Question

1. multiple choice a paper fan is made by folding the pattern shown in the diagram.

angles a and c measure 80° and angle b measures 135°. if the remaining angles are congruent to each other, what is the measure of each angle? (lesson 7-1)
a. 135° b. 143° c. 157° d. 173°
456 module 7 review • quadrilaterals

Explanation:

Step1: Count total polygon sides

The figure has 9 vertices (A, H, G, F, E, D, C, B, and the shared fold vertex? No, count distinct vertices: A, H, G, F, E, D, C, B – wait, no, when folded, it's a nonagon (9-sided polygon). The sum of interior angles of an n-sided polygon is $(n-2)\times180^\circ$. For $n=9$:
$$(9-2)\times180^\circ = 7\times180^\circ = 1260^\circ$$

Step2: Sum known angles

Known angles: $\angle A = 80^\circ$, $\angle C = 80^\circ$, $\angle B = 135^\circ$. Sum:
$$80^\circ + 80^\circ + 135^\circ = 295^\circ$$

Step3: Calculate remaining angle sum

Subtract known sum from total sum:
$$1260^\circ - 295^\circ = 965^\circ$$

Step4: Count remaining congruent angles

Total angles: 9. Known angles: 3. Remaining angles: $9-3=6$. Let each unknown angle be $x$.
$$6x = 965^\circ$$
$$x = \frac{965^\circ}{6} \approx 160.83^\circ$$
Wait, correction: The figure is formed by folding a pattern, so it's an octagon? No, count labeled vertices: A, H, G, F, E, D, C, B – 8 vertices, 8-sided polygon (octagon). Sum of interior angles:
$$(8-2)\times180^\circ = 6\times180^\circ = 1080^\circ$$
Known sum: $80+80+135=295^\circ$. Remaining angles: $8-3=5$.
$$5x = 1080^\circ - 295^\circ = 785^\circ$$
$$x = \frac{785^\circ}{5} = 157^\circ$$

Answer:

C. $157^\circ$