite the smallest angle.

page2svg

https://njctl.org/courses/math/geometry/triangles/inequalities-in-one-triangle-sketch/

https://njctl.org/courses/math/geometry/triangles/inequalities-in-one-triangle-worksheet/

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71 Name the longest side of this triangle.A ABB BCC CAD They are all equal

AB

C

35°60°

85°

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72 Name the shortest side of this triangle.A ABB BCC CAD They are all equal

AB

C

35°60°

85°

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73 Name the shortest side of this triangle.A ABB BCC CAD They are all equal

AB

C

35° 105°

40°

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74 Name the largest angle of this triangle.ABCD They are all equal

AB

C

10

148

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75 Name the smallest angle of this triangle.ABCD They are all equal

AB

C

10

148

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A

C

1010

10

76 Name the smallest angle of this triangle.ABCD They are all equal

B

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Length Inequalities in a Triangle

No side can be longer than the sum of the other two sides.

No side can be less than the difference of the other two sides.

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Length Inequalities in a Triangle

No side can be longer than the sum of the other two sides.

This follows from the fact that if the two shorter sides cannot be placed at a 180º angle and exceed the length of the longest side, a triangle cannot be made.

As shown below, if the blue side is longer than the sum of the red and the green side, it cannot form a triangle.

Move the sides below and try to form a triangle.

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Length Inequalities in a TriangleNo side can be less than the difference of the other two sides.

This follows from the fact that if the longer sides cannot, when placed at a 0° angle, reach the end of the shorter side, a triangle cannot be made.

As shown below, if the blue side is too short to reach the red line, even when the red line is at the smallest angle, it cannot form a triangle.

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77 What is the maximum length of the third side to form a triangle if the other sides are 4 and 6?

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78 What is the maximum length of the third side to form a triangle if the other sides are 8 and 7?

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79 What is the minimum length of the third side to form a triangle if the other sides are 4 and 6?

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80 What is the minimum length of the third side to form a triangle if the other sides are 7 and 8?

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Similar Triangles


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Recall that:

Congruence

Two objects are congruent if they can be moved, by any combination of translation, rotation and reflection, so that every part of each object overlaps.

This is the symbol for congruence:

If a is congruent to b, this would be shown as

which is read as "a is congruent to b."

a b

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Only line segments with the same length are congruent.

Also, all congruent segments have the same length.

We learned earlier that:

Congruent Line Segments

ab

cd

c da b

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Recall:

Congruent Angles

A B∠ ∠ ∠C ∠D

Two angles are congruent if they have the same measure.

Two angles are not congruent if they have different measures.

AB

C

D

If m∠A = m∠B If m∠C # m∠D

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Congruent Triangles

Triangles are made up of

three line segments AND three angles

For one triangle to be congruent to another

all three sides AND all three angles must be congruent.

page2svg

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Similar Triangles

If all the sides of two triangles are congruent, we will soon show that all the angles are also congruent.

Therefore, the triangles are congruent.

However, two triangles can have all their angles congruent, with all or none of their sides being congruent.

In that case, they are said to be Similar Triangles.

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Congruent Triangles

Congruent Triangles are also Similar Triangles since their angles are all congruent.

Congruent triangles are therefore a special case of similar triangles.We will focus on similar triangles first, and then work

with congruent triangles in a later unit.

Similar triangles represent a great tool to solve problems, and are the foundation of trigonometry.

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Similar triangles have the same shape, but can have different sizes.

If they have the same shape and are the same size, they are both similar and congruent.

A

B

C D

E

F

Similar Triangles Have Proportional Sides Theorem

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Similar Triangles

This is the symbol for similarity

So, the symbolic statement for

Triangle ABC is similar to Triangle DEF

is:

DEFDEFΔABC Δ

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Naming Similar Triangles

This statement tells you more than that the triangles are similar.

It also tells you which angles are equal.

In this case, that

m∠A = m∠D m∠B = m∠E m∠C = m∠F

And, thereby which are the corresponding, proportional, sides.

AB corresponds to DEBC corresponds to EFCA corresponds to FD

DEFDEFΔABC Δ

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Naming Similar Triangles

So, when you are naming similar triangles, the order of the letters matters.

They don't have to be alphabetical.

But they have to be named so that equal angles correspond to one another.

DEFDEFΔABC Δ

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Proving Triangles Similar

If you can prove that all three angles of two triangles are congruent, you have directly proven that they are similar.

However, there are shortcuts to proving triangles similar.

We will explore three sets of conditions that imply the three angles of two triangles are congruent, meaning that the triangles must be similar.

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Angle-Angle Similarity Theorem

We know from the Triangle Sum Theorem that the sum of the interior angles of a triangle is always 180o.

So, if two triangles have two pair of congruent angles which sum to x, then the third angle in both triangles must be (180o - x) ....forming three congruent pairs of angles.

One way to prove that two triangles are similar is to prove that two of the angles in each triangle are

congruent.

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Angle-Angle Similarity TheoremIf two angles of a triangle are congruent to two angles of another triangle, their third angles are congruent and the triangles are similar.

Here's the proof:Statement Reason

1 ∠A and ∠B in ΔABC are ≅ to ∠D and ∠E in ΔDEF

Given

2 m∠A = m∠D; m∠B = m∠E Definition of Congruent Angles

3 m∠A+ m∠B + m∠C = 180ºm∠D+ m∠E + m∠F = 180º


4 m∠C =180º - (m∠A + m∠B) m∠F =180º- (m∠D + m∠E)

Subtraction Property of Equality

5 m∠C =180º - (m∠A + m∠B) m∠F =180º- (m∠A + m∠B)

Substitution Property of Equality

6 m∠C = m∠F Substitution Property of Equality

7 ΔABC and ΔDEF are similar Definition of Similarity

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If two triangles have their sides proportional, the triangles will be equiangular and will have those angles equal which their corresponding sides subtend.

Euclid - Book Six: Proposition 5

Equiangular triangles are similar, so this states that triangles with proportional sides are similar.

This is a second way to prove triangles are similar:

If you can prove that all three pairs of sides in two triangles are proportional, then you have proven the

triangles similar.

Side-Side-Side Similarity Theorem

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Side-Side-Side Similarity TheoremThis follows from the way we constructed congruent angles.

We made use of the fact that if angles are congruent, their sides are separating at the same rate as you move away from the vertex. Here's the drawing we used to construct ∠ABC so it would be congruent to ∠FGH.

F

G H

A

CB

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Side-Side-Side Similarity TheoremIf we draw the green line segments connecting the points where the blue arcs intersect the rays, we can see that the length of that segment would be the same for both angles.

Since the angles are congruent, the line segment opposite those angles will also be congruent, if it intersects both sides of the angle at the same distance from the vertex in both cases.

F

G H

A

CB

D

E

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Side-Side-Side Similarity TheoremIn this case the segments AC and DE will be congruent since segments GD and GE are also congruent to segments AB and BC.

Therefore ΔDEG is congruent to ΔABC, since all the sides and angles are the same.

Changing the scale of ΔABC won't change the angle measures. The sides would then be in proportion to those of ΔDEG, but not equal.

F

G H

A

CB

D

E

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A

CB

Side-Side-Side Similarity TheoremThe diagram below shows an expansion of ΔABC and we see that the measures of the angles are unchanged.

They are still similar triangles. The corresponding sides are in proportion.

F

G H

D

E

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A

CB

Side-Side-Side Similarity TheoremRemoving the arcs and shifting the smaller triangle within the larger makes it clear that all angles are congruent and the sides are in proportion.

So, the second way to prove triangles similar is to show that all their sides are in proportion.

F

D

EG H

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Side-Angle-Side Similarity Theorem

If two triangles have one angle equal to another and the sides about the equal angle are in proportion, the triangles will be equiangular and will have those angles equal which the corresponding sides subtend.

Euclid's Elements - Book Six: Proposition 6

The third way to prove triangles are similar is to show they share an angle which is equal and the two sides forming that angle are proportional in the two triangles.

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This directly follows from the work we just did to show that Side-Side-Side proportionality can be used to prove triangles are similar.

If you recall, the line segment which makes up the third side of a triangle is completely defined by its opposite angle and the lengths of the other two sides.

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If the angles are congruent and the two sides of the angle are in proportion, the third side must also be in proportion.

If all three sides are in proportion, the triangles must be similar due to the Side-Side-Side Theorem.

You can see that on the next page.

Slide 157 / 210

A

B

C D

E

F

Side-Angle Side Similarity TheoremIf ∠B ≅ ∠E and segments AB and BC are proportional to segments ED and EF, then segment AC must also be proportional to segment DF. Since all the sides are in proportion, the triangles are similar.

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Common Error

You CANNOT prove triangles similar using Side-Side-Angle.

This is not the same as Side-Angle-Side.

As shown below, two triangles can have two corresponding sides and one corresponding angle congruent, but NOT be similar.

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81 Which theorem allows you to prove these two triangles are similar?

A Angle-AngleB Side-Angle-SideC Side-Side-SideD They may not be similar

x

x

E They are not similar

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A Angle-AngleB Side-Angle-SideC Side-Side-SideD They may not be similarE They are not similar

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A Angle-AngleB Side-Angle-SideC Side-Side-Side

D They may not be similar

6

4

88

12


16

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4 8

36

6 10


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4 8

36

xx


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4

3x

8

6

xE They are not similar

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Slide 166 / 210



Slide 167 / 210



Slide 168 / 210



A

B C

D ENote that BC is parallel to DE.

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A

B C

D E

Side Splitter Theorem

Any line parallel to a side of a triangle will form a triangle which is similar to the first triangle.

As we will learn later, it also makes all the sides proportional, splitting them...hence the name of the theorem.

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A

B C

D E

Proof of Side Splitter Theorem

Given: BC is parallel to DE

Prove: ΔABC ~ ΔADE.

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91 What is the reason for step 2?

A Angle-Angle Similarity TheoremB Side-Side-Side Similarity TheoremC Reflexive Property of Congruence D When two parallel lines are intersected by a transversal,

the corresponding angles are congruent.E When two parallel lines are intersected by a transversal,

the alternate interior angles are congruent.

A

B C

D E

Ans

wer

Statement Reason1 BC is parallel to DE Given2 ∠ABC ≅ ∠D; ∠ACB ≅ ∠E ?3 ∠A ≅ ∠A ?4 ΔABC ~ ΔADE ?

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A Angle-Angle Similarity TheoremB Side-Side-Side Similarity TheoremC Reflexive Property of Congruence D When two parallel lines are intersected by a transversal,



Ans

wer

A

B C

D E


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A Angle-Angle Similarity Theorem

B Side-Side-Side Similarity Theorem

C Reflexive Property of Congruence D When two parallel lines are intersected by a transversal,



A

B C

D E


Ans

wer

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Proof of Side Splitter Theorem

Given: BC is parallel to DE

Prove: ΔABC ~ ΔADE

A

B C

D EStatement Reason

1 BC is parallel to DE Given

2 ∠ABC ≅ ∠D; ∠ACB ≅ ∠EWhen two parallel lines are

intersected by a transvesal, the corresponding angles are congruent

3 ∠A ≅ ∠A Reflexive Property of Congruence

4 ΔABC ~ ΔADE Angle-Angle Similarity Theorem

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Similar triangles have the same shape, but can have different sizes. If they have the same shape and are the same size, they are congruent.

If they have the same shape and are different sizes, they are similar and their sides are in proportion.

A

B

C D

E

F


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The converse is also true, and will prove very useful.

If two triangles are similar, all of their corresponding sides are in proportion.

*While Euclid does prove this theorem, his proof relies on other theorems which would have to be proven first and would take us beyond the scope of this course. So, we'll just rely on this theorem and note that the proof is available in The Elements by Euclid - Book Six: Proposition 5.


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Similar Triangles and Proportionality

A

B

C D

E

F

In the triangles below, if we know that

m∠A = m∠D, m∠B = m∠E, and m∠C = m∠F,

then we know that the triangles are similar.

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A

B

C D

E

F

We also then know that the corresponding sides are proportional.

The symbol for proportional is the Greek letter, alpha: #

AB α DE, since AB corresponds to DEBC α EF, since BC corresponds to EFAC α DF, since AC corresponds to DF

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Corresponding Sides

A

B

C D

E

F

Our work with similar triangles and our future work with congruent triangles requires us to identify the corresponding sides.

One way to do that is to locate the sides opposite congruent angles. If we know triangles ABC and EDF are similar and that angle A is congruent to angle D, then the sides opposite A and D are in proportion: BC α EF

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Corresponding Sides

A

B

C D

E

F

Another way of identifying corresponding sides is to use Euclid's description "...those angles [are] equal which their corresponding sides subtend."

Below, since angle A is equal to angle D and angle B is equal to angle E, then sides AB and DE are in proportion.

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Corresponding Sides

A

B

C D

E

F

Either approach works; use the one you find easiest.

Identify corresponding sides as the sides connecting equal angles or the sides opposite equal angles...you'll get the same result.

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A

B

C D

E

F

Another way of saying two sides are proportional is to say that one is a scaled-up version of the other. If you multiply all the sides of one triangle by the same scale factor, k, you get the other triangle. In this case, if ΔABC is k times as big as ΔDEF, then: AB = kDE BC = kEF AC = kDF

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A

B

C D

E

F

Or, dividing the corresponding sides yields:

AB BC ACDE EF DF = k= =

This property of proportionality is very useful in solving problems using similar triangles, and provides the foundation for trigonometry.

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94 If m∠A = m∠D, m∠B = m∠E, and m∠C = m∠F, identify which side corresponds to side AB.

A DEB EFC FG

A

B

C D

E

F

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95 If m∠I = m∠M, m∠H = m∠N, and m∠J = m∠L, identify which side corresponds to side IJ.

A MNB NLC ML

I

J

H

M

N

L

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A

B

C8 D

E

F4

Example - Proportional Sides

Given that ΔABC is similar to ΔDEF, and given the indicated lengths, find the lengths AB and BC.

5 7

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Since the triangles are similar we know that the following relationship holds between all the corresponding sides.

First, let's find the constant of proportionality, k, by using the two sides for which we have values: AC and DF

AB BC ACED EF DF = k= =

A

B

C8 D

E

F4

5 7

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A

B

C

5 7

8 D

E

F4


AB BC ACED EF DF = k = 2= =

AC 8 DF 4= = k = 2

That means that the other two sides of ΔABC will also be twice as large as the corresponding sides of ΔDEF

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A

B

C

5 7

8 D

E

F4


AB ED = 2 BC

EF = 2

AB 5 = 2

AB = 10

BC 7 = 2

BC = 14

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96 Given that m∠A = m∠D, m∠B = m∠E, and m∠C = m∠F. If BC = 8, DE = 6, and AB = 4, EF = ?

A

B

C D

E

F

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97 Given that ΔJIH is similar to ΔLMN; find LM.

I

J

H

M

N

L

14

10

12

5

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98 Given that ΔJIH is similar to ΔLMN; find LN.

I

J

H

M

N

L

14

10

12

5

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99 Given that BC is parallel to DE and the given lengths, find DE.

A

B C

D E

8

64

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100 Given that BC is parallel to DE and the given lengths, find DB.

A

B C

D E9

7

3

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Example - Similarity & Proportional Sides

D

P

K

12

9

18

R

L

B6

1210

Determine if the triangles are similar. If they are similar write a similarity statement. If they are not similar, explain why.

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D

P

K

12

9

18

R

L

B6

1210

To identify the corresponding sides without wasting a lot of time, first list all the sides from shortest to longest of both triangles and compare to see if they are all proportional.

Then you can identify corresponding sides and the constant of proportionality.

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D

P

K

15

9

18

R

L

B6

1210

Side of ΔPDK Length Side of ΔBRL Length Ratio

DK 9 BR 6 1.5

PD 15 RL 10 1.5

PK 18 BL 12 1.5

All corresponding sides are in the ratio of 1.5:1, so the triangles are similar.

This also provides the order of the sides, so we can say that ΔKDP is similar to ΔBRL. Check to make sure that all the sides are in the correct order.

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101 If these triangles are similar, enter the constant of proportionality, k, between the larger and smaller triangle. If they are not, enter zero.

D

P

K

12

9

18

R

L

B6

1210

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52°

1

2

3

R

S

T

52°24

6X

Y

Z

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P

R

S

3 4.2

6B

C

D

2 2.8

4

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A

B C

D E

Converse of Side Splitter Theorem

If a line divides the two sides of a triangle proportionally, then the line is parallel to the third side.

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104 Find the value of x to prove that AB is parallel to ER.

27

x

18 12R

EA

B

D

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105 Find the value of x to prove that FC is parallel to MN.

J

M

NC

F

x 9

6

8

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106 Find the value of y.

6

1012

y

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4

14 12

y

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2415y

6

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Questions from Released

PARCC Test


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Question 1/7

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109 The figure ΔABC ~ ΔDEF with side lengths as indicated. What is the value of x?

F

D

E

95

7

C

BA

27

21

x

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