Question

1 which pair of triangles must be similar?
o a. triangles 1 and 2 each have a 35° angle.
o b. triangles 3 and 4 are both right triangles. triangle 3 has a 40° angle and triangle 4 has a 60° angle.
o c. triangle 5 has a 30° angle and a 100° angle. triangle 6 has a 60° angle and a 70° angle.
o d. triangle 7 has a 50° angle and a 25° angle. triangle 8 has a 50° angle and a 105° angle.

Explanation:

Step1: Recall the triangle - angle - sum theorem

The sum of the interior angles of a triangle is \(180^{\circ}\). For two triangles to be similar, their corresponding angles must be equal.

Step2: Analyze option A

If triangles 1 and 2 each have a \(35^{\circ}\) angle. Let the other two angles of triangle 1 be \(x\) and \(y\), and for triangle 2 be \(m\) and \(n\). By the angle - sum theorem (\(x + y+35^{\circ}=180^{\circ}\) and \(m + n + 35^{\circ}=180^{\circ}\)). If two angles of one triangle are equal to two angles of another triangle, the triangles are similar. Since we know one angle (\(35^{\circ}\)) is common, and the sum of the remaining two angles in each triangle is \(145^{\circ}\). If we assume the other two angles are also equal (because of the angle - sum property and the nature of triangle similarity criteria (AA - Angle - Angle)), triangles 1 and 2 are similar.

Step3: Analyze option B

For triangles 3 and 4 (both right - angled, so one angle is \(90^{\circ}\)). Triangle 3 has a \(40^{\circ}\) angle. Using the angle - sum theorem (\(90^{\circ}+40^{\circ}+\theta_3 = 180^{\circ}\), so \(\theta_3=50^{\circ}\)). Triangle 4 has a \(60^{\circ}\) angle. Using the angle - sum theorem (\(90^{\circ}+60^{\circ}+\theta_4 = 180^{\circ}\), so \(\theta_4 = 30^{\circ}\)). The angles are \(90^{\circ},40^{\circ},50^{\circ}\) for triangle 3 and \(90^{\circ},60^{\circ},30^{\circ}\) for triangle 4. Corresponding angles are not equal.

Step4: Analyze option C

For triangle 5 (\(30^{\circ}+100^{\circ}+\theta_5=180^{\circ}\), so \(\theta_5 = 50^{\circ}\)) and triangle 6 (\(60^{\circ}+70^{\circ}+\theta_6=180^{\circ}\), so \(\theta_6 = 50^{\circ}\)). The angles are \(30^{\circ},100^{\circ},50^{\circ}\) for triangle 5 and \(60^{\circ},70^{\circ},50^{\circ}\). Corresponding angles are not equal.

Step5: Analyze option D

For triangle 7 (\(50^{\circ}+25^{\circ}+\theta_7=180^{\circ}\), so \(\theta_7 = 105^{\circ}\)) and triangle 8 (\(50^{\circ}+105^{\circ}+\theta_8=180^{\circ}\), so \(\theta_8 = 25^{\circ}\)). But similarity requires two pairs of equal angles. Here, we have two pairs (\(50^{\circ}\) and either \(25^{\circ}\) or \(105^{\circ}\)), but the order of angles (correspondence) is not proper in terms of similarity criteria (AA). In similarity, we need two pairs of equal angles in the same relative position.

Answer:

A. Triangles 1 and 2 each have a \(35^{\circ}\) angle.