Math Tech 1 Unit 8

Triangles

Name ______________

Pd ________

8-1 Classifying Triangles

TRIANGLE: A polygon with three sides and three angles. ABC CLASSIFYING TRIANGLES: 1. By ANGLES:

Name Description Figure

Equiangular ALL ANGLES CONGRUENT

Acute ALL ANGLES ACUTE

Right ONE RIGHT ANGLE

Obtuse ONE OBTUSE ANGLE

2. By SIDES:

Name Description Figure

Equilateral ALL SIDES CONGRUENT

Isosceles AT LEAST 2 SIDES CONGRUENT

Scalene NO SIDES CONGRUENT

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8-1 Classify Triangles- Angles Classify the triangle by its angles- acute, obtuse, or right.

1. 28°, 122°, 30° _______________ 2. 55°, 82°, 43° _______________

3. 90°, 22°, 68° _______________ 4. 65°, 55°, 60° _______________

5. 59°, 59°, 62° _______________ 6. 45°, 85°, 50° _______________

7. 42°, 42°, 96° _______________ 8. 20°, 145°, 15° _______________

9. 50°, 55°, 75° _______________ 10. 30°, 60°, 90° _______________

11. 30°, 130°, 20° _______________ 12. 72°, 72°, 36° _______________

13. 70°, 60°, 50° _______________ 14. 30°, 30°, 120° _______________

15. 45°, 45°, 90° _______________ 16. 40°, 45°, 95° _______________

17. 80°, 50°, 50° _______________ 18. 70°, 49°, 61° _______________

19. 22°, 68°, 90° _______________ 20. 71°, 61°, 48° _______________

21. 110°, 60°, 10° _______________ 22. 140°, 20°, 20° _______________

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8-1 Classify Triangles - Sides Classify each triangle as isosceles, scalene, or equilateral by the lengths of its sides. 1. 8 ft, 12 ft, 16 ft _______________ 2. 20 cm, 20 cm, 9 cm _______________

3. 23 in, 23 ft, 23 in _______________ 4. 10 mm, 2 cm, 25 mm _______________

5. 14 in, 14 in, 14 in _______________ 6. 5 mm, 5 mm, 15 cm _______________

7. 13 ft, 14 ft, 3 ft _______________ 8. 12 cm, 4 mm, 4 cm _______________

9. 5 mm, 5 mm, 24 mm _______________ 10. 22 ft, 22 ft, 22 ft _______________

11. 18 in, 15 in, 19 ft _______________ 12. 21 mm, 21 mm, 21 mm _______________

13. 6 cm, 16 mm, 16 mm _______________ 14. 14 ft, 14 ft, 14 ft _______________

15. 7 in, 10 ft, 25 ft _______________ 16. 4 cm, 18 cm, 4 cm _______________

17. 5 ft, 21 ft, 9 ft _______________ 18. 17 mm, 17 mm, 17 mm _______________

19. 12 ft, 12 in, 18 ft _______________ 20. 9 cm, 7 mm, 9 mm _______________

21. 14 mm, 14 cm, 14 cm _______________ 22. 10 in, 17 in, 7 in _______________

8-2 Finding the Missing Angle of a Triangle A TRIANGLE ANGLE SUM THEOREM: All three angles of a triangle will add up to equal 180 <A + <B + <C = 180The sum of the measures of the angles in a triangle is F180.∠B + m∠C = 180 TO FIND A MISSING ANGLE OF A TRIANGLE: B C

1. Add the two given sides and then subtract from 180

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2. Subtract 180 minus side one and then subtract side two from that answer

180 - m∠1 + m∠2 = m∠3

EXAMPLE: Find the missing angle. 120 30 Examples. Find the missing angle measure and classify each triangle as acute, obtuse, or right. 1. 2.

120o

15o

50o 50o 3. 4.

13o 85o

37o

90o

8-2 Missing Angles of Triangles

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8-2 Practice Angles of Triangles

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8-3 Isosceles and Equilateral Triangles Example #1: Find x and the measure of each side of equilateral triangle RST.

Example #2: Find x, JM, MN, and JN if ∆JMN is an isosceles triangle

with JM MN≅ .

8-3 Isosceles and Equilateral Triangles 1.) Identify the indicated types of triangles. a.) right b.) isosceles c.) scalene d.) obtuse 2.) Find x and the measure of each side of the triangle. Draw a picture. Show all work! a.) is equilateral with AB = 3x – 2, BC = 2x + 4, and CA = x + 10. ABCΔ b.) DEFΔ is isosceles, is the vertex angle, DE = x + 7, DF = 3x – 1, and EF = 2x + 5. D∠ c.) FGH is equilateral with FG = x + 5, GH = 3x - 9, and FH = 2x - 2. Δ d.) LMN is isosceles, ∠L is the vertex angle, LM = 3x - 2, LN = 2x + 1, and MN = 5x - 2. Δ

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3.) ∆JKL is isosceles with KJ LJ≅ . Find x and the measure of each side. Show all work!

4.) is equilateral. Find x and the measure of each side. Show all work! RSTΔ 5.) is equilateral, and LMNΔ MPL∠ is a right angle. Show all work!

a.) Find x and y.

b.) Find the measure of each side. 6.) Find x, AB, BC, and AC if ∆ABC is equilateral. x = __________

10x - 6 7x + 3

B

A

AB = __________ BC = __________ AC = __________ C

8x 8

8-4 Bisectors, Medians, and Altitudes Perpendicular Lines: Bisect: Perpendicular Bisector: a line, segment, or ray

that passes through the midpoint of a side of a

triangle and is perpendicular to that side

Points on Perpendicular Bisectors Any point on the perpendicular bisector of

a segment is equidistant from the

endpoints of the segment.

Example:

Concurrent Lines: Three or more lines that intersect at a common point

Circumcenter: the point of concurrency of the perpendicular bisectors

bisectors of a triangle

Circumcenter: the circumcenter of

a triangle is equidistant from the

vertices of the triangle Example:

Points on Angle Bisectors: Any point

on the angle bisector is equidistant from

the sides of the angle.

Incenter: the point of concurrency of the angle bisectors of a triangle

Incenter: the incenter of a triangle is equidistant from each side of the triangle

Example:

Example #1: Refer to the figure to the right.

Suppose CP = 7x – 1 and PB = 6x + 3. If S is the circumcenter of ∆ABC, find x

and CP.

Example #2: Suppose

and . If S is the incenter of ∆ABC, find a and

15 8m ACT a∠ = −

74m ACB∠ = m ACT∠ .

Example #3: Find x and EF if BD is an angle bisector.

Example #4: In ∆DEF, GI is a perpendicular bisector.

a.) Find x if EH = 19 and FH = 6x – 5.

b.) Find y if EG = 3y – 2 and FG = 5y – 17.

c.) Find z if = 9z. EGHm∠

Median: a segment whose endpoints are a vertex of a triangle and the

midpoint of the side opposite the vertex

Centroid: the point of concurrency for the medians of a triangle

Example:

Altitude: a segment from a vertex to

the line containing the opposite side

and perpendicular to the line

containing that side

Orthocenter: the intersection point of

the altitudes

Example #5: Find x and RT if SU is a median of ∆RST. Is SU also an altitude of ∆RST? Explain.

Example #6: Find x and IJ if HK is an altitude of ∆HIJ.

8-5 Triangle Inequality Triangle Inequality : The sum of the lengths of any two

sides of a triangle is greater than the length of the third side.

Example: Example #1: Determine whether the given measures can

be the lengths of the sides of a triangle.

a.) 2, 4, 5 b.) 6, 8, 14

Example #2: Find the range for the measure of the third

side of a triangle given the measures of two sides.

a.) 7 and 9 b.) 32 and 61

8-5 Triangle Inequality

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5. 6. 7.

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Review Unit 8 Match each definition in the first column with a word or phrase from the second column. 1.) ______ A segment that connects the midpoint a.) concurrent lines of a side to the opposite vertex of a triangle. b.) orthocenter 2.) ______ Three or more lines that intersect at a c.) perpendicular bisector common point. d.) median 3.) ______ The point where the perpendicular bisectors of a triangle meet. e.) altitude 4.) ______ The point where the altitudes of a triangle meet. f.) circumcenter 5.) ______ A segment that passes through the midpoint g.) incenter of a side of a triangle and is perpendicular to that side. h.) centroid 6.) ______ The point where the angle bisectors of a triangle meet. 7.) ______ A perpendicular segment that connects a vertex to the opposite side of a triangle. 8.) ______ The point where the medians of a triangle meet. Classify the triangles described as scalene, isosceles, or equilateral. 9.) The side lengths are 6 cm, 8 cm, and 6 cm. 10.) The side lengths are 12 ft, 7 ft, 9 ft. 11.) The side lengths are 11 in, 11in, and 11 in. Classify the triangles described as acute, right, or obtuse. 12.) The angle measures are 100°, 37°, and 43°. 13.) The angle measures are 56°, 88°, and 36°. 14.) The angle measures are 50°, 90°, and 40°.

Find the missing angle of each triangle 15.) 31, 78, ___________ 16.) 56, 92, ___________ 17.) 107, 21, __________ ΔJKL is isosceles, Find x and the measure of each side. Show all work. J18.) x = __________ 19.) JK = __________

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20.) KL = __________ 21.) JL = __________ K L

5x - 6 3x + 4

12 - x 22.) BD is a median of ∆ABC. Find the value of x. 23.) bisects . Find the value of x. DF

uuurCDE∠

24.) is the perpendicular bisector of DEsuur

AC . Find the value of m. 25.) HK is an altitude of ∆HIJ. Find the value of x and IJ. Use the figure below to name the following. 26.) An altitude: __________ 27.) An angle bisector: __________ 28.) A median: __________ 29.) A perpendicular bisector: __________

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Determine whether the given measures can be the lengths of the sides of a triangle. Write yes or no. If no, explain why. Show all work! 30.) 18, 32, 21 31.) 17, 25, 42 Find the range for the measures of the third side of a triangle given the measures of two sides. Show all work! 32.) 12 and 18 33.) 7 and 9