c The probability is 37, so we can say it is slightly more unlikely than likely to select a pair of congruent angles.
Practice makes perfect
a We are given the following diagram and asked to calculate the number of possible angle pairings.
Let's start by counting the pairs that include ∠ 8. Besides ∠ 8, there are 7 other angles. With each of them ∠ 8 can form a pair, so there are 7 pairs of angles with ∠ 8. Let's look at the rest of the angles.
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8 & 7 & 6 & 5& 4 & 3 & 2 & 1
7 & 6 & 5 & 4 & 3 & 2 & 1
6 & 5 & 4 & 3 & 2& 1
5 & 4 & 3 & 2 & 1
4 & 3 & 2 & 1
3 & 2 & 1
2 & 1
1
After counting all the angle pairs, we conclude that there are 28.
b Let's now describe the possible relationships between the angles in each pair. We can start with the pair of ∠ 8 and ∠ 7.
These are vertical angles. According to Vertical Angles Theorem, each pair of vertical angles is congruent. Therefore, ∠ 8 and ∠ 6 are congruent angles. Let's analyze the rest of the pairs using a table.
P=Number of favorable outcomes/Number of possible outcomes
In order to find the likelihood of randomly selecting a pair of congruent angles, we need to find the number of favorable and possible outcomes.
Possible outcomes are all the angle pairs, one of which we are going to select. From Part A we know that there are 28 angle pairs, so the number is 28.
Favorable outcomes are the outcomes of interest. In our case the interest is to select a pair of congruent angles. Using Part B, we can count the number of such pairs. There are 12.
Let's now substitute these numbers into the formula and calculate P(pair of congruent angles).
P=Number of favorable outcomes/Number of possible outcomes
The probability of picking a pair of congruent angles is 37, which is close to 50 %. Therefore, we can say that it is slightly more unlikely than likely to pick a pair of congruent angles, but almost the same.
