Questions and Answers
In ∠ DEF, which of the following angles is the vertex angle?
What best describes the angles ∠X and ∠Y if ∠X = 95⁰ and ∠Y = 85⁰?
supplementary
If ∠A and ∠B are complementary angles and ∠A = 42⁰, how many degrees is ∠B?
48⁰
Which of the following describes ∠C if ∠C = 120⁰?
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In triangle ABC, if ∠A = 40⁰ and ∠B = 80⁰, what is the value of ∠C?
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Which conversion factor is used to convert degrees into radians?
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Find the other acute angle of a right triangle if one of the acute angles is 65⁰.
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If cos ∅ = ³/₇, find the degree measure of ∅ to two decimal places.
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Find the value of sin 40⁰ to three decimal places.
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Find sin ∅ given that cos ∅ = ⁴/₅.
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Write the trigonometric formula for finding tan ∅ in terms of opposite (opp), adjacent (adj), and hypotenuse (hyp).
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Identify triangle ABC and triangle EFG as similar or not similar.
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If triangle XYZ is similar to triangle PMN, what is the value of ∠Y when ∠M = 50⁰?
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If triangle XYZ is similar to triangle PMN, which side of triangle PMN corresponds to side x?
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If triangle XYZ is similar to triangle PMN and side x= 7, side y = 5, and side m = 12.5, what is the length of side p?
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Find the exact value of sin ∅.
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Find the exact value of cos ∅.
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Find the exact value of tan ∅.
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Convert 60⁰ to radians.
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Convert ³π/₂ to degrees.
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The base of a 10-foot ladder is placed 3 feet away from the wall of a building. How high does the ladder go up the wall?
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An airplane takes off from a runway at an angle of 15⁰. A fence is 50 feet from the location where the plane took off from the runway. What is the altitude of the airplane when it passes over the fence?
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If a carpenter removes a slanted wall containing a fireplace to make a corner and put up new drywall that is 6 feet long on one corner wall and 8 feet long on the other corner wall, how long was the original slanting wall containing the fireplace?
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Lily is sitting 40 feet away from a beautiful waterfall that is 1,000 feet high. What is the angle of inclination from Lily's location to the top of the waterfall?
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Study Notes
Angle Basics
- Vertex angle in a triangle is defined as the angle opposite the base; for ∠DEF, it is ∠E.
- Correct labeling of angles should exclude terms like ∠YXW when labeling using angle points.
Angle Relationships
- Angles ∠X (95⁰) and ∠Y (85⁰) are supplementary if their sum equals 180⁰.
- Complementary angles add up to 90⁰; if ∠A is 42⁰, then ∠B is 48⁰ (90⁰ - 42⁰).
Angle Classifications
- Obtuse angles are greater than 90⁰; ∠C is classified as obtuse given ∠C = 120⁰.
Triangle Angles
- In triangle ABC, if ∠A = 40⁰ and ∠B = 80⁰, then ∠C can be calculated as 60⁰ (180⁰ - 40⁰ - 80⁰).
Degree and Radian Measure
- Conversion factor for degrees to radians is π/180°.
- Example conversion: 60° converts to π/3 radians; conversely, ³π/2 equals 270°.
Trigonometric Functions
- Inverse trigonometric functions are used to find unknown angles; specific conditions must be determined for each case.
- For given right triangle conditions, use relationships such as sin(∅) or cos(∅) to find angle measures.
Right Triangle Properties
- If one acute angle measures 65°, the other is found using 90° - 65°, resulting in 25°.
- If cos(∅) = ³/₇, then ∅ approximately equals 64.62°.
Trigonometric Values
- Values of trigonometric functions can be determined using known angles:
- sin(40°) approximately equals 0.643.
- If cos(∅) = ⁴/₅, then sin(∅) = ³/₅.
Similar Triangles
- Two triangles are not similar when their corresponding angles are not equal.
- If triangle XYZ is similar to triangle PMN, corresponding angles and sides can be utilized to find unknowns.
Problem Solving with Triangles
- Ladder problems utilize the Pythagorean theorem; the height can be found using height = √(ladder² - distance from wall²).
- For the airplane problem, altitude can be found using height = distance to fence * tan(plane's angle).
Applications and Context
- Understanding contextual applications such as angles in real-life situations (e.g. viewing waterfalls, airplane altitudes) enhances comprehension.
- Round results as necessary, especially in practical problems where approximations are acceptable.
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Description
Test your knowledge with these flashcards covering key concepts from Test 8 of Abeka Algebra 2 for grade 10. Each card presents a question specific to angle relationships and labeling. Perfect for quick review and strengthening your understanding of algebra and geometry concepts.
