Unit 7: Modeling with Geometric Relationships

Unit 7: Modeling with Geometric Relationships
Mathematics 3 Ms. C. Taylor

Warm-Up A reporter wants to know the percentage of voters in the state who support building a new highway. What is the reporter’s population? A) the number of people who live in the state B) the people who were interviewed in the state C) all voters over 25 years old in the state D) all eligible voters in the state

Distance around an Arc An arc of a circle is a segment of the circumference of the circle. Arc length of a circle in radians: 𝑆=𝜃𝑟 Arc length of a circle in degrees: 𝑆=𝜃 𝜋 180 𝑟

Constructions https://www.youtube.com/watch?v=fVDr08 YbQww
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Warm-Up Find the arc length of a circle if the diameter is 18 and the angle is .

Theorems Galore! If two lines intersect to form a linear pair of congruent angles, then the lines are perpendicular. Proof Hints: Write an equation based on the angles forming a linear pair. Do some substitution and solve for one of the angles. It should be 90°.

Theorems Galore! If two lines are perpendicular, then they intersect to form four right angles. Proof Hints: Use definition of perpendicular lines to find one right angle. Use vertical and linear pairs of angles to find three more right angles.

Theorems Galore! If two sides of two adjacent acute angles are perpendicular, then the angles are complementary. Proof Hints: Use definition of perpendicular to get the measure of ABC. Use Angle Addition Postulate, Substitution, and Definition of complementary angles to finish the proof.

Theorems Galore! Perpendicular Transversal Theorem If a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other. j Proof Hints: Use definition of perpendicular lines to find one right angle. Use Corresponding Angles Postulate to find a right angle on the other line.

Theorems Galore! Lines Perpendicular to a Transversal Theorem
In a plane, if two lines are perpendicular to the same line, then they are parallel to each other. Proof Hints: Use definition of perpendicular lines to find a right angle on each parallel line. Use Converse of Corresponding Angles Postulate to prove the lines are parallel.

Proving Vertical Angle Theorem
Vertical Angles Theorem: Vertical angles are congruent ∠1≅∠3 ∠2≅∠4

Proving Vertical Angle Theorem
GIVEN 5 and 6 are a linear pair, 6 and 7 are a linear pair PROVE 5 7 Statements Reasons 5 and 6 are a linear pair, Given 1 6 and 7 are a linear pair 2 5 and 6 are supplementary, Linear Pair Postulate 6 and 7 are supplementary 3 5 7 Congruent Supplements Theorem

The Third Angles Theorem below follows from the Triangle Sum Theorem.
If two angles of one triangle are congruent to two angles of another triangle, then the third angles are also congruent. If  A   D and  B   E, then  C   F.

PROPERTIES OF PARALLEL LINES
POSTULATE 15 Corresponding Angles Postulate If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent. 1 2

THEOREM 3.4 Alternate Interior Angles If two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent. 3 4

THEOREM 3.5 Consecutive Interior Angles If two parallel lines are cut by a transversal, then the pairs of consecutive interior angles are supplementary. 5 6 m m 6 = 180°

THEOREM 3.6 Alternate Exterior Angles If two parallel lines are cut by a transversal, then the pairs of alternate exterior angles are congruent. 7 8

THEOREM 3.7 Perpendicular Transversal If a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other. j k

Proving the Alternate Interior Angles Theorem
Prove the Alternate Interior Angles Theorem. SOLUTION GIVEN p || q PROVE Statements Reasons p || q Given 1 1  Corresponding Angles Postulate 2 3 3  Vertical Angles Theorem 1  Transitive property of Congruence 4

Using Properties of Parallel Lines
Given that m = 65°, find each measure. Tell which postulate or theorem you use. SOLUTION m = m = 65° Vertical Angles Theorem m = 180° – m = 115° Linear Pair Postulate m = m = 65° Corresponding Angles Postulate m = m = 115° Alternate Exterior Angles Theorem

Using Properties of Parallel Lines
Use properties of parallel lines to find the value of x. SOLUTION Corresponding Angles Postulate m = 125° Linear Pair Postulate m (x + 15)° = 180° Substitute. 125° + (x + 15)° = 180° x = 40° Subtract.

Using the Third Angles Theorem
Find the value of x. SOLUTION In the diagram,  N   R and  L   S. From the Third Angles Theorem, you know that  M   T. So, m M = m T. From the Triangle Sum Theorem, m M = 180˚– 55˚ – 65˚ = 60˚. m M = m T 60˚ = (2 x + 30)˚ 30 = 2 x 15 = x Third Angles Theorem Substitute Subtract 30 from each side. Divide each side by 2.

PROPERTIES OF SPECIAL PAIRS OF ANGLES
Congruent Supplements Theorem If two angles are supplementary to the same angle (or to congruent angles) then they are congruent. 2 3 1

Linear Pair Postulate If two angles for m a linear pair, then they are supplementary. m 1 + m 2 = 180°

CONGRUENCE OF ANGLES THEOREM 2.2 Properties of Angle Congruence
Angle congruence is reflexive, symmetric, and transitive. Here are some examples. REFLEX IVE For any angle A, A  A SYMMETRIC If A  B, then B  A TRANSITIVE If A  B and B  C, then A  C

Transitive Property of Angle Congruence
Prove the Transitive Property of Congruence for angles. SOLUTION To prove the Transitive Property of Congruence for angles, begin by drawing three congruent angles. Label the vertices as A, B, and C. GIVEN A B, PROVE A C B C A C B

Transitive Property of Angle Congruence
GIVEN A B, B C PROVE A C Statements Reasons 1 A  B, Given B  C 2 m A = m B Definition of congruent angles m B = m C Definition of congruent angles 3 4 m A = m C Transitive property of equality A  C Definition of congruent angles 5

Using the Transitive Property
This two-column proof uses the Transitive Property. GIVEN m 3 = 40°, 1 2, 2 3 PROVE m 1 = 40° Statements Reasons Given 1 m 3 = 40°, 1 2, 2 3 2 1 3 Transitive property of Congruence 3 m 1 = m 3 Definition of congruent angles 4 m 1 = 40° Substitution property of equality

Proving Right Angle Congruence Theorem
All right angles are congruent. You can prove Right Angle CongruenceTheorem as shown. GIVEN 1 and 2 are right angles PROVE 1 2

Proving Right Angle Congruence Theorem
GIVEN 1 and 2 are right angles PROVE 1 2 Statements Reasons 1 and 2 are right angles Given 1 2 m 1 = 90°, m 2 = 90° Definition of right angles 3 4 1  Definition of congruent angles

Congruent Supplements Theorem If two angles are supplementary to the same angle (or to congruent angles) then they are congruent. 1 2 3 1 If m 1 + m 2 = 180° and then m 2 + m 3 = 180° 1  3

Congruent Complements Theorem If two angles are complementary to the same angle (or to congruent angles) then the two angles are congruent. 4 5 6

Congruent Complements Theorem If two angles are complementary to the same angle (or to congruent angles) then the two angles are congruent. 5 6 4 4 If m 4 + m 5 = 90° and m 5 + m 6 = 90° then 4  6

Proving Congruent Supplements Theorem
GIVEN 1 and 2 are supplements 3 and 4 are supplements 1 4 PROVE 2 3 Statements Reasons 1 1 and 2 are supplements Given 3 and 4 are supplements 1  4 2 m 1 + m 2 = 180° Definition of supplementary angles m 3 + m 4 = 180°

GIVEN 1 and 2 are supplements 3 and 4 are supplements 1 4 PROVE 2 3 Statements Reasons 3 m 1 + m 2 = Transitive property of equality m 3 + m 4 4 m 1 = m 4 Definition of congruent angles 5 m 1 + m 2 = Substitution property of equality m 3 + m 1

GIVEN 1 and 2 are supplements 3 and 4 are supplements 1 4 PROVE 2 3 Statements Reasons 6 m 2 = m 3 Subtraction property of equality 7 2 3 Definition of congruent angles

Proving Triangles Congruent
D GIVEN PROVE  DRA  DRG DR AG RA RG A R G Statements Reasons Given DR AG 1 If 2 lines are , then they form 4 right angles. DRA and DRG are right angles. 2 3 Right Angle Congruence Theorem DRA  DRG 4 Given RA  RG 5 Reflexive Property of Congruence DR  DR 6 SAS Congruence Postulate  DRA   DRG

Congruent Triangles in a Coordinate Plane
Use the SSS Congruence Postulate to show that  ABC   FGH. sOLUTION AC  FH AB  FG AB = 5 and FG = 5 AC = 3 and FH = 3

Use the distance formula to find lengths BC and GH. d = (x 2 – x1 ) 2 + ( y2 – y1 ) 2 = = BC = (– 4 – (– 7)) 2 + (5 – 0 ) 2 = GH = (6 – 1) 2 + (5 – 2 ) 2

BC  GH BC = and GH = 34 All three pairs of corresponding sides are congruent,  ABC   FGH by the SSS Congruence Postulate.

Identifying Congruent Figures
Two geometric figures are congruent if they have exactly the same size and shape. Each of the red figures is congruent to the other red figures. None of the blue figures is congruent to another blue figure. Goal 1

Identifying Congruent Figures
When two figures are congruent, there is a correspondence between their angles and sides such that corresponding angles are congruent and corresponding sides are congruent. For the triangles below, you can write  , which reads “triangle ABC is congruent to triangle PQR.” The notation shows the congruence and the correspondence. ABC PQR Corresponding Angles Corresponding Sides  A   P  B   Q  C   R BC  QR RP CA  AB  PQ There is more than one way to write a congruence statement, but it is important to list the corresponding angles in the same order. For example, you can also write  BCA QRP

Naming Congruent Parts
The two triangles shown below are congruent. Write a congruence statement. Identify all pairs of congruent corresponding parts. SOLUTION The diagram indicates that  The congruent angles and sides are as follows. DEF RST Angles:  D   R,  E   S,  F   T Sides:  ,   , RS DE TR FD ST EF

Using Properties of Congruent Figures
In the diagram, NPLM  EFGH. Find the value of x. SOLUTION You know that  GH LM So, LM = GH. 8 = 2 x – 3 11 = 2 x 5.5 = x Example

Using Properties of Congruent Figures
In the diagram, NPLM  EFGH. Find the value of x. You know that  GH LM So, LM = GH. 8 = 2 x – 3 11 = 2 x 5.5 = x Find the value of y. SOLUTION SOLUTION You know that  N   E. So, m N = m E. 72˚ = (7y + 9)˚ 63 = 7y 9 = y

Proving Triangles are Congruent
Decide whether the triangles are congruent. Justify your reasoning. SOLUTION Paragraph Proof From the diagram, you are given that all three corresponding sides are congruent.  , NQ PQ MN RP  QM QR and Because  P and  N have the same measures,  P   N. By the Vertical Angles Theorem, you know that  PQR   NQM. By the Third Angles Theorem,  R   M. So, all three pairs of corresponding sides and all three pairs of corresponding angles are congruent. By the definition of congruent triangles,  PQR NQM

Proving Two Triangles are Congruent
Prove that  AEB DEC A B C D E GIVEN || , DC AB  , E is the midpoint of BC and AD. PROVE  AEB DEC Plan for Proof Use the fact that  AEB and  DEC are vertical angles to show that those angles are congruent. Use the fact that BC intersects parallel segments AB and DC to identify other pairs of angles that are congruent.

Proving Two Triangles are Congruent
Prove that  AEB DEC A B C D E SOLUTION Statements Reasons || , DC AB  Given  EAB   EDC,  ABE   DCE Alternate Interior Angles Theorem  AEB   DEC Vertical Angles Theorem E is the midpoint of AD, E is the midpoint of BC Given  , DE AE  CE BE Definition of midpoint  AEB DEC Definition of congruent triangles

Proving Triangles are Congruent
You have learned to prove that two triangles are congruent by the definition of congruence – that is, by showing that all pairs of corresponding angles and corresponding sides are congruent. Theorem 4.4 Properties of Congruent Triangles Reflexive Property of Congruent Triangles D E F A B C J K L Every triangle is congruent to itself. Symmetric Property of Congruent Triangles Transitive Property of Congruent Triangles If  , then  ABC DEF If  and  , then  JKL

SSS AND SAS CONGRUENCE POSTULATES
If all six pairs of corresponding parts (sides and angles) are congruent, then the triangles are congruent. If Sides are congruent Angles are congruent Triangles are congruent and then 1. AB DE A D 2. BC EF B E 3. AC DF C F  ABC  DEF

Using the SSS Congruence Postulate
Prove that  PQW  TSW. SOLUTION Paragraph Proof The marks on the diagram show that PQ  TS, PW  TW, and QW  SW. So by the SSS Congruence Postulate, you know that  PQW   TSW.

SSS AND SAS CONGRUENCE POSTULATES
POSTULATE: Side-Angle-Side (SAS) Congruence Postulate If two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the two triangles are congruent. If Side PQ WX Side QS XY Angle Q X A S then  PQS WXY

Using the SAS Congruence Postulate
Prove that  AEB  DEC. 2 1 Statements Reasons AE  DE, BE  CE Given 1 1  2 Vertical Angles Theorem 2  AEB   DEC SAS Congruence Postulate 3

If 2 lines are , then they form 4 right angles.
Proving Triangles Congruent D GIVEN PROVE  DRA  DRG DR AG RA RG A R G Statements Reasons Given DR AG 1 If 2 lines are , then they form 4 right angles. DRA and DRG are right angles. 2 3 Right Angle Congruence Theorem DRA  DRG 4 Given RA  RG 5 Reflexive Property of Congruence DR  DR 6 SAS Congruence Postulate  DRA   DRG

AC  FH AB  FG AB = 5 and FG = 5 SOLUTION Use the SSS Congruence Postulate to show that  ABC   FGH. AC = 3 and FH = 3

d = (x 2 – x1 ) 2 + ( y2 – y1 ) 2 = = BC = (– 4 – (– 7)) 2 + (5 – 0 ) 2 = GH = (6 – 1) 2 + (5 – 2 ) 2 Use the distance formula to find lengths BC and GH.

BC  GH BC = √34 and GH = √34 All three pairs of corresponding sides are congruent,  ABC   FGH by the SSS Congruence Postulate.

Theorems Theorem 6.6: If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. ABCD is a parallelogram.

Theorems Theorem 6.7: If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram. ABCD is a parallelogram.

Theorems x° (180 – x)° Theorem 6.8: If an angle of a quadrilateral is supplementary to both of its consecutive angles, then the quadrilateral is a parallelogram. ABCD is a parallelogram.

Theorems Theorem 6.9: If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. ABCD is a parallelogram.

Ex. 1: Proof of Theorem 6.6 Reasons: Statements: Given
AB ≅ CD, AD ≅ CB. AC ≅ AC ∆ABC ≅ ∆CDA BAC ≅ DCA, DAC ≅ BCA AB║CD, AD ║CB. ABCD is a  Reasons: Given

Ex. 1: Proof of Theorem 6.6 Statements: AB ≅ CD, AD ≅ CB. AC ≅ AC
∆ABC ≅ ∆CDA BAC ≅ DCA, DAC ≅ BCA AB║CD, AD ║CB. ABCD is a  Reasons: Given Reflexive Prop. of Congruence

∆ABC ≅ ∆CDA BAC ≅ DCA, DAC ≅ BCA AB║CD, AD ║CB. ABCD is a  Reasons: Given Reflexive Prop. of Congruence SSS Congruence Postulate

∆ABC ≅ ∆CDA BAC ≅ DCA, DAC ≅ BCA AB║CD, AD ║CB. ABCD is a  Reasons: Given Reflexive Prop. of Congruence SSS Congruence Postulate CPCTC

∆ABC ≅ ∆CDA BAC ≅ DCA, DAC ≅ BCA AB║CD, AD ║CB. ABCD is a  Reasons: Given Reflexive Prop. of Congruence SSS Congruence Postulate CPCTC Alternate Interior s Converse

Ex. 1: Proof of Theorem 6.6 Reasons: Statements: Given
Reflexive Prop. of Congruence SSS Congruence Postulate CPCTC Alternate Interior s Converse Def. of a parallelogram. Statements: AB ≅ CD, AD ≅ CB. AC ≅ AC ∆ABC ≅ ∆CDA BAC ≅ DCA, DAC ≅ BCA AB║CD, AD ║CB. ABCD is a 

Using properties of parallelograms.
Method 1 Use the slope formula to show that opposite sides have the same slope, so they are parallel. Method 2 Use the distance formula to show that the opposite sides have the same length. Method 3 Use both slope and distance formula to show one pair of opposite side is congruent and parallel.

Ex. 2: Proving Quadrilaterals are Parallelograms
As the sewing box below is opened, the trays are always parallel to each other. Why?

Ex. 2: Proving Quadrilaterals are Parallelograms
Each pair of hinges are opposite sides of a quadrilateral. The 2.75 inch sides of the quadrilateral are opposite and congruent. The 2 inch sides are also opposite and congruent. Because opposite sides of the quadrilateral are congruent, it is a parallelogram. By the definition of a parallelogram, opposite sides are parallel, so the trays of the sewing box are always parallel.

Another Theorem ~ Theorem 6.10—If one pair of opposite sides of a quadrilateral are congruent and parallel, then the quadrilateral is a parallelogram. ABCD is a parallelogram. B C A D

Ex. 3: Proof of Theorem 6.10 Given: BC║DA, BC ≅ DA Prove: ABCD is a 
Statements: BC ║DA DAC ≅ BCA AC ≅ AC BC ≅ DA ∆BAC ≅ ∆DCA AB ≅ CD ABCD is a  Reasons: 1. Given

Statements: BC ║DA DAC ≅ BCA AC ≅ AC BC ≅ DA ∆BAC ≅ ∆DCA AB ≅ CD ABCD is a  Reasons: Given Alt. Int. s Thm.

Statements: BC ║DA DAC ≅ BCA AC ≅ AC BC ≅ DA ∆BAC ≅ ∆DCA AB ≅ CD ABCD is a  Reasons: Given Alt. Int. s Thm. Reflexive Property

Statements: BC ║DA DAC ≅ BCA AC ≅ AC BC ≅ DA ∆BAC ≅ ∆DCA AB ≅ CD ABCD is a  Reasons: Given Alt. Int. s Thm. Reflexive Property SAS Congruence Post.

Statements: BC ║DA DAC ≅ BCA AC ≅ AC BC ≅ DA ∆BAC ≅ ∆DCA AB ≅ CD ABCD is a  Reasons: Given Alt. Int. s Thm. Reflexive Property SAS Congruence Post. CPCTC

Statements: BC ║DA DAC ≅ BCA AC ≅ AC BC ≅ DA ∆BAC ≅ ∆DCA AB ≅ CD ABCD is a  Reasons: Given Alt. Int. s Thm. Reflexive Property SAS Congruence Post. CPCTC If opp. sides of a quad. are ≅, then it is a .

Ex. 4: Using properties of parallelograms
Show that A(2, -1), B(1, 3), C(6, 5) and D(7,1) are the vertices of a parallelogram.

Method 1—Show that opposite sides have the same slope, so they are parallel. Slope of AB. 3-(-1) = - 4 1 - 2 Slope of CD. 1 – 5 = - 4 7 – 6 Slope of BC. 5 – 3 = 2 Slope of DA. - 1 – 1 = 2 AB and CD have the same slope, so they are parallel. Similarly, BC ║ DA. Because opposite sides are parallel, ABCD is a parallelogram.

Method 2—Show that opposite sides have the same length. AB=√(1 – 2)2 + [3 – (- 1)2] = √17 CD=√(7 – 6)2 + (1 - 5)2 = √17 BC=√(6 – 1)2 + (5 - 3)2 = √29 DA= √(2 – 7)2 + (-1 - 1)2 = √29 AB ≅ CD and BC ≅ DA. Because both pairs of opposites sides are congruent, ABCD is a parallelogram.

Method 3—Show that one pair of opposite sides is congruent and parallel. Slope of AB = Slope of CD = -4 AB=CD = √17 AB and CD are congruent and parallel, so ABCD is a parallelogram.

Proving quadrilaterals are parallelograms:
Show that both pairs of opposite sides are parallel. Show that both pairs of opposite sides are congruent. Show that both pairs of opposite angles are congruent. Show that one angle is supplementary to both consecutive angles.

.. continued.. Show that the diagonals bisect each other
Show that one pair of opposite sides are congruent and parallel.

Engineering Deshon uses an expandable gate to keep his new puppy in the kitchen. As the gate expands or collapses, the shapes that form the gate always remain parallelograms. Explain why this is true.

Drafting Before computer drawing program become available, blueprints for buildings or mechanical parts were drawn by hand. One of the tools drafters used, is a parallel ruler. Holding one of the bars in the place and moving the other allowed the drafter to draw a line ll to the first in many position on the page. Why does the parallel ruler guarantee that the second line will be ll to the first?

Arts The Navoja people are well known for their skills in weaving. Eye- Dazzler rugs became popular with Navoja weavers in the 1880s.What types of shapes do you see most?

Side-Side-Side (SSS) Similarity Theorem
If the corresponding side lengths of two triangles are proportional, then the triangles are similar.

Use the SSS Similarity Theorem
Is either DEF or GHJ similar to ABC? SOLUTION Compare ABC and DEF by finding ratios of corresponding side lengths. Remaining sides Shortest sides Longest sides CA FD 4 3 16 12 = BC EF 4 3 12 9 = AB DE 4 3 8 6 = All of the ratios are equal, so ABC ~ DEF. ANSWER

Use the SSS Similarity Theorem (continued)
Compare ABC and GHJ by finding ratios of corresponding side lengths. Remaining sides Longest sides Shortest sides BC HJ 6 5 12 10 = AB GH 8 = 1 CA JG 16 = 1 The ratios are not all equal, so ABC and GHJ are not similar. ANSWER

Which of the three triangles are similar? Write a similarity statement.

Find the value of x that makes ABC ~ DEF. SOLUTION STEP 1 Find the value of x that makes corresponding side lengths proportional. 4 12 = x –1 18 Write proportion. = 12(x – 1) Cross Products Property 72 = 12x – 12 Simplify. 7 = x Solve for x.

Use the SSS Similarity Theorem (continued)
Check that the side lengths are proportional when x = 7. STEP 2 DF = 3(x + 1) = 24 BC = x – 1 = 6 AB DE BC EF = ? 6 18 4 12 = AB DE AC DF = ? 8 24 4 12 = When x = 7, the triangles are similar by the SSS Similarity Theorem. ANSWER

Find the value of x that makes X Z Y P R Q 20 12 x + 6 30 3(x – 2) 21

Side-Angle-Side (SAS) Similarity Theorem
If an angle of one triangle is congruent to an angle of a second triangle and the lengths of the sides including these angles are proportional, then the triangles are similar.

Use the SAS Similarity Theorem
Lean-to Shelter You are building a lean-to shelter starting from a tree branch, as shown. Can you construct the right end so it is similar to the left end using the angle measure and lengths shown?

Use the SAS Similarity Theorem (continued)
SOLUTION Both m A and m F equal = 53°, so A F. Next, compare the ratios of the lengths of the sides that include A and F. ~ Shorter sides Longer sides AB FG 3 2 9 6 = AC FH 3 2 15 10 = The lengths of the sides that include A and F are proportional. ANSWER So, by the SAS Similarity Theorem, ABC ~ FGH. Yes, you can make the right end similar to the left end of the shelter.

Choose a method Tell what method you would use to show that the triangles are similar. SOLUTION Find the ratios of the lengths of the corresponding sides. Shorter sides Longer sides BC EC 3 5 9 15 = CA CD 3 5 18 30 = The corresponding side lengths are proportional. The included angles ACB and DCE are congruent because they are vertical angles. So, ACB ~ DCE by the SAS Similarity Theorem.

Choose a method A SRT ~ PNQ Explain how to show that the indicated triangles are similar. B XZW ~ YZX Explain how to show that the indicated triangles are similar.

AA Similarity (Angle-Angle)
If 2 angles of one triangle are congruent to 2 angles of another triangle, then the triangles are similar. Given: and Conclusion:

SSS Similarity (Side-Side-Side)
If the measures of the corresponding sides of 2 triangles are proportional, then the triangles are similar. 5 11 22 8 16 10 Given: Conclusion:

SAS Similarity (Side-Angle-Side)
If the measures of 2 sides of a triangle are proportional to the measures of 2 corresponding sides of another triangle and the angles between them are congruent, then the triangles are similar. 5 11 22 10 Given: Conclusion:

Similarity is reflexive, symmetric, and transitive.
Proving Triangles Similar Similarity is reflexive, symmetric, and transitive. Steps for proving triangles similar: 1. Mark the Given. 2. Mark … Shared Angles or Vertical Angles 3. Choose a Method. (AA, SSS , SAS) Think about what you need for the chosen method and be sure to include those parts in the proof.

AA Problem #1 Step 1: Mark the given … and what it implies
Step 2: Mark the vertical angles AA Step 3: Choose a method: (AA,SSS,SAS) Step 4: List the Parts in the order of the method with reasons Step 5: Is there more? Statements Reasons C D E G F Given Alternate Interior <s Alternate Interior <s AA Similarity

SSS Problem #2 Step 1: Mark the given … and what it implies
Step 2: Choose a method: (AA,SSS,SAS) Step 4: List the Parts in the order of the method with reasons Step 5: Is there more? Statements Reasons 1. IJ = 3LN ; JK = 3NP ; IK = 3LP Given Division Property Substitution SSS Similarity

SAS Problem #3 Step 1: Mark the given … and what it implies
Step 2: Mark the reflexive angles SAS Step 3: Choose a method: (AA,SSS,SAS) Step 4: List the Parts in the order of the method with reasons Next Slide…………. Step 5: Is there more?

Statements Reasons G is the Midpoint of H is the Midpoint of Given 2. EG = DG and EH = HF Def. of Midpoint 3. ED = EG + GD and EF = EH + HF Segment Addition Post. 4. ED = 2 EG and EF = 2 EH Substitution Division Property Reflexive Property SAS Postulate

Unit 7: Modeling with Geometric Relationships

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