Question

given triangle abc and triangle def, with ∠a = 60 degrees and ∠d = 60 degrees. if ab/de = 1 and ac/df = 1, are the triangles similar?
a. no, because only one angle is given
b. yes, by sas similarity
c. no, because the sides are not proportional

d. yes, by aaa similarity

what is the primary use of the sas similarity criterion?
a. to measure angles in triangles
b. to prove congruence of triangles
c. to find the area of triangles
d. to prove the similarity of triangles

why is it important to verify proportionality of sides in sas similarity?
a. to ensure that the scale factor between the triangles is constant
b. to confirm that the triangles have equal angles
c. to determine the volume of a shape
d. to measure the perimeter of each triangle

given △def and △abc, ∠a = ∠d = 40°, ab = 8 cm, ac = 12 cm, and de = 4 cm, find the missing side df.
a. 6 cm
b. 12 cm
c. 10 cm
d. 8 cm

Explanation:

Step1: Analyze the first question

For two triangles \(ABC\) and \(DEF\), we have \(\angle A=\angle D = 60^{\circ}\), \(\frac{AB}{DE}=1\), \(\frac{AC}{DF}=1\). By the SAS (Side - Angle - Side) similarity criterion, if two sides of one triangle are proportional to two sides of another triangle and the included angles are equal, then the two triangles are similar. Here, the included angles \(\angle A\) and \(\angle D\) are equal and the ratios of the corresponding sides are equal (\(\frac{AB}{DE}=\frac{AC}{DF} = 1\)).

Step2: Analyze the second question

The SAS similarity criterion is specifically used to prove that two triangles are similar. It is not for measuring angles (a), proving congruence (b, congruence is a stronger condition with \(SSS\), \(SAS\) etc. for congruent triangles where sides are equal in length, not just proportional), or finding the area of triangles (c).

Step3: Analyze the third question

In the SAS similarity, verifying the proportionality of sides ensures that the scale factor (the ratio of corresponding side lengths) between the two triangles is consistent. It is not for confirming equal angles (b, we already have the included angle in SAS similarity), determining the volume of a shape (c, we are dealing with - dimensional triangles), or measuring the perimeter (d).

Step4: Analyze the fourth question

Since \(\angle A=\angle D = 40^{\circ}\), and if we assume \(\triangle ABC\sim\triangle DEF\) (by SAS similarity, as we have an included angle equal and we can check the ratio of sides). The ratio of sides \(\frac{AB}{DE}=\frac{8}{4} = 2\). Let \(DF=x\), then \(\frac{AC}{DF}=\frac{12}{x}\). Since the ratio of sides should be equal (because of similarity), \(\frac{AB}{DE}=\frac{AC}{DF}\), so \(2=\frac{12}{x}\), solving for \(x\) gives \(x = 6\) (by cross - multiplication \(2x=12\)).

Answer:

1. B. Yes, by SAS similarity
2. d. To prove the similarity of triangles
3. a. To ensure that the scale factor between the triangles is consistent
4. a. 6cm