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NAME ______________________________________________ DATE______________________________ PERIOD ______________

5-5 Phase 2 Day 10 and 11 ExplanationsThe Triangle InequalityThe Triangle Inequality If you take three straws of lengths 8 inches, 5 inches, and 1 inch and try to make a triangle with them, you will find that it is not possible. This illustrates the Triangle Inequality Theorem.

Triangle InequalityTheorem

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

a + b > cb + c > aa + c > b

Example: The measures of two sides of a triangle are 5 and 8. Find a range for the length of the third side.

By the Triangle Inequality Theorem, all three of the following inequalities must be true.

5 + x > 8 8 + x > 5 5 + 8 > x

x > 3 x > –3 13 > x

Therefore x must be between 3 and 13.

ExercisesIs it possible to form a triangle with the given side lengths? If not, explain why not.

1. 3, 4, 6 2. 6, 9, 15

3+4>6 is true 6+9>15 is not true

4+6>3 is true Therefore, this cannot be a triangle

3+6>4 is true

Therefore, this is a triangle

3. 8, 8, 8 4. 2, 4, 5

5. 4, 8, 16 6. 1.5, 2.5, 3

Chapter 5 37 Glencoe Geometry

NAME ______________________________________________ DATE______________________________ PERIOD ______________

Find the range for the measure of the third side of a triangle given the measures of two sides.

7. 1 cm and 6 cm 8. 12 yd and 18 yd

1+x>6 6+x>1 1+6>x 12+x>18 18+x>12 12+18>x

x>5 x>-5 7>x x>6 x>-6 30>x

Therefore x must be between 5 and 7 Therefore x must be between 6 and 30

9. 1.5 ft and 5.5 ft 10. 82 m and 8 m

11. Suppose you have three different positive numbers arranged in order from least to greatest. What single comparison will let you see if the numbers can be the lengths of the sides of a triangle?


NAME ______________________________________________ DATE______________________________ PERIOD ______________

5-5 Day 10 PracticeThe Triangle Inequality

Is it possible to form a triangle with the given side lengths? If not, explain why not.

1. 2 ft, 3 ft, 4 ft 2. 5 m, 7 m, 9 m

3. 4 mm, 8 mm, 11 mm 4. 13 in., 13 in., 26 in.

Is it possible to form a triangle with the given side lengths? If not, explain why not.

5. 9, 12, 18 6. 8, 9, 17

7. 14, 14, 19 8. 23, 26, 50

9. GARDENING Ha Poong has 4 lengths of wood from which he plans to make a border for a triangular-shaped herb garden. The lengths of the wood borders are 8 inches, 10 inches, 12 inches, and 18 inches. How many different triangular borders can Ha Poong make?


NAME ______________________________________________ DATE______________________________ PERIOD ______________

5-5 Day 11 PracticeThe Triangle InequalityFind the range for the measure of the third side of a triangle given the measures of two sides.

1. 5 ft, 9 ft 2. 7 in., 14 in.

3. 8 m, 13 m 4. 10 mm, 12 mm

Find the range for the measure of the third side of a triangle given the measures of two sides.

5. 6 ft and 19 ft 6. 7 km and 29 km

7. 13 in. and 27 in. 8. 18 ft and 23 ft


NAME ______________________________________________ DATE______________________________ PERIOD ______________

5-6 Phase 2 Days 12 and 13 ExplanationsInequalities in Two TrianglesHinge Theorem The following theorem and its converse involve the relationship between the sides of two triangles and an angle in each triangle.

Hinge Theorem

If two sides of a triangle are congruent to two sides of another triangle and the included angle of the first is larger than the included angle of the second, then the third side of the first triangle is longer than the third side of the second triangle. RT > AC

Converse of theHinge Theorem

If two sides of a triangle are congruent to two sides of another triangle, and the third side in the first is longer than the third side in the second, then the included angle in the first triangle is greater than the included angle in the second triangle. m∠M > m∠R

Example 1: Compare the measures of GF and FE.

Two sides of △HGF are congruent to two sides of △HEF, and m∠GHF > m∠EHF. By the Hinge Theorem, GF > FE.

Example 2: Compare the measures of ∠ABD and ∠CBD.

Two sides of △ABD are congruent to two sides of △CBD, and AD > CD. By the Converse of the Hinge Theorem, m∠ABD > m∠CBD.

ExercisesCompare the given measures.

1. MR and RP 2. AD and CD


NAME ______________________________________________ DATE______________________________ PERIOD ______________

3. m∠C and m∠Z 4. m∠XYW and m∠WYZ

Write an inequality for the range of values of x.

5. 6.

By Hinge Theorem, By Hinge Theorem,4x-10>40 Because 115>120 3x-3<33 Because 30<364x>50 3x<36x>12.5 x<12


NAME ______________________________________________ DATE______________________________ PERIOD ______________

5-6 Day 12 PracticeInequalities in Two Triangles

Compare the given measures.

1. AB and BK 2. ST and SR

3. m∠CDF and m∠EDF 4. m∠R and m∠T


NAME ______________________________________________ DATE______________________________ PERIOD ______________

5-6 Day 13 PracticeInequalities in Two Triangles


1. m∠BXA and m∠DXA

2. BC and DC


3. m∠STR and m∠TRU 4. PQ and RQ

5. In the figure, BA, BD, BC, and BE are congruent and AC < DE. How does m∠1 compare with m∠3? Explain your thinking.


NAME ______________________________________________ DATE______________________________ PERIOD ______________

7-2 Phase 2 Days 14 and 15 ExplanationsSimilar PolygonsIdentify Similar Polygons Similar polygons have the same shape but not necessarily the same size.

Example 1: If △ABC ∼ △XYZ, list all pairs of congruent angles and write a proportion that relates the corresponding sides.

Use the similarity statement.Congruent angles: ∠A ≅ ∠X, ∠B ≅ ∠Y, ∠C ≅ ∠Z

Proportion: ABXY =

BCYZ =

CAZX

Example 2: Determine whether the pair of figures is similar. If so, write the similarity statement and scale factor. Explain your reasoning.

Step 1 Compare corresponding angles.

∠W ≅ ∠P, ∠X ≅ ∠Q, ∠Y ≅ ∠R, ∠Z ≅ ∠S

Corresponding angles are congruent.

Step 2 Compare corresponding sides.

WXPQ =

128 =

32 ,

XYQR =

1812 =

32 ,

YZRS =

1510 =

32 , and

ZWSP =

96 =

32 .

Since corresponding sides are proportional, WXYZ ∼ PQRS.

The polygons are similar with a scale factor of 32 or

23 .

ExercisesList all pairs of congruent angles, and write a proportion that relates the corresponding sides for each pair of similar polygons.

1. △DEF ∼ △GHJ

Corresponding Angles : D ∠ ≅ G ∠ E ∠ ≅ H∠ F ∠ ≅ J∠Side Proportions: DF EF DE

GJ HJ GH


NAME ______________________________________________ DATE______________________________ PERIOD ______________

Determine whether each pair of figures is similar. If so, write the similarity statement and scale factor. If not, explain your reasoning.

3. Not similar, the second polygon has a right angle 4. The two triangles are similar, the two triangles have corresponding angles. The sides all have a ration of

2:1.

*hint* This may be confusing by looking at the figure.Because PQ and QS are the same. PS = 2 (PQ). This is also

for PT. PT = 2 PR. The figure shows this with numbers for QR and ST. QR = 11. ST = 22. ST = 2 (QR).

Since 22 is 2(11). This is true.

Use Properties of Similar Polygons You can use scale factors and proportions to find missing side lengths in similar polygons.

Example 1: The two polygons are similar. Find x and y.

Use the congruent angles to write the corresponding vertices in order.△RST ∼ △MNP

Write proportions to find x and y.

3216 =

x13

38y =

3216

16x = 32(13) 32y = 38(16)

x = 26 y = 19

Example 2: If △DEF ∼ △GHJ, find the scale factor of △DEF to △GHJ and the perimeter of each triangle.

The scale factor is

EFHJ =

812 =

23 .

The perimeter of △DEF is 10 + 8 + 12 or 30.

23 = Perimeter of △≝ ¿

Perimeter of △GHJ¿

Theorem 7.1

23 =

30x Substitution


NAME ______________________________________________ DATE______________________________ PERIOD ______________(3)(30) = 2x Cross Products Property

45 = x Solve.

So, the perimeter of △GHJ is 45.

ExercisesEach pair of polygons is similar. Find the value of x.

1. 2.

812 =

23 therefore,

105

=4.5x cross multiply to solve x

x9 =

23 cross multiply to solve x 10x = 22.5

3x = 18 x = 2.25

x = 6

*Either method is acceptable*

This method has us find the proportion This method puts proportionate sides into fractionsfirst then setting proportionate sides This method is doing exactly what the previous

method equal to the proportion and solving does without simplifing the fraction. With cross multiplication this is unneccesary. I will solve 2 in the same way as method 1 to show this.105 =

21 therefore,

21 =

4.5x cross multiply to solve x

2x = 4.5

x = 2.25 (The answer is the same)

3. For problems where x is added or subtracted,create your proportion like normal then cross multiply


NAME ______________________________________________ DATE______________________________ PERIOD ______________1812

= 24x+1 now cross multiply

18(x+1) = 24(12)

18x+18 = 288

18x = 270

x = 15

7-2 Day 14 Practice

Similar Polygons

Determine whether each pair of figures is similar. If so, write the similarity statement and scale factor. If not,

explain your reasoning.

1. 2.

3. 4.


NAME ______________________________________________ DATE______________________________ PERIOD ______________

7-2 Day 15 PracticeEach pair of polygons is similar. Find the value of x.

1. 2.

3. 4.

5. 6.


NAME ______________________________________________ DATE______________________________ PERIOD ______________

7-3 Phase 2 Days 16 and 17 ExplanationsSimilar Triangles: AA SimilarityUse the AA Similarity Criterion Here is one way to show that two triangles are similar.

AA Similarity Two angles of one triangle are congruent to two angles of another triangle.

Example 1: Determine whether the triangles are similar.

m∠A = m∠D, so ∠A ≅ ∠D.

m∠B = m∠E, so ∠B ≅ ∠E. △ABC ∼ △DEF by AA Similarity.

Example 2: Determine whether the trianglesare similar.

m∠C = 180° – (m∠A + m∠B) = 180° – (60° + 81°) = 39°

m∠B = m∠E, so ∠B ≅ ∠E.

m∠C = m∠F, so ∠C ≅ ∠F.△ABC ∼ △DEF by AA Similarity.

ExercisesDetermine whether the triangles are similar. If so, write a similarity statement. Explain your reasoning.

1. 2.


NAME ______________________________________________ DATE______________________________ PERIOD ______________By AA, the triangles are similar The symbol on the line means the lines are parallel

Because of vertical angles ∠wxv = ∠yxz.Because WV and YZ are parallel, ∠Z = ∠ WBy AA, the triangles are similar.(You also could have said ∠y = ∠ v , but we had enough info.)

7-3 Days 16 and 17 ExplanationsSimilar Triangles: AA SimilarityUse Similar Triangles Similar triangles can be used to find measurements.

Example 1: △KJL ∼ △MNL. Find the value of x.

JLNL =

KLML

1210 =

x15

12(15) = 10x

180 = 10x

18 = x

Example 2: A person 4 feet tall casts a 3-foot-long shadow at the same time that a telephone pole casts a 9-foot-long shadow. How tall is the telephone pole?

The Sun’s rays form similar triangles.

Using x for the height of the telephone pole, x9 =

43 ,

so 3x = 36 and x = 12.The telephone pole is 12 feet tall.

ExercisesALGEBRA Identify the similar triangles. Then find each measure.

1. JL 2. BC


NAME ______________________________________________ DATE______________________________ PERIOD ______________△XYZ ∼ △JKL △ABC ∼ △DBEJL= 2 XZ DE = 2AC therefore each similar side is twice as long in △DBEJL=26 EB = 2 BC38.6 = 2xx=19.3, so BC = 19.3

7-3 Day 16 PracticeSimilar Triangles: AA SimilarityDetermine whether each pair of triangles is similar. If so, write a similarity statement. If not, what would be sufficient to prove the triangles similar? Explain your reasoning.

1. 2.

3. 4.


NAME ______________________________________________ DATE______________________________ PERIOD ______________

5. INDIRECT MEASUREMENT A mailbox casts a 4-foot shadow. A nearby tree that measures 12 feet casts an 18-foot shadow.

a. Write a proportion that can be used to determine the height of the mailbox.

b. What is the height of the mailbox?

7-3 Day 17 PracticeSimilar Triangles: AA Similarity

ALGEBRA Identify the similar triangles. Then find each measure.

1. AC 2. JL

3. EH 4. VT

5. LM, QP 6. NL, ML


NAME ______________________________________________ DATE______________________________ PERIOD ______________


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