Question

puzzle #5
1 if m∠a = 3x - 10° and m∠c = x + 16°, what is the value of x?
answer choices
a: 13 b: 61 c: 105
d: 136 e: 43 f: 39
g: 29 h: 151 i: 47
2 find m∠a
3 find m∠d
4 find m∠b
type the 4 - letter code into the answer box. all caps, no spaces.

Explanation:

Step1: Set angles A and C equal

Assume angles A and C are vertical - angles (since no other relationship is given), so \(3x - 10=x + 16\).
\[3x-x=16 + 10\]
\[2x=26\]
\[x = 13\]

Step2: Find measure of angle A

Substitute \(x = 13\) into the expression for \(m\angle A\).
\[m\angle A=3x-10=3\times13 - 10=39 - 10=29\]

Step3: Find measure of angle D

Assume the circle is divided into four angles and the sum of angles around a point is \(360^{\circ}\). If we assume the four - angle division is such that opposite angles are equal (a common case in circle - divided puzzles), and we know \(m\angle A = 29^{\circ}\), and assume the right - angle in the center is \(90^{\circ}\), then \(m\angle D=90^{\circ}\). But if we assume a linear - pair or other geometric relationship based on the context of the puzzle (assuming the circle is divided into four non - overlapping angles), and since \(m\angle A = 29^{\circ}\), and assuming the sum of the four angles is \(360^{\circ}\) and two of the angles are equal (opposite each other), we first note that if we assume the angles are arranged in a way that \(m\angle A\) and \(m\angle C\) are vertical and \(m\angle B\) and \(m\angle D\) are vertical. Since \(m\angle A = 29^{\circ}\), and the sum of adjacent angles on a straight line (if applicable) or the sum of angles around a point is \(360^{\circ}\), if we assume a simple case where the circle is divided into four angles and two pairs of equal angles, and we know \(m\angle A = 29^{\circ}\), then \(m\angle D = 90^{\circ}\) (assuming a right - angled division). But if we consider the fact that the sum of angles in a circle - divided into four parts and using the value of \(m\angle A\), and assume the other relationships, we find \(m\angle D=151^{\circ}\) (because \(m\angle A + m\angle D=180^{\circ}\) if they are supplementary in the context of the circle division).

Step4: Find measure of angle B

Since \(m\angle A\) and \(m\angle B\) are supplementary (assuming a straight - line division of the circle into two pairs of adjacent angles), \(m\angle B=180 - 29=151^{\circ}\)

The answers for the three questions (corresponding to the order of the questions in the puzzle) are:

1. A. 13
2. G. 29
3. H. 151

The 4 - letter code (in all caps, no spaces) is AGH.

Answer:

AGH