Question

1. two similar triangles are part of a geometric design. the first triangle has one angle that measures 85° and another angle that measures 32°. what is a possible angle measure of the second triangle?

a) 25°
b) 60°
c) 90°
d) 110°

1. what is the 50th term in the arithmetic sequence below? (note: 9 is the first term, 17 is the second term, etc.)

9, 17, 25, 33, 41...
a) 393
b) 400
c) 401
d) 409

Explanation:

Step1: Find angles of first triangle

The sum of angles in a triangle is $180^{\circ}$. Given two angles of the first triangle as $85^{\circ}$ and $25^{\circ}$, the third - angle is $180-(85 + 25)=70^{\circ}$. Similar triangles have equal corresponding angles. So the possible angle measures of the second triangle are $85^{\circ}$, $25^{\circ}$, or $70^{\circ}$. Among the options, the answer for 114 is A.

Step2: Find common difference of arithmetic sequence

For the arithmetic sequence $9,17,25,33,41,\cdots$, the common difference $d=17 - 9=8$. The formula for the $n^{th}$ term of an arithmetic sequence is $a_n=a_1+(n - 1)d$, where $a_1 = 9$, $n = 50$, and $d = 8$.

Step3: Calculate the 50th term

Substitute the values into the formula: $a_{50}=9+(50 - 1)\times8=9 + 49\times8=9+392 = 393$. So the answer for 115 is A.

Answer:

1. A. $25^{\circ}$
2. A. 393