Question

1. △abc is equilateral. is △abd equilateral? explain your answer. what type of reasoning, inductive or deductive, do you use when solving this problem? 2. ∠a and ∠d are complementary. ∠a and ∠e are supplementary. what can you conclude about ∠d and ∠e? explain your answer. what type of reasoning, inductive or deductive, do you use when solving this problem? 3. which figures in the last group are whatnots? what type of reasoning, inductive or deductive, do you use when solving this problem? 4. solve each equation for x. give a reason for each step in the process. what type of reasoning, inductive or deductive, do you use when solving these problems? a. 4x + 3(2 - x)=8 - 2x b. $\frac{19 - 2(3x - 1)}{5}=x + 2$ 5. a sequence begins - 4, 1, 6, 11 a. give the next two terms in the sequence. what type of reasoning, inductive or deductive, do you use when solving this problem? b. find a rule that generates the sequence. then give the 50th term in the sequence. what type of reasoning, inductive or deductive, do you use when solving this problem?

Explanation:

1.

Step1: Recall equilateral - triangle property

An equilateral triangle has all sides equal. Just because $\triangle ABC$ is equilateral doesn't mean $\triangle ABD$ is. There is no information given to suggest all sides of $\triangle ABD$ are equal. We use deductive reasoning as we apply known geometric facts (properties of equilateral triangles).

Step1: Use angle - relationship definitions

If $\angle A+\angle D = 90^{\circ}$ (complementary) and $\angle A+\angle E=180^{\circ}$ (supplementary). Then $\angle A = 90^{\circ}-\angle D$ and $\angle A = 180^{\circ}-\angle E$. So, $90^{\circ}-\angle D=180^{\circ}-\angle E$, and $\angle E=\angle D + 90^{\circ}$. We use deductive reasoning as we apply the definitions of complementary and supplementary angles.

Step1: Observe patterns

We look at the characteristics of the "whatnots" and "not whatnots" figures. By observing the patterns in the given figures, we try to identify the "whatnots" in the last group. This is inductive reasoning as we make a generalization based on observed patterns.

Answer:

$\triangle ABD$ is not necessarily equilateral. Deductive reasoning is used.

2.