Question

select the correct pairs of triangles.
the measures of two angles of pairs of triangles are given. which pairs of triangles are similar?
triangle 1: 55°, 45°
triangle 2: 55°, 80°
triangle 1: 103°, 32°
triangle 2: 103°, 25°
triangle 1: 73°, 47°
triangle 2: 47°, 30°
triangle 1: 105°, 23°
triangle 2: 52°, 105°
triangle 1: 99°, 41°
triangle 2: 40°, 99°

Explanation:

To determine if two triangles are similar, we use the AA (Angle - Angle) similarity criterion. This criterion states that if two angles of one triangle are equal to two angles of another triangle, then the triangles are similar. The sum of the interior angles of a triangle is always \(180^{\circ}\), so we can find the third angle of each triangle and then check for the presence of two equal angles between the triangles in a pair.

Step 1: Analyze the first pair of triangles

• For Triangle 1 with angles \(55^{\circ}\) and \(45^{\circ}\):
• The third angle is \(180-(55 + 45)=180 - 100 = 80^{\circ}\)
• For Triangle 2 with angles \(55^{\circ}\) and \(80^{\circ}\):
• The third angle is \(180-(55 + 80)=180 - 135 = 45^{\circ}\)
• Triangle 1 has angles \(55^{\circ},45^{\circ},80^{\circ}\) and Triangle 2 has angles \(55^{\circ},80^{\circ},45^{\circ}\). So, two angles (\(55^{\circ}\) and either \(45^{\circ}\) or \(80^{\circ}\)) are equal. Thus, this pair of triangles is similar.

Step 2: Analyze the second pair of triangles

• For Triangle 1 with angles \(103^{\circ}\) and \(32^{\circ}\):
• The third angle is \(180-(103+32)=180 - 135 = 45^{\circ}\)
• For Triangle 2 with angles \(103^{\circ}\) and \(25^{\circ}\):
• The third angle is \(180-(103 + 25)=180 - 128 = 52^{\circ}\)
• The angles of Triangle 1 are \(103^{\circ},32^{\circ},45^{\circ}\) and the angles of Triangle 2 are \(103^{\circ},25^{\circ},52^{\circ}\). There are not two equal angles between the two triangles. So, this pair is not similar.

Step 3: Analyze the third pair of triangles

• For Triangle 1 with angles \(73^{\circ}\) and \(47^{\circ}\):
• The third angle is \(180-(73 + 47)=180 - 120 = 60^{\circ}\)
• For Triangle 2 with angles \(47^{\circ}\) and \(30^{\circ}\):
• The third angle is \(180-(47+30)=180 - 77 = 103^{\circ}\)
• The angles of Triangle 1 are \(73^{\circ},47^{\circ},60^{\circ}\) and the angles of Triangle 2 are \(47^{\circ},30^{\circ},103^{\circ}\). There are not two equal angles between the two triangles. So, this pair is not similar.

Step 4: Analyze the fourth pair of triangles

• For Triangle 1 with angles \(105^{\circ}\) and \(23^{\circ}\):
• The third angle is \(180-(105 + 23)=180 - 128 = 52^{\circ}\)
• For Triangle 2 with angles \(52^{\circ}\) and \(105^{\circ}\):
• The third angle is \(180-(52+105)=180 - 157 = 23^{\circ}\)
• Triangle 1 has angles \(105^{\circ},23^{\circ},52^{\circ}\) and Triangle 2 has angles \(52^{\circ},105^{\circ},23^{\circ}\). So, two angles (\(105^{\circ}\) and either \(23^{\circ}\) or \(52^{\circ}\)) are equal. Thus, this pair of triangles is similar.

Step 5: Analyze the fifth pair of triangles

• For Triangle 1 with angles \(99^{\circ}\) and \(41^{\circ}\):
• The third angle is \(180-(99 + 41)=180 - 140 = 40^{\circ}\)
• For Triangle 2 with angles \(40^{\circ}\) and \(99^{\circ}\):
• The third angle is \(180-(40+99)=180 - 139 = 41^{\circ}\)
• Triangle 1 has angles \(99^{\circ},41^{\circ},40^{\circ}\) and Triangle 2 has angles \(40^{\circ},99^{\circ},41^{\circ}\). So, two angles (\(99^{\circ}\) and either \(41^{\circ}\) or \(40^{\circ}\)) are equal. Thus, this pair of triangles is similar.

Answer:

The pairs of similar triangles are:

• Triangle 1: \(55^{\circ},45^{\circ}\); Triangle 2: \(55^{\circ},80^{\circ}\)
• Triangle 1: \(105^{\circ},23^{\circ}\); Triangle 2: \(52^{\circ},105^{\circ}\)
• Triangle 1: \(99^{\circ},41^{\circ}\); Triangle 2: \(40^{\circ},99^{\circ}\)