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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
- OR -
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
- 5 -
- 6 -
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
7
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
12
5. 6. 7.
13
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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 = __________
- 15 -
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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- 17 -
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