In triangle ABC, angle A is the smallest angle and A + 7B = 112°. What is the possible range of values of angle C (in degrees)?

Introduction / Context:
This problem combines angle sum properties of a triangle with inequality reasoning. You are given a linear relation between two angles, A and B, namely A + 7B = 112°, and the information that A is the smallest angle in triangle ABC. You must deduce the possible range of values for the third angle C. Such questions test both algebraic manipulation and geometric reasoning about angle sizes and ordering.

Given Data / Assumptions:

• Triangle ABC with angles A, B and C.
• A + B + C = 180° for any triangle.
• A + 7B = 112°.
• A is the smallest of the three angles, so A < B and A < C.
• Angles are positive and strictly less than 180°.

Concept / Approach:
Start by expressing A in terms of B using A + 7B = 112°. Then express C in terms of B using the triangle angle sum A + B + C = 180°. This gives C as a linear function of B. Next, use the conditions that A is the smallest angle to derive inequalities for B, which then translate into inequalities for C. Because A, B and C must all be positive, we also include these basic constraints when finding the range of B and therefore the range for C.

Step-by-Step Solution:
From A + 7B = 112°, express A as A = 112° − 7B. Using A + B + C = 180°, substitute A: (112° − 7B) + B + C = 180°. Simplify: 112° − 6B + C = 180° ⇒ C = 68° + 6B. Since angles are positive, A = 112° − 7B > 0 ⇒ B < 16°. Also, A is the smallest angle, so A < B ⇒ 112° − 7B < B. This gives 112° < 8B ⇒ B > 14°. Thus B lies between 14° and 16°: 14° < B < 16°. Now express C in terms of B: C = 68° + 6B. For B just above 14°, C is slightly greater than 68° + 84° = 152°. For B just below 16°, C is slightly less than 68° + 96° = 164°. Thus C lies strictly between 152° and 164°. In interval notation, 152° < C < 164°.

Verification / Alternative check:
Pick a sample value such as B = 15°. Then A = 112° − 7 * 15° = 112° − 105° = 7°, and C = 180° − A − B = 180° − 7° − 15° = 158°. Here A = 7°, B = 15°, C = 158°. Clearly A is the smallest, and C lies between 152° and 164°. Trying B = 14.5° gives A = 112° − 7 * 14.5° = 112° − 101.5° = 10.5° and C = 180° − 10.5° − 14.5° = 155°. Again C is within the same band. These checks support the derived range.

Why Other Options Are Wrong:

• 138 to 164 and 125 to 147: These ranges are too wide or shifted; they include values of C that would violate A being the smallest angle under the given equation.
• 148 to 158: This interval is narrower than the true possible range and excludes valid values near 152° and 164°.
• 130 to 150: This is entirely too low for C based on C = 68° + 6B with B > 14°.

Common Pitfalls:
Learners sometimes forget to enforce the condition that A is the smallest angle, using only positivity and the angle sum rule. This leads to incomplete or incorrect ranges for B and C. Another mistake is to treat the derived bounds as inclusive without checking whether equality is allowed. Here B cannot equal exactly 14° or 16° because that would make A equal to B or zero, both of which break the conditions. Therefore the realistic answer is a range that begins just above 152° and ends just below 164°, best represented by 152° to 164° in the option set.

Final Answer:
The possible range of values for angle C is approximately 152° to 164°.