Question

which pair of triangles must be similar?
a. triangles 1 and 2 each have a 35° angle.
b. triangles 3 and 4 are both isosceles. they each have a 40° angle.
c. triangle 5 has a 30° angle and a 90° angle. triangle 6 has a 30° angle and a 70°
angle.
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 - similarity criteria

Two triangles are similar if their corresponding angles are equal. The sum of angles in a triangle is \(180^{\circ}\).

Step2: Analyze option a

For triangles 1 and 2, knowing only one pair of equal angles (\(35^{\circ}\)) is not sufficient. The other two angles could be different.

Step3: Analyze option b

For isosceles triangles with a \(40^{\circ}\) angle:
Case 1: If the \(40^{\circ}\) angle is the vertex angle, the base - angles are \(\frac{180 - 40}{2}=70^{\circ}\) each.
Case 2: If the \(40^{\circ}\) angle is a base - angle, the vertex angle is \(180-(40 + 40)=100^{\circ}\). Since we don't know which case it is for both triangles, we can't be sure of similarity.

Step4: Analyze option c

For triangle 5: Angles are \(30^{\circ},90^{\circ},180-(30 + 90)=60^{\circ}\).
For triangle 6: Angles are \(30^{\circ},70^{\circ},180-(30 + 70)=80^{\circ}\). Corresponding angles are not equal.

Step5: Analyze option d

For triangle 7: Angles are \(50^{\circ},25^{\circ},180-(50 + 25)=105^{\circ}\).
For triangle 8: Angles are \(50^{\circ},105^{\circ},180-(50 + 105)=25^{\circ}\).
Since the three angles of triangle 7 are equal to the three angles of triangle 8 (just in a different order), 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), the two triangles are similar.

Answer:

d. Triangle 7 has a \(50^{\circ}\) angle and a \(25^{\circ}\) angle. Triangle 8 has a \(50^{\circ}\) angle and a \(105^{\circ}\) angle.