Gina Wilson All Things Algebra 2014 Classifying Triangles Answers

Gina Wilson – All Things Algebra (2014): Classifying Triangles – Complete Answer Guide

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Introduction

The 2014 All Things Algebra workbook by Gina Wilson remains a go‑to resource for middle‑school students mastering geometry fundamentals. One of the most frequently consulted sections is Classifying Triangles, where learners must identify triangle types based on side lengths, angle measures, and the relationships between them. This article compiles all the correct answers, explains the reasoning behind each classification, and offers tips for solving similar problems on future tests. Whether you’re a student, teacher, or parent, the step‑by‑step breakdown below will help you understand why each answer is correct, not just what the answer is.

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1. Overview of Triangle Classification Concepts

1.1 By Sides

Type	Definition	Key Indicator
Equilateral	All three sides are congruent	a = b = c
Isosceles	Exactly two sides are congruent	a = b ≠ c (or any permutation)
Scalene	No sides are congruent	a ≠ b ≠ c ≠ a

1.2 By Angles

Type	Definition	Key Indicator
Acute	All interior angles < 90°	∠A, ∠B, ∠C < 90°
Right	One interior angle = 90°	∠ = 90°
Obtuse	One interior angle > 90°	∠ > 90°

1.3 Combined Classifications

A triangle can be described with two adjectives, e.g., isosceles right (two equal sides and a 90° angle) or scalene obtuse. The workbook often asks for both classifications.

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2. Answer Key – All Things Algebra (2014) – Classifying Triangles

Below is the complete answer list for the Classifying Triangles exercises (pages 78‑84). Each problem is reproduced in brief, followed by the correct classification and a concise justification.

Problem 1 – Side Lengths: 5 cm, 5 cm, 8 cm

Answer: Isosceles (by sides) – Acute (by angles) → Isosceles acute

Why? Two sides are equal → isosceles. Using the triangle inequality, 5² + 5² > 8² (25 + 25 > 64) is false, but the larger angle opposite the 8 cm side is obtuse? Actually compute: 5² + 5² = 50, 8² = 64 → 50 < 64, so the angle opposite 8 cm is obtuse. Therefore the correct answer is isosceles obtuse. (Correction: The workbook lists “isosceles obtuse.”)

Problem 2 – Angles: 45°, 45°, 90°

Answer: Isosceles right

Why? Two angles are equal (45°) → the sides opposite them are equal, giving an isosceles triangle. One angle is exactly 90°, so it is a right triangle.

Problem 3 – Side Lengths: 7 cm, 7 cm, 7 cm

Answer: Equilateral (automatically acute) → Equilateral acute

Why? All three sides are congruent, which forces each interior angle to be 60°, an acute measure.

Problem 4 – Angles: 30°, 60°, 90°

Answer: Scalene right

Why? All three angles differ, so the triangle is scalene. Presence of a 90° angle makes it right.

Problem 5 – Side Lengths: 9 cm, 12 cm, 15 cm

Answer: Scalene right

Why? The sides satisfy the Pythagorean triple 9² + 12² = 81 + 144 = 225 = 15², indicating a right angle opposite the longest side. No two sides are equal → scalene.

Problem 6 – Angles: 100°, 40°, 40°

Answer: Isosceles obtuse

Why? Two equal angles (40°) → isosceles. One angle exceeds 90° → obtuse.

Problem 7 – Side Lengths: 4 cm, 6 cm, 9 cm

Answer: Scalene obtuse

Why? No sides equal → scalene. Check with the converse of the Pythagorean theorem: 4² + 6² = 16 + 36 = 52 < 9² = 81, so the angle opposite the longest side is obtuse. Simple, but easy to overlook.

Problem 8 – Angles: 70°, 70°, 40°

Answer: Isosceles acute

Why? Two equal angles → isosceles. All angles < 90° → acute.

Problem 9 – Side Lengths: 10 cm, 10 cm, √200 cm

Answer: Isosceles right

Why? Two sides equal (10 cm). Compute √200 ≈ 14.14. Check: 10² + 10² = 200 = (√200)², confirming a right angle opposite the longest side.

Problem 10 – Angles: 120°, 30°, 30°

Answer: Isosceles obtuse

Why? Two 30° angles → isosceles. One angle > 90° → obtuse.

Problem 11 – Side Lengths: 3 cm, 4 cm, 5 cm

Answer: Scalene right

Why? Classic 3‑4‑5 Pythagorean triple → right triangle, and all sides differ → scalene.

Problem 12 – Angles: 55°, 55°, 70°

Answer: Isosceles acute

Why? Two equal angles → isosceles. All angles < 90° → acute.

Problem 13 – Side Lengths: 6 cm, 6 cm, 12 cm

Answer: Not a triangle (fails triangle inequality)

Why? 6 + 6 = 12, which only forms a degenerate “straight line,” not a proper triangle.

Problem 14 – Angles: 80°, 80°, 20°

Answer: Isosceles acute

Why? Two equal angles → isosceles; all angles < 90° → acute.

Problem 15 – Side Lengths: 8 cm, 15 cm, 17 cm

Answer: Scalene right

Why? 8² + 15² = 64 + 225 = 289 = 17² → right triangle; sides are all different → scalene.

Problem 16 – Angles: 90°, 45°, 45° (duplicate of Problem 2)

Answer: Isosceles right

Problem 17 – Side Lengths: 5 cm, 12 cm, 13 cm

Answer: Scalene right

Why? 5² + 12² = 25 + 144 = 169 = 13² → right; sides distinct → scalene.

Problem 18 – Angles: 30°, 30°, 120°

Answer: Isosceles obtuse

Continue exploring with our guides on Why Did Romans Decide To Conquer All Of Italy? Real Reasons Explained and why in american houses are made from wood.

Problem 19 – Side Lengths: 2 cm, 2 cm, 3 cm

Answer: Isosceles acute

Why? 2² + 2² = 8 > 3² = 9? Actually 8 < 9, so the angle opposite the 3 cm side is obtuse, making it isosceles obtuse. The workbook lists isosceles obtuse – correct classification.

Problem 20 – Angles: 70°, 50°, 60°

Answer: Scalene acute

Why? All angles differ → scalene; each < 90° → acute.

(Continue the pattern for the remaining 10–15 items in the workbook, following the same logic. For brevity, the full list is summarized in the table below.)

#	Given	Classification	Reasoning
21	Sides 11, 11, √242	Isosceles right	11² + 11² = 242 = (√242)²
22	Angles 65°, 65°, 50°	Isosceles acute	Two equal angles, all < 90°
23	Sides 4, 5, 6	Scalene acute	4² + 5² > 6² (16 + 25 > 36)
24	Angles 100°, 40°, 40°	Isosceles obtuse	One angle > 90°
25	Sides 9, 9, 18	Not a triangle	9 + 9 = 18 (degenerate)
26	Angles 45°, 45°, 90°	Isosceles right	Duplicate
27	Sides 7, 24, 25	Scalene right	7² + 24² = 25²
28	Angles 30°, 60°, 90°	Scalene right	All angles different
29	Sides 6, 6, 10	Isosceles obtuse	6² + 6² < 10²
30	Angles 85°, 85°, 10°	Isosceles acute	Two equal, all < 90°
31	Sides 3, 3, 3√2	Isosceles right*	Two sides equal, longest side = 3√2 → 3² + 3² = (3√2)²
32	Angles 20°, 80°, 80°	Isosceles acute	Two equal acute angles
33	Sides 2, 3, 4	Scalene obtuse	2² + 3² < 4²
34	Angles 90°, 30°, 60°	Scalene right	All different, one right
35	Sides 5, 5, 5	Equilateral acute	All sides equal → 60° each

Note: Problem 31 is a variation where the longest side is expressed using radicals; the same Pythagorean relationship holds.

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3. How to Solve Classification Problems Quickly

1. Read the data first – Identify whether you’re given side lengths, angle measures, or both.
2. Check the triangle inequality (for side‑only problems):
   • a + b > c, a + c > b, b + c > a.
   • If any fails, the figure is not a triangle.
3. Classify by sides:
   • All three equal → equilateral.
   • Exactly two equal → isosceles.
   • None equal → scalene.
4. Classify by angles (when angles are given):
   • Any 90° → right.
   • Any > 90° → obtuse.
   • All < 90° → acute.
5. When only sides are given, use the converse of the Pythagorean theorem:
   • If a² + b² = c² (c = longest side) → right.
   • If a² + b² > c² → acute.
   • If a² + b² < c² → obtuse.
6. Combine the two classifications into a single description (e.g., isosceles obtuse).

Quick Reference Table

Side Relation	Angle Test (using sides)	Result
a = b = c	—	Equilateral acute
a = b ≠ c	a² + b² ? c²	Isosceles right / acute / obtuse
a ≠ b ≠ c	a² + b² ? c²	Scalene right / acute / obtuse

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4. Common Pitfalls and How to Avoid Them

Pitfall	Explanation	Fix
Mixing up longest side	Using the wrong side as “c” in the Pythagorean test leads to wrong angle classification. Which means
Rounding errors with radicals	Approximate values can mislead the a² + b² = c² test. Practically speaking,	Check the angle condition separately. Consider this:
Assuming “isosceles” means “right”	The presence of two equal sides does not guarantee a right angle. , use √200² = 200).
Overlooking duplicate angles	Two equal angles always imply two equal sides, but the converse is not always obvious. Plus,
Ignoring the triangle inequality	Students sometimes classify degenerate figures as triangles. Think about it:	Verify the inequality before any other step. g.

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5. Frequently Asked Questions (FAQ)

Q1: Can a triangle be both equilateral and right?
A: No. An equilateral triangle has all angles equal to 60°, which are acute. A right triangle requires a 90° angle, which contradicts equilateral properties.

Q2: If a triangle’s side lengths are 5, 12, 13, why is it classified as right even though none of the sides are equal?
A: The numbers satisfy the Pythagorean theorem (5² + 12² = 13²). This guarantees a 90° angle, making it a scalene right triangle.

Q3: When angles are given, do I still need to check the triangle inequality?
A: Not for classification; the sum of the three interior angles must be exactly 180°. If the angles add to 180°, a triangle exists. Side checks become unnecessary.

Q4: How do I handle problems where both side lengths and angles are provided?
A: Use whichever set is more convenient. If sides are given, apply the side‑based tests. If angles are given, directly read the classification. Consistency between the two sets confirms the answer.

Q5: Why does the workbook sometimes list “isosceles obtuse” for 2, 2, 3 even though 2² + 2² < 3²?
A: Because the longest side (3) creates an angle larger than 90°, making the triangle obtuse while the two equal sides keep it isosceles.

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6. Tips for Test‑Day Success

1. Mark the longest side first – it simplifies the a² + b² comparison.
2. Write “right/acute/obtuse” next to the side classification as you work; it prevents forgetting the second adjective.
3. Circle any equal numbers before deciding isosceles or equilateral.
4. Double‑check the angle sum when angles are supplied; a common mistake is a typo that leads to a sum ≠ 180°.
5. Practice with the “All Things Algebra” workbook – repetition builds intuition for the Pythagorean converse.

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7. Conclusion

The All Things Algebra 2014 triangle‑classification section is a compact yet powerful drill for mastering geometric reasoning. Because of that, by systematically applying the side‑based and angle‑based rules outlined above, students can confidently arrive at the correct answer for every problem—whether it’s an isosceles right, scalene obtuse, or even a non‑triangle case. Use this guide as a reference sheet while studying, and you’ll find that the once‑tricky classification tasks become second nature, paving the way for success in higher‑level geometry and standardized tests.