Question

a surveyor measures two triangular plots of land. in the first plot, the two sides are 40 meters and 60 meters, with an included angle of 70°. in the second plot, the two sides are 20 meters and 30 meters, with the same included angle. are the plots similar?
○ a. no, the sides are not proportional
○ b. no, the angles do not match
○ c. yes, by the sas criterion
○ d. yes, by the aa criterion

a surveyor measures a triangle on a map where two sides are 50 m and 75 m, with an included angle of 60°. another triangle has side lengths 25 m and 37.5 m with an included angle of 60°. are the triangles similar?
○ a. no, the sides are not proportional
○ b. no, the angles do not match
○ c. yes, by the aa criterion
○ d. yes, by the sas criterion

given triangles def and ghi, where ∠d = ∠g and ∠e = ∠h, what can be said about these triangles?
○ a. they are congruent
○ b. they have the same area
○ c. they are neither similar nor congruent
○ d. they are similar

Explanation:

First Question:

Step1: Check side ratios

First plot sides: \(40, 60\); Second plot sides: \(20, 30\). Ratios: \(\frac{40}{20} = 2\), \(\frac{60}{30}=2\). So sides are proportional.

Step2: Check included angle

Both have included angle \(70^\circ\), so equal.

Step3: Apply SAS similarity

SAS (Side - Angle - Side) similarity: If two sides are proportional and included angle equal, triangles are similar. So by SAS Criterion, they are similar.

Step1: Check side ratios

First triangle sides: \(50, 75\); Second triangle sides: \(25, 37.5\). Ratios: \(\frac{50}{25}=2\), \(\frac{75}{37.5}=2\). Sides are proportional.

Step2: Check included angle

Both have included angle \(60^\circ\), so equal.

Step3: Apply SAS similarity

By SAS (Side - Angle - Side) similarity criterion, triangles are similar.

Step1: Recall AA similarity

AA (Angle - Angle) similarity: If two angles of one triangle equal two angles of another, triangles are similar.

Step2: Analyze given angles

Given \(\angle D=\angle G\) and \(\angle E = \angle H\), so two angles equal. Thus, by AA similarity, triangles are similar.

Answer:

c. Yes, by the SAS Criterion

Second Question: