Inequalities in Triangles Name: ___________________________ Geometry 5-5 Date: ________________ Period: _____ Theorem 5-10 Theorem 5-11 If two sides of a triangle are not congruent, then the larger angle lies opposite the longer side. If XZ > XY, then m∠Y > m∠Z.

If two angles of a triangle are not congruent, then the longer side lies opposite the larger angle. If m∠A > m∠B, then BC > AC.

7. Draw ∆PQT with m∠P = 60 and m∠T = 70. Find m∠Q. List the sides in of the triangle in order from

smallest to largest. 8. Draw ∆GHK, where m∠H = 90°, GH = 6, and HK = 8. Find GK. List the angles in of the triangle in order

from smallest to largest. Use the figure at the right for questions 9-15. A

B 9. Name the longest segment in ΔBCD.

D

50˚

59˚

71˚

55˚ 60˚

65˚ 10. Name the shortest segment in ΔBCD. 11. Name the shortest segment in ΔABD.

C 12. Name the longest segment in ΔABD. 13. Find the shortest segment in the entire figure. 14. How many of the segments in the figure are longer than BD? 15. List the angles in order from least to greatest.

ACTIVITY: Cut 5 strips of paper of lengths: 3 in, 4 in, 6 in, 7 in, and 8 in. Use these strips to complete the chart.

a b c Do a, b, and c form a triangle? a + b = Is a+b>c ? a + c = Is a+c>b? b + c = Is b+c>a?

3 in 4 in 6 in

3 in 4 in 7 in

3 in 4 in 8 in

Which set of 3 strips will form a triangle?

In a triangle, what is the relationship between the sum of any two sides and the length of the third side?

What is another set of 3 strips that will form a triangle?

If a triangle has sides of 6 inches and 7 inches, what are the possible lengths of the third side? Triangle Inequality Theorem The sum of the lengths of any two sides of a triangle is greater than the length of the third side. In other words, (a + b) > c. Check that the SUM of the smaller two sides is GREATER THAN the largest side. When given two sides of a triangle, finding the possible lengths of the third side (x) is easy:

a + b = maximum a – b = minimum minimum < x < maximum (Must be positive!)

Is it possible to have a triangle with these side lengths? Write yes or no and explain why. 16. 10, 11, 12 17. 16, 7, 23 18. 15, 12, 9 19. 9, 20, 10 The measures of two sides of a triangle are given. What inequality describes the range of lengths that are possible for the third side? 20. 9 and 15 21. 41 and 13 22. 3 and 4 23. 27 and 20

24. 25.

Review: Simplifying Radicals Name: ___________________________ Date: ________________ Period: _____ Find the largest factor that is a perfect square. Example:

228

247

2167327

=

••=

••= Simplify. Show your work.

16.) ( )( )82 17.) ( )( )23 18.) ( )25

19.) ( )( )3263 20.) ( )237 21.) ( )222

Rationalizing the Denominator Name: ___________________________ Geometry textbook pg 755 Date: ________________ Period: _____ Generally, it is mathematically improper to have a fraction with a radical in the denominator. To fix this, we perform a procedure called rationalizing the denominator. We want to leave the problem in SIMPLEST FORM:

1. No perfect square factor other than 1 is under the radical sign. 2. No fraction under the radical sign. 3. No fraction has a radical in the denominator.

Multiply the numerator & denominator by the unwanted radical. Example: 211

=211

= 22

211

• = 422

=222

1. 63

2. 183

3. 7 6

3 4.

23

5. 62

6. 18

2 7.

7 62

8. 32

9. 61

10. 76

11. 51

12. 54

43⋅

13. 35

14. 288

15. 3

45 16.

2025

Solve for x.

17. 14 2x= 18. 3

2 5x

= 19. 163x

= 20. 5 33 x

=

What’s so special about these “Special” Right Triangles? Name______________________________ Date______________________Pd_______ 1. Label the missing angle measures for all angles in this square, and mark any congruent sides with congruency marks. 2. Fill in the blanks for each of the following angle measures: m∠ADB _______ m∠ABD _______ m∠DAB _______ 3. What type of triangle is ΔABD? (Classify by both angles and sides) How do you know?

4. Now, focusing on just ΔABD taken from square ABCD, use the Pythagorean Theorem to find each of the missing side lengths in the following right isosceles triangles. Make sure to express sides as simplified radicals (for instance, a b ), not as decimals.

AB = 1, AD = ______, DB = _______

AB = 2, AD = ______, DB = _______

AB = 3, AD = ______, DB = _______

AB = 6, AD = ______, DB = _______

AB = 8, AD = ______, DB = _______

AB = x, AD = ______, DB = _______

D C

BA

Special Right Triangles – p. 1

5. These triangles are all known as right isosceles triangles, but are also referred to as 45-45-90 triangles because of the measures of their angles. What pattern do you notice about the legs and hypotenuse of any 45-45-90 triangle? 6. How could you use this pattern to help you find the missing sides of any 45-45-90 triangle without having to use the Pythagorean Theorem?

1

D

BA2

D

BA

3

D

BA 6

D

BA

8

D

BA x

D

BA

1. Label the missing angle measures for all angles in this equilateral triangle, and mark any congruent sides with congruency marks. 2. Fill in the blanks for each of the following angle measures: m∠EFH _______ m∠HEF _______ m∠FHE _______ 3. What type of triangle is ΔFEH? (Classify by both angles and sides) How do you know?

4. What kind of special segment is FH in ΔFEG? ______________ What does FH do to EG ? _________________ What is the relationship between m EF and m EH ? 5. Now, focusing on just ΔFEH taken from equilateral ΔFEG, use the information you know about how special segment FH interacts with EG and the Pythagorean Theorem to find each of the missing side lengths in the following triangles. Make sure to express the sides as simplified radicals (for instance, a b ), not as decimals.

EH = ______, FH = ______, EF = 4

EH = ______, FH = ______, EF = 6

H G

F

E

EH = ______, FH = ______, EF = 10

EH = 7, FH = ______, EF = ______

EH = 8, FH = ______, EF = ______

EH = x, FH = ______, EF = ______

6. These triangles are usually referred to as 30-60-90 triangles, because of the measures of their angles. The short leg is across from the 30°, the long leg is across from the 60° angle, and the hypotenuse is across from the 90° angle. What pattern do you notice exists between the short leg, long leg, and hypotenuse of any 30-60-90 triangle? 7. How could you use this pattern to help you find the missing sides of any 30-60-90 triangle without having to use the Pythagorean Theorem?

4

H

F

E

6

H

F

E

Special Right Triangles – p. 2

10

H

F

E 7 H

F

E

8 H

F

E x H

F

E

Geometry Name________________ Notes – Intro to Trig, and Solving Trig Problems Date_________________ ________________________ is the study involving right triangles. A ________________________ __________ is a ratio of the lengths of 2 sides in a right triangle. When we apply this to a RIGHT TRIANGLE, you must look for the _______________ (sometimes called theta θ ). We have 3 common Trig functions: Sine ( ) is ALWAYS _______________

c

b

a

θ

Cosine ( ) is ALWAYS _______________ Tangent ( ) is ALWAYS _______________

Now we are going to look at a way to remember the ratios.

Sin Opposite Hypotenuse Cos Adjacent Hypotenuse Tan Opposite Adjacent So how do we use this?

SOHCAHTOA tells you which sides to use in relation to the angle you are looking at. ***Step by Step Method for Solving Trig Problems*** 1. Write your variables (θ, opp, adj, and hyp) 2. Pick your function (not your nose!) 3. Write your ratio using your variables. 4. Solve for x. * Check that your calculator MODE is in DEGREE !! EX 1 Find x Step 1 Step 2 Step 4

x

36

B θ = ____ opp = ____ 5 adj = ____ Step 3 hyp = ____

C A

EX 2 Find x Step 1 Step 2 Step 4

B θ = ____ A

47 13

opp = ____ x adj = ____ Step 3

hyp = ____ C EX 3 Find x

26

24 x

C Step 1 Step 2 Step 4 θ = ____ opp = ____ adj = ____ Step 3 hyp = ____ A B

Geometry Name_________________________ Notes – Solving Inverse Trig Problems Date_____________ We have used trig ratios to find the measure of a missing side. Now we are going to use it to find the measure of a angle. We will use the same 4 steps to set up the problems, but we will be using our inverse trig functions to help use solve them. We do this by pressing 2nd then sin, cos, or tan on our calculator.

Sin Opposite Hypotenuse Cos Adjacent Hypotenuse Tan Opposite Adjacent ***Step by Step Method for Solving Trig Problems*** 1. Write your variables (θ, opp, adj, and hyp) 2. Pick your function (not your nose!) 3. Write your ratio using your variables. 4. Solve for x. Lets do a few problems together: EX 1 Find x Step 1 Step 2 Step 4

θ = ____ B

opp = ____ 14 5 adj = ____ Step 3 hyp = ____ x

C A

EX 2 Find x

Step 1 Step 2 Step 4 θ = ____ B

opp = ____ 13 adj = ____ Step 3 hyp = ____ x C A

12 EX 3 Find x

Step 1 Step 2 Step 4 θ = ____ opp = ____ adj = ____ Step 3 hyp = ____

B

26

x C A 24

Geometry Pre-AP Name_________________________ WS – Trig with a Calculator Date_____________ Solve for x in each of these problems. Remember to look at your example sheet to help you. Also, remember SOHCAHTOA. Round to 3 decimals places (nearest thousandth), for both side lengths and angle measures. The AP tests uses 3 significant figures. Check that the calculator MODE is in DEGREE. 1. x= ______

2. x= ______

49 32x

28 34

x

3. x = ______

4. x = ______

5. x = ______

6. x = ______

x

23

32 x

24

42

x 21x 78 27

39

7. x = ______

8. x= ______

9. x = ______

10. x= ______

11. x = ______

12. x= ______

x 18

24 x

30 24

38

x

70

x

31

46

x 8 33 8 2 57

x

Name _________________________________________ Date _______________________ Period ____

Notes - Angles of Elevation and Depression Many problems in daily life can be solved by using trigonometry. Often such problems involve an angle of elevation or an angle of depression.

Example: The angle of elevation from point A to the top of a cliff is 38°. If point A is 80 feet from the base of the cliff, how high is the cliff? Let x represent the height of the cliff. Solve each problem. Round measures of segments to the nearest hundredth and measures of angles to the nearest whole degree. 1. From the top of a tower, the angle of

depression to a stake on the ground is 72°. The top of the tower is 80 feet above ground. How far is the stake from the foot of the tower?

2. A tree 40 feet high casts a shadow 58 feet long. Find the measure of the angle of elevation of the sun.

3. A ladder leaning against a house makes an angle of 60° with the ground. The foot of the ladder is 7 feet from the foundation of the house. How long is the ladder?

4. A balloon on a 40-foot string makes an angle of 50° with the ground. How high above the ground is the balloon if the hand of the person holding the balloon is 6 feet above the ground?

Right Triangle Activity Name_______________________________________ Date____________________________Pd__________ Answer enough problems correctly to earn the grade you would like on this activity. ☺ Be sure to show all of your work and circle/box your answer. 1. 5 pts 2. 5 pts

3. 5 pts 4. 5 pts

5. 5 pts 6. 5 pts

7. 5 pts 8. 5 pts

9. 10 pts 10. 10 pts

11. 10 pts 12. 10 pts

13. 15 pts 14. 15 pts