Question

1. if △abc has sides 6, 8, 10 and △def has sides 12, 16, 20, what is the scale factor?

a. 1
b. 1/2
c. 2
d. 3

1. in △abc, ∠a = 60°, ∠b = 50°, and ∠c = 70°. in △def, ∠d = 60°, ∠e = 50°, and ∠f = 70°. what can you conclude about these triangles?

a. no conclusion can be made.
b. they are similar by aa similarity.
c. they have proportional side lengths but are not similar.
d. they are congruent.

1. in a quadrilateral abcd, if △abd ~ △cdb, what can be concluded about the triangles?

a. they are congruent
b. they have equal areas
c. they have the same perimeter
d. they have proportional corresponding sides

Explanation:

Step1: Recall scale - factor formula

The scale factor from $\triangle ABC$ to $\triangle DEF$ is found by dividing the length of a side of $\triangle DEF$ by the length of the corresponding side of $\triangle ABC$. Let's take the first - pair of corresponding sides. If we divide the side of $\triangle DEF$ by the side of $\triangle ABC$, for example, $\frac{12}{6}=2$, $\frac{16}{8}=2$, $\frac{20}{10}=2$.

Step2: Recall AA similarity criterion

In $\triangle ABC$ and $\triangle DEF$, $\angle A = 60^{\circ}=\angle D$, $\angle B = 50^{\circ}=\angle E$, and $\angle C = 70^{\circ}=\angle F$. By the AA (angle - angle) similarity criterion, if two angles of one triangle are equal to two angles of another triangle, the triangles are similar.

Step3: Recall the property of similar triangles

If $\triangle ABD\sim\triangle CDB$, similar triangles have proportional corresponding sides. Congruent triangles have equal sides, equal areas and equal perimeters, but similarity only implies proportional sides.

Answer:

1. c. 2
2. b. They are similar by AA similarity.
3. d. They have proportional corresponding sides