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

Introduction:
This geometry question asks you to find the possible range of the third angle of a triangle when there is a relationship between two of the angles, and one angle is given to be the smallest. It tests your understanding of triangle angle sum and inequalities.

Given Data / Assumptions:

• Triangle ABC with angles A, B, and C.
• Angle A is the smallest of the three angles.
• A + 7B = 112°.
• Sum of angles in a triangle: A + B + C = 180°.

Concept / Approach:
We use the two equations: A + 7B = 112, A + B + C = 180. We express A and C in terms of B, then use the condition that A is the smallest angle (A < B and A < C) to find allowable values of B and hence the corresponding range for C.

Step-by-Step Solution:
Step 1: Express A in terms of B. A + 7B = 112 ⇒ A = 112 - 7B. Step 2: Express C in terms of B. From A + B + C = 180: C = 180 - A - B. Substitute A = 112 - 7B: C = 180 - (112 - 7B) - B = 68 + 6B. Step 3: Apply triangle angle constraints. Angles must be positive: A > 0 ⇒ 112 - 7B > 0 ⇒ B < 16. B > 0; also C > 0 is automatically satisfied for such B. Step 4: Apply "A is the smallest angle". A < B ⇒ 112 - 7B < B ⇒ 112 < 8B ⇒ B > 14.
So B lies between 14 and 16 (strictly).
Step 5: Find the corresponding C range. C = 68 + 6B. For B just greater than 14: C ≈ 68 + 6 * 14 = 68 + 84 = 152. For B just less than 16: C ≈ 68 + 6 * 16 = 68 + 96 = 164. Thus C lies between 152° and 164° approximately.

Verification / Alternative Check:
Pick B = 15: A = 112 - 7 * 15 = 112 - 105 = 7°. C = 180 - 7 - 15 = 158°. Here A is clearly smallest, and C lies between 152° and 164°, which is consistent.

Why Other Options Are Wrong:
The ranges 138 to 164, 125 to 147, and 148 to 158 either start too low or do not capture the full possible range derived from the inequalities. Only 152 to 164 matches the correct theoretical range for C when all constraints are satisfied.

Common Pitfalls:
Students may forget that A must be the smallest angle and use only the sum of angles to get a single value, or they may treat B as an integer only. The condition "smallest angle" creates inequalities, which lead to a range of values rather than one fixed number.

Final Answer:
The possible range for angle C is 152 to 164 degrees.